Embedding surfaces in 4-manifolds

We prove a surface embedding theorem for 4-manifolds with good fundamental group in the presence of dual spheres, with no restriction on the normal bundles. The new obstruction is a Kervaire-Milnor invariant for surfaces and we give a combinatorial formula for its computation. For this we introduce the notion of band characteristic surfaces.

The vanishing of the intersection and self-intersection numbers of F in Theorem 1.2 is equivalent to the existence of a collection of Whitney discs that pair all double points of F . These may be chosen to be a convenient collection, meaning that all Whitney discs are framed, that they have disjointly embedded boundaries, and that their interiors are transverse to F (Definition 2.24). By definition, km(F ) vanishes if and only if, after finitely many finger moves taking F to some F , there is a convenient collection of Whitney discs for F whose interiors are disjoint from F .
A group is said to be good if it satisfies the π 1 -null disc property [FT95] (see also [KOPR21]); we shall not repeat the definition. In practice, it suffices to know that virtually solvable groups and groups of subexponential growth are good, and that the class of good groups is closed under taking subgroups, quotients, extensions, and colimits [FT95,KQ00]. We will define the other notions used in Theorem 1.2 in more detail in Section 2.
If Σ is a union of discs or spheres, Theorem 1.2 follows from [FQ90, Theorem 10.5(1)]. The latter theorem contained an error found by Stong [Sto94] (cf. Theorem 5.8), but this is not relevant to Theorem 1.2 because of the way we defined the Kervaire-Milnor invariant. It is, however, very relevant to how to compute the Kervaire-Milnor invariant, and Stong's correction will be one of the ingredients in our results (see Section 1.1).
For an arbitrary Σ, one could try to prove Theorem 1.2 by using general position to embed the 0and 1-handles of Σ (relative to ∂Σ) and removing a small open neighbourhood thereof from M . This gives a new connected 4-manifold M 0 with the same fundamental group as M , and only the 2-handles {h i : (D 2 , S 1 ) (M 0 , ∂M 0 )}, one for each component Σ i of Σ, remain to be embedded. One then hopes to apply [FQ90, Theorem 10.5(1)] (i.e. Theorem 1.2 for Σ a union of discs) to these maps of 2-handles to produce the desired embedded surface. The original algebraically dual spheres {g i } for {f i } perform the same rôle for {h i } in M 0 . The intersection and self-intersection numbers λ and µ remain unchanged, hence they also vanish for {h i }. However, the Kervaire-Milnor invariant may behave differently. That is, it may become nonzero for the embedding problem for the discs {h i }, whereas it was trivial for the original F . We show that this phenomenon can occur in Example 8.3. This difference stems from the fact that in applying [FQ90,Theorem 10.5(1)] we fix an embedding of the 1-skeleton of Σ and try to extend it across the 2-handles. As usual in obstruction theory, it might be advantageous to go back and change the solution of the problem on the 1-skeleton.
1.1. Computing the Kervaire-Milnor invariant. The strength of Theorem 1.2 versus the above strategy using [FQ90, Theorem 10.5(1)] lies in our computation of the Kervaire-Milnor invariant for arbitrary compact surfaces. In the case of discs and spheres, Stong showed that the Kervaire-Milnor invariant vanishes in more situations than claimed by Freedman-Quinn, due to the ambiguity arising from sheet choices when pairing up double points by Whitney discs, when the associated fundamental group element has order two. As we recall in Theorem 5.8, Stong [Sto94] introduced the notion of an r-characteristic surface, short for RP 2 -characteristic surface (Definition 5.6), to give a criterion, in terms of immersed copies of RP 2 in the ambient manifold M , to decide whether the sheet changing move is viable. Combined with the work of Freedman and Quinn, this enabled the computation of the Kervaire-Milnor invariant, and therefore answered the embedding problem for every finite union of discs or spheres with algebraically dual spheres, in an ambient 4-manifold with good fundamental group (see Remark 5.9 for more details).
In order to compute the Kervaire-Milnor invariant km(F ) for general surfaces, we extend the notion of r-characteristic surfaces to a notion of b-characteristic surfaces, short for band characteristic (Definition 5.16), defined using maps of bands (annuli and Möbius bands) into M . The next theorem is a generalisation of Stong's computation of km(F ) to arbitrary compact surfaces, and is our main result.
Let F denote the restriction of F to the subsurface Σ ⊆ Σ consisting of those connected components whose images do not admit framed algebraically dual spheres (see Definition 5.1). Consider a convenient collection W of Whitney discs that pair all the intersections and self-intersections of F , and define t(F , W ) ∈ Z/2 to be the mod 2 count of intersections between F and the interiors of the Whitney discs in W (Definition 5.4). The main novelty in this theorem lies in finding the right condition on F that makes the combinatorial formula t(F , W ) independent of the choice of Whitney discs, namely that F is b-characteristic. Note that if π 1 (M ) is good, then for km(F ) = 0 in the above theorem we can apply Theorem 1.2 to obtain an embedding. We view this as providing a complete answer for whether or not an immersed surface with algebraically dual spheres in an ambient manifold with good fundamental group is regularly homotopic to an embedding. In practice, it can often be easy to determine if a given surface is b-characteristic, as demonstrated by the following corollaries, derived in Section 8 as consequences of the more general Proposition 8.1.
Corollary 1.4. If M is a simply connected 4-manifold and Σ is a connected, oriented surface with positive genus, then any generic immersion F : (Σ, ∂Σ) (M, ∂M ) with vanishing self-intersection number is not b-characteristic. Thus if F has an algebraically dual sphere then km(F ) = 0, and since π 1 (M ) is good the map F is regularly homotopic, relative to ∂Σ, to an embedding.
1.2. Band characteristic maps. We briefly explain how the notion of a map being b-characteristic arises in the context of embedding general surfaces. Given F : Σ M as in Convention 1.1, assume that the intersections and self-intersections are paired by a convenient collection of Whitney discs W. Then the interiors of the discs in W could be tubed into spheres in M , potentially changing the count t from Theorem 1.3. The condition that F is s-characteristic, short for spherically characteristic (Definition 5.2), precisely ensures that the count is preserved under this move. Similarly, consider a band, i.e. an annulus or Möbius band, immersed in M with boundary lying on F (Σ) minus the double points, as in Figure 1.1. Then as shown in the figure we may perform a finger move on F along a fibre of the band, creating F with two new intersections, paired by a new Whitney disc W B arising from the band B (see Figure 1.1). We call this move the band fibre finger move and give further details in Construction 6.2. Adding W B to W might in principle change the count t, but the requirement that F is b-characteristic maps ensures it does not. In the case that Σ has only simply connected components, the boundary of the band is null-homotopic in Σ, and therefore the band can be closed off by discs to produce either a sphere (from an annulus) or an RP 2 (from a Möbius band). Thus in this case it suffices to consider r-characteristic maps.
However, for general Σ there may exist a band in M with a boundary curve that is nontrivial in π 1 (Σ). This necessitates the new notion of b-characteristic maps, which by definition requires that a function Θ : B(F ) → Z/2 vanishes (Definitions 5.12 and 5.13), where B(F ) consists of the homology classes in H 2 (M, Σ ; Z/2) that can be represented by certain immersed bands in M with boundary on Σ (Definition 5.10). Roughly speaking, the vanishing of Θ means that every band with boundary on Σ intersects Σ evenly many times in its interior. Intersections among the boundary components of the bands and a relative Euler number also play a rôle: see Sections 5 and 6 for details. If Θ ≡ 0 then for every band B, adding W B does not change the t-count, and in fact t is well-defined if and only if F is b-characteristic (Lemma 6.3). See Example 8.3 for a map which is r-characteristic but not b-characteristic.
The first step for deciding whether F is b-characteristic is to determine the subset B(F ). In general this could be difficult, but in practice it is often soluble. If this can be done, then, as shown in Figure 1.2, by computing λ Σ | ∂B(F ) and Θ : B(F ) → Z/2, we can decide whether F is b-characteristic. Both of these are functions on a finite group, so in principle these computations are manageable.
1.3. An embedding obstruction without dual spheres. Irrespective of whether F has algebraically dual spheres, we obtain a secondary embedding obstruction.
(i) The number t(F , W ) ∈ Z/2 is independent of the choice of convenient collection of Whitney discs W pairing all the intersections and self-intersections of F . Denote the resulting invariant by t(F ) ∈ Z/2. (ii) If H is regularly homotopic to F then H is b-characteristic and t(H ) = t(F ) ∈ Z/2. (iii) If km(F ) = 0, for example if F is an embedding, then t(F ) = 0.
As part of our analysis of the obstruction t, we shall prove the following additivity properties.
Corollary 1.8. For any g, there exists a smooth, closed 4-manifold M g , a closed, oriented surface Σ g of genus g, and a smooth, b-characteristic, generic immersion F : Σ g M g with F = F and t(F ) = 0, and therefore km(F ) = 0.
By contrast, we show in Example 8.5 that every map of a closed surface to S 1 × S 3 is homotopic to an embedding. One could ask whether there exists a 4-manifold M and immersions Σ g M with nontrivial Kervaire-Milnor invariant, for every g. However, as a partial negative answer we will show in Proposition 8.7 that a b-characteristic generic immersion F : Σ M from a closed surface Σ to a compact 4-manifold M with abelian fundamental group with n generators must satisfy χ(Σ) ≥ −2n.
1.4. Homotopy versus regular homotopy. Theorems 1.2, 1.3, and 1.6 together give a framework for deciding whether or not an immersed surface is regularly homotopic to an embedding, as illustrated by the flowchart in Figure 1.2. However, in the first sentence of the article, we began by considering whether a given continuous map is homotopic to an embedding. We explain now how to extend the framework of Figure 1.2 to decide this, for maps of surfaces that admit algebraically dual spheres.
For a map f from a connected surface to a 4-manifold, we will show in Theorem 2.27 that there are either infinitely many or precisely two regular homotopy classes in the homotopy class of f , according to whether f * (w 1 (M )) is trivial or nontrivial respectively. Our strategy is to make a judicious choice of regular homotopy class to which we apply our previous theory.
Begin with a continuous map F : Σ → M that restricts to an embedding on ∂Σ and satisfies F −1 (∂M ) = ∂Σ, where Σ, M are as in Convention 1.1. Denote the components of F by f i : (Σ i , ∂Σ i ) → (M, ∂M ), and suppose that F has algebraically dual spheres. Note that homotopies preserve the intersection numbers λ(f i , f j ), but might not preserve the self-intersection number µ(f i ), since adding a local cusp in f i changes µ(f i ) 1 , the coefficient of 1 ∈ π 1 (M ), by ±1. Depending on the behaviour of the orientation characters of M and Σ, the coefficient µ(f i ) 1 lies in either Z or Z/2, and is preserved under regular homotopy (see Lemma 2.21 and Proposition 2.22). Now, in order to decide whether F is homotopic to an embedding, we will either find a generic immersion in the homotopy class of F which is regularly homotopic to an embedding, or we will show that this is impossible. First, by performing a homotopy we may assume without loss of generality that F is a generic immersion such that µ(f i ) 1 = 0 for every component f i of F . If λ(f i , f j ) for all i = j and µ(f i ) for all i are not all trivial, then F is not homotopic to an embedding. On the other hand if λ(f i , f j ) = µ(f i ) = 0 for all i = j, we have the two following cases. Below, (f i ) • is the map induced on fundamental groups by f i using some choice of basing path connecting f i (Σ i ) to the basepoint of M .
By Theorem 2.27 the regular homotopy class of F is uniquely determined by the condition that µ(f i ) 1 = 0 for each i with f i ∈ F . Run the analysis in Figure 1.2 on F to determine whether it is regularly homotopic to an embedding. Note that the outcome of this analysis only depends on the regular homotopy class of F , rather than all of F . In particular, if π 1 (M ) is good then F is homotopic to an embedding if and only if F is regularly homotopic to an embedding.
In this case we use the following theorem.
Then there exists a generic immersion F homotopic to F with µ(f j ) = 0 for all j, and a choice of Whitney discs W such that t((F ) , (W ) ) = 0. Thus if F has algebraically dual spheres then km(F ) = 0, and if moreover π 1 (M ) is good then F is regularly homotopic, relative to ∂Σ, to an embedding.
Using this theorem, we can immediately conclude that our F as in Case 2 is homotopic to an embedding, as long as π 1 (M ) is good. Notably, it is not relevant in this case whether F is b-characteristic. This completes the analysis of whether a given continuous map of a surface into a 4-manifold is homotopic to an embedding.
We now indicate the proof of Theorem 1.9. By the vanishing of the intersection and self-intersection numbers, there is a convenient collection of Whitney discs W for F and therefore for F . In case t(F , W ) = 0 the proof is completed by setting F = F . In case t(F , W ) = 1, we use Construction 5.19 to find another generic immersion F homotopic to F . Briefly, Construction 5.19 involves creating four new double points in the component f i with nontrivial w 1 (Σ)| ker(fi)• using local cusps, and then cancelling them using a suitable choice of Whitney arcs and discs. Theorem 1.9 also has the following immediate corollaries. These are the nonorientable analogues of Corollaries 1.4 and 1.5. They provide homotopies to embeddings rather than regular homotopies, and again it is not relevant whether F is b-characteristic.
Corollary 1.10. If M is a simply connected 4-manifold and Σ is a connected, nonorientable surface, then a generic immersion F : (Σ, ∂Σ) (M, ∂M ) with vanishing self-intersection number and an algebraically dual sphere is homotopic, relative to ∂Σ, to an embedding.
Corollary 1.11. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1, with Σ connected and π 1 (M ) good. Suppose that F has vanishing self-intersection number and an algebraically dual sphere. If F is obtained from F by an ambient connected sum with any embedding RP 2 → S 4 , then F is homotopic, relative to ∂Σ, to an embedding.
As in our analysis for the embedding problem up to regular homotopy, our techniques are primarily applicable in the presence of algebraically dual spheres and good fundamental group of the ambient space. It is however sometimes possible to conclude that a map without algebraically dual spheres is homotopic to an embedding. For example we show in Example 8.5 that every map of a closed surface to S 1 × S 3 is homotopic to an embedding.
1.5. Applications to knot theory. Theorem 1.2 can be applied to the problem of finding embedded surfaces in general 4-manifolds bounded by knots in their boundary. Given a closed 4-manifold M , let M • denote the punctured manifold M \D 4 . The M -genus of a knot K ⊆ S 3 = ∂M • , denoted by g M (K), is the minimal genus of an embedded orientable surface bounding K in M • . If M is smooth, we also consider the smooth M -genus, denoted by g Diff M (K), the minimal genus of a smoothly embedded orientable surface with boundary K. The quantities g S 4 and g Diff S 4 coincide with the topological and smooth slice genus of knots in D 4 respectively. Note that g M (K) = g M (K), so (2) below implies a corresponding result for CP 2 .
The smooth CP 2 -genus has been studied by [Yas91,Yas92,AN09,Pic19], and differs dramatically from the topological result in Corollary 1.12 (2). It is not known whether g Diff CP 2 can be arbitrarily high; the largest value in the literature is g Diff CP 2 (T (2, 17)) = 7 [AN09, Theorem 1.2]. Corollary 1.12 (1) is straightforward to prove if M topologically splits as a connected sum with S 2 ×S 2 or S 2 ×S 2 , because g S 2 ×S 2 (K) = g S 2 ×S 2 (K) = 0 for all K by the Norman trick [Nor69, Corollary 3, Remark]. For the K3 surface, this implies that g K3 (K) = 0 for all knots K. On the other hand, it is an open question whether there exists a K with g Diff K3 (K) = 0 [MMP20, Question 6.1]. Proof of Corollary 1.12. Let K ⊆ S 3 be an arbitrary knot and let M be an arbitrary closed, simply connected 4-manifold. Let ∆ be a generically immersed disc bounded by K in a collar S 3 × [0, 1] of ∂M • . Since M is simply connected, every class in H 2 (M ; Z) ∼ = π 2 (M ) is represented by a generically immersed sphere. By assumption, M is not homeomorphic to S 4 and thus H 2 (M ; Z) is nontrivial. Since M is closed, every primitive class α ∈ H 2 (M ; Z) has an algebraic dual β ∈ H 2 (M ; Z), i.e. λ(α, β) = 1. Represent α and β by generically immersed spheres, and tube the interior of ∆ into β to obtain ∆. Add local cusps to arrange µ(∆) = 0.
To prove part (2), take M = CP 2 , α = β = [CP 1 ], and use Corollary 1.5 to show that the connected sum of ∆ with an unknotted torus is homotopic to an embedding. Corollary 1.5 applies since ∆ has the algebraically dual sphere α and π 1 (M ) = 1 is a good group. The same proof implies that g M (K) ≤ 1 for every M ∼ = S 4 .
Next we deduce part (1), that in fact g M (K) = 0, assuming that M ∼ = S 4 , CP 2 . In this case H 2 (M ; Z) has rank at least 2. Hence, in the argument of the previous paragraph, we can choose α to be a primitive class satisfying λ(α, α) ∈ 2Z. To do this, note that H 2 (M ; Z) has a summand isomorphic to Z ⊕ Z, so the classes x, y, and x + y, for the generators x, y of the Z-factors, are primitive, and at least one of λ(x, x), λ(y, y), or λ(x + y, x + y) is even. Tube ∆ into a generically immersed sphere representing β which is algebraically dual to α. Since λ(α, α) is even, the disc ∆ is not b-characteristic, and therefore ∆ is homotopic rel. boundary to an embedding.
Given a knot K ⊆ S 3 and an integer n ∈ Z, we build the n-trace X n (K) by attaching a 2-handle D 4 along K with framing n. The minimal genus of an embedded surface representing a generator of H 2 (X n (K); Z) is called the n-shake genus of K, denoted by g sh n (K). Similarly, the smooth n-shake genus of K is denoted by g sh,Diff n (K). We recover the following result of [FMN + 21].
Outline of the paper. In Section 2 we describe the primary embedding obstructions arising from the theory of equivariant intersection numbers for surfaces in 4-manifolds. In Section 3 we define the Kervaire-Milnor invariant carefully. We prove Theorem 1.2 in Section 4. In Section 5 we explain our combinatorial method for computing the Kervaire-Milnor invariant, and define b-characteristic surfaces, postponing almost all proofs to Section 6. In Section 7, we prove Theorems 1.3 and 1.6. Finally, in Section 8, we prove Corollaries 1.4 and 1.5, and give further applications and examples.

Generic immersions and intersection numbers
2.1. Topological generic immersions. We start with the definition of an immersion of manifolds in the topological setting. For m ≥ 0 let R m For k ≤ n we consider the standard inclusions: Definition 2.1. A continuous map F : Σ k → M n between topological manifolds of dimensions k ≤ n is an immersion if locally it is a flat embedding, that is if for each point p ∈ Σ there is a chart ϕ around p and a chart Ψ around F (p) fitting into one of the following commutative diagrams. The first diagram is for p ∈ Int Σ and F (p) ∈ Int M , the second diagram is for p ∈ ∂Σ and F (p) ∈ Int M , and the third is for p ∈ ∂Σ and F (p) ∈ ∂M . In particular F is required to map interior points of Σ to interior points of M .
Some authors prefer to call this notion a locally flat immersion.
Definition 2.2. A (linear) normal bundle for an immersion F : Σ k → M n is an (n − k)-dimensional real vector bundle π : ν F → Σ, together with an immersion F : ν F → M that restricts to F on the zero section s 0 , i.e. F • s 0 = F , and such that each point p ∈ Σ has a neighbourhood U such that F | π −1 (U ) is an embedding.
We now restrict to the relevant dimensions for this paper, k = 2 and n = 4, and take M to be a connected topological 4-manifold as in Convention 1.1. The singular set of an immersion F : Σ → M is the set Recall that a continuous map is said to be proper if the inverse image of every compact set in the codomain is compact.
Definition 2.3. Let Σ be a surface, possibly noncompact. A continuous, proper map F : Σ → M is said to be a (topological) generic immersion, denoted F : Σ M , if it is an immersion and the singular set is a closed, discrete subset of M consisting only of transverse double points, each of whose preimages lies in the interior of Σ. In particular whenever m ∈ S(F ), there are exactly two points p 1 , p 2 ∈ Σ with F (p i ) = m, and there are disjoint charts ϕ i around p i , for i = 1, 2, where ϕ 1 is as in the left-most diagram of (2.1), and ϕ 2 is the same, with respect to the same chart Ψ around m, but with ι replaced by ι : Theorem 2.4. A generic immersion F : Σ M , for possibly noncompact Σ, has a normal bundle as in Definition 2.2 with the additional property that F is an embedding outside a neighbourhood of F −1 (S(F )), and near the double points F plumbs two coordinate regions π −1 (ϕ i (R 2 )) ∼ = ϕ i (R 2 ) × R 2 , i = 1, 2, together i.e. F • (ϕ 1 (x), y) = F • (ϕ 2 (y), x).
Proof. Let ∂ 1 Σ ⊆ ∂Σ denote the union of the components of ∂Σ mapped to ∂M . Then since F | ∂1Σ is an embedding of a 1-manifold in a 3-manifold, it has a normal bundle. We extend this to a collar neighbourhood of F (∂ 1 Σ) contained in a collar neighbourhood of ∂M . To do this, first note that since (i) ∂M is closed in M , (ii) S(F ) is closed and contained in Int(M ), and (iii) manifolds are normal, it follows that there is an open neighbourhood of ∂M disjoint from S(F ). Then argue as in Connelly's proof [Con71] that boundaries of manifolds have collars, to obtain a homeomorphism of pairs The normal bundle over the boundary extends to a collar in the codomain, hence its pull-back extends to a collar in the domain.
Next, let ∂ 2 Σ ⊆ ∂Σ denote the union of the components of ∂Σ mapped to Int M . We see that F (∂ 2 Σ) has a normal bundle by [FQ90,Theorem 9.3], as a submanifold of M . Let F be the embedding of the total space, as in Definition 2.2. By using the inward pointing normal for ∂ 2 Σ in Σ, we obtain an orthogonal decomposition of each fibre as ν ∂2Σ →Σ ⊕ V , where V is a 2-dimensional subspace. Then translates of V in the direction of the inward pointing normal give rise to a normal bundle on the intersection of a collar of ∂ 2 Σ with the image of the normal bundle of ∂ 2 Σ under F . Now we want to extend the normal bundle that we have just constructed on a neighbourhood of ∂Σ to the rest of Σ. First we will produce a normal bundle in a neighbourhood of both preimages of each double point, and then finally we will extend the normal bundle to the rest of the interior of Σ.
Let U m 1 and U m 2 be open neighbourhoods in Σ of p 1 and p 2 , contained within the images of ϕ 1 and ϕ 2 above, respectively. Define Σ := Σ \ m∈S(F ) (U m 1 ∪ U m 2 ). Then the restriction of F gives an embedding of Σ in M . We already have a normal bundle defined on a neighbourhood of ∂Σ . Apply [FQ90,Theorem 9.3A] to extend the given normal bundle on U m 1 ∪ U m 2 and ∂Σ to all of Σ and therefore we have a normal bundle on all of Σ.
Remark 2.5. Freedman-Quinn [FQ90,Theorem 9.3] produce an extendable normal bundle for every submanifold of a 4-manifold. The extendability condition is technical with an important consequence: extendable normal bundles are unique up to isotopy. One can always find an extendable normal bundle embedded in the total space of any given normal bundle.
The proof of Theorem 2.4 also applies in the following more general setting where ∂ 2 Σ is not embedded in M , but F | ∂2Σ factors as a composition of generic immersions ∂ 2 Σ S M , for some surface S. We will use this case in the definition of b-characteristic maps in Section 5, so we introduce nomenclature.  (M, S) be a generic immersion of pairs. Then H admits a normal bundle, i.e. a normal bundle for B in M such that the restriction to C contains a normal bundle for C in S.
Proof. Note that C has a normal bundle in S, and then the sum of this with the normal bundle of S in M guaranteed by Theorem 2.4 gives rise to a normal bundle for C in M . The rest of the proof proceeds as before.
Observe that smooth generic immersions are topological immersions. Next we show that when both notions make sense they coincide, which justifies the terminology. We know that M has a smooth structure by [FQ90,Theorem 8 Proof. Fix a smooth structure on ∂M such that the generic immersion F restricted to those connected components of ∂Σ that map to ∂M is a smooth embedding. To find such a smooth structure, first use the standard smooth structure on the normal bundle of F | ∂Σ , and then extend this to a smooth structure on all of ∂M . Since any two smooth structures on a 3-manifold are isotopic this could also be arranged by an isotopy of ∂Σ, but our aim is to use the given map without isotoping it.
By Theorem 2.4 there is a normal bundle (ν F , F ) for F . Let D(ν F ) → Σ be the (closed) disc bundle. This yields a regular neighbourhood N (F ) := F (D(ν F )) of F (Σ), a codimension zero submanifold of M . The regular neighbourhood N (F ) can be identified with a smooth manifold obtained from D(ν F ) after the requisite plumbing operations and smoothing corners. Use such an identification to fix a smooth structure on N (F ) ⊆ M . With respect to this smooth structure the map Σ → N (F ) is a smooth generic immersion.
In addition, the boundary of N (F ) inherits a smooth structure. The complement in M of Int N (F )∪ (N (F ) ∩ ∂M ) is a connected, noncompact 4-manifold with a prescribed smooth structure on its boundary. Then the interior has a compatible smooth structure by [FQ90, Theorem 8.2; Qui82, Corollary 2.2.3], giving rise to a smooth structure on all of M . Since the smooth structure on N (F ) is unaltered, F become a smooth generic immersion, as desired. This is motivated by the smooth result which states that two generic immersions are ambiently isotopic (in the sense of Definition 2.9 but with homeomorphism replaced by diffeomorphism in the definition of an isotopy) if and only if they are connected by a path in the space of generic immersions [GG73, Chapter III, Theorem 3.11]. Note that for embeddings one does not need the isotopy H Σ .
Mirroring the smooth notion, a generic homotopy between generically immersed surfaces in a 4manifold is a sequence of ambient isotopies, finger moves, Whitney moves, and cusp homotopies. The moves in question are defined in local coordinates exactly as in the smooth setting. A regular homotopy between generically immersed surfaces in a 4-manifold is by definition a sequence of ambient isotopies, finger moves, and Whitney moves. The following proposition explains that maps of surfaces in a 4manifold can be assumed to be generic immersions, and homotopies between generic immersions may be assumed to be generic as well. (1) Every map (Σ, ∂Σ) → (M, ∂M ) is homotopic (relative to the embedded boundary) to a generic immersion.
Briefly, the proposition is proven as follows. Homotope the maps away from a point of M using cellular approximation, remove that point, choose a smooth structure on the complement of the point, and then apply the smooth theory of generic immersions, combining [ 2.2. Intersection numbers. We define intersection numbers between compact surfaces in 4-manifolds. We call a manifold X based if it comes equipped with basepoints p i ∈ X i for each connected component X i ⊆ X, together with a local orientation at each p i . For this and the next subsection, let M be a connected, based 4-manifold and let Σ and Σ be based, compact, connected surfaces. Our generic immersions F : Σ M will also be based, i.e. equipped with a whisker, namely a path in M from the basepoint of M to F (p), for p the basepoint of Σ.
Let f : Σ → M and g : Σ → M be based maps that are transverse, i.e. around each intersection point f (s) = g(s ), s ∈ Σ, s ∈ Σ , there are coordinates that make f and g (in a neighbourhood of s in Σ and a neighbourhood of s in Σ ) resemble the standard inclusions R 2 × {0} and {0} × R 2 into R 4 respectively, as in (2.1). We assume that these intersections are the only singularities between f and g and that f (∂Σ) and g(∂Σ ) are disjoint.
Let v f and v g be whiskers for f and g. The intersection number λ(f, g) is the sum of signed fundamental group elements λ(f, g) := p∈f g ε(p) · η(p) as follows. A priori this is the formal sum of a list of elements of the set {±1} × π 1 (M ). It will ultimately give rise to an element of a quotient of Z[π 1 (M )], given in Definition 2.11, after we factor out the effect of finger and Whitney moves and the effect of the choice of the paths γ p f and γ p g in the first bullet point below. Fix p ∈ f g. Next we define ε(p) ∈ {±1} and η(p) ∈ π 1 (M ). We use * to denote concatenation of paths.
• Let γ p f be a path in Σ from the basepoint to f −1 (p) and let γ p g be a path in Σ from the basepoint to g −1 (p).
• The sign ε(p) ∈ {±1} is determined as follows. Transport the local orientation of Σ at the basepoint to f −1 (p) along γ p f , and the local orientation of Σ at the basepoint to g −1 (p) along γ p g . This induces a local orientation at p, by ordering f before g. Another local orientation is obtained by transporting the local orientation at the basepoint of M to p along the concatenated path v g * (g • γ p g ). We define ε(p) = +1 when the two local orientations match at p, and −1 otherwise.
where f + is a push-off of f along a section of its normal bundle transverse to the zero section. In case the embedding f | ∂Σ is equipped with a specified framing for its normal bundle, then f + is defined to be a push-off of f along a section restricting to the first vector of that framing on ∂Σ.
If f t is a homotopy of f that is transverse to g for all t then λ(f t , g) is independent of t as a set of signed fundamental group elements, assuming the above choices of γ p ft are made carefully. However, if f t describes a finger move of f into g, there is a single time t 0 at which f t0 and g are not transverse, because there is a tangency. After the tangency, two new intersection points p and q arise. These have the same group element η(p) = η(q) and opposite signs ε(p) = −ε(q), with appropriate choices of γ p ft and γ q ft . Similarly, a Whitney move reduces the intersections between f and g by such a pair. To get a regular homotopy invariant notion, it is thus important to specify the home of λ(f, g) carefully.
For Σ and Σ simply connected, the sum λ(f, g) is usually considered as an element of Z[π 1 (M )] and is independent of the choice of {γ p f } p and {γ p g } p . In the abelian group Z[π 1 (M )] the relations −a+a = 0 for each a ∈ π 1 (M ) are built in, and if one identifies the sign ε(p) with the inverse in this abelian group then finger moves and Whitney moves do not change λ(f, g) as an element in the group ring.
Remark 2.12. In the case that M , Σ, and Σ are all oriented, Γ f,g is the free abelian group generated by the double coset quotient of π 1 (M ) by left and right multiplication by the images of loops in Σ and Σ respectively. In general, due to the signs introduced by the orientation characters, there may be torsion in Γ f,g . For example, consider f : RP 2 M with f • (RP 1 ) = 1. Then for any g : Σ → M , the group Γ f,g is 2-torsion, due to the relations γ = w RP 2 (RP 1 ) · f • (RP 1 ) * γ = −γ for every γ ∈ π 1 (M ), arising from setting α := RP 1 and β := 1 in (2.2).
To understand Γ f,g better, we introduce some notation. Write [a] ∈ Γ f,g for the equivalence class of a ∈ Z[π 1 (M )], and let ∼ denote the equivalence relation on π 1 (M ) induced by the composition π 1 (M ) → Z[π 1 (M )] Γ f,g , i.e. for a, b ∈ π 1 (M ), a ∼ b if and only if the images of a and b in Γ f,g coincide. The following lemma is immediate from the definitions.
Let pm : π 1 (M )/∼ f • π 1 (Σ)\π 1 (M )/g • π 1 (Σ ) be the quotient map. We write |γ| := pm(γ). Here one should think that pm stands for dividing out "plus-minus". Note that pm has fibres of order 1 or 2 and we can decompose the double coset as a disjoint union B 1 B 2 according to this distinction, where pm gives a bijection pm −1 (B 1 ) ↔ B 1 while pm −1 (B 2 ) → B 2 is 2-1. Choose a section s of pm. For each s(b) ∈ pm −1 (B 2 ) we denote the other element of pm −1 (b) by −s(b). Their images in Γ f,g are indeed inverse to one another, which motivates the notation.
Lemma 2.14. Fix a section s for pm as above. The abelian group Γ f,g is a direct sum Γ f,g = F A ⊕ V of a free abelian group F A on the set s(B 2 ) ⊆ π 1 (M )/∼ ⊆ Γ f,g and a Z/2-vector space V with basis pm −1 (B 1 ) ⊆ π 1 (M )/∼.
Reading off the coefficients in this decomposition gives homomorphisms c s(b) : Γ f,g → Z for each b ∈ B 2 , and c b : Γ f,g → Z/2 for each b ∈ B 1 , yielding a decomposition of Γ f,g as a direct sum of copies of Z and Z/2.
Proof. Starting with the free abelian group with basis π 1 (M ), a relator in (2.2) does one of the following three things.
The subgroup V and its basis clearly do not depend on our choice of section, but the basis of F A depends on this choice. If we change the section s at a point b ∈ B 2 to s so that s (b) = −s(b), the associated basis element changes to its inverse. It follows that the subgroup F A of Γ f,g does not depend on the choice of s. It also follows that the coefficient maps c s(b) only depend on s up to sign and satisfy For a given a ∈ π 1 (M )/ ∼ we can choose a section s as above with s(pm(a)) := a and hence we get a coefficient map c a that is independent of the other values of s. For example, we can take a := [γ] ∈ π 1 (M )/∼ with γ ∈ π 1 (M ) to get c γ .
Proposition 2.16. Let f : Σ → M and g : Σ → M be based maps that are transverse. The intersection number λ(f, g) is preserved by homotopies in the interior of f g. The intersection number is not preserved by a homotopy pushing an intersection point off the boundary of Σ or Σ .
Remark 2.17. The geometric definition of λ given above has a well known algebraic version in the case that f and g correspond to classes in H 2 (M, ∂M ; Z[π 1 (M )]). This extends to the case of positive genus, as we now sketch. We restrict ourselves to the case that M , Σ, and Σ are closed and oriented for convenience.
Choose a basepoint in the universal cover of M , lifting the basepoint of M . The maps f and g lift uniquely (with respect to this choice of basepoint) to covers M and M , corresponding to the subgroups f • (π 1 (Σ)) and g • (π 1 (Σ )) respectively. These lifts represent classes . By Poincaré duality, this yields an element in which is isomorphic as an abelian group to Finally we have the isomorphism We shall not prove that this formulation agrees with the geometric definition.
2.3. Self-intersection numbers. Next we turn to the self-intersection number for a based generic immersion f : Σ M with whisker v f . The definition of µ, given below, is similar to that of λ in the previous subsection, except that there is no longer a clear choice of which sheet to consider first at a given double point. Consequently, the values of µ lie in a further quotient of the group Γ f,f from Definition 2.11. The self-intersection number for annuli was considered by Schneiderman in [Sch03].
We write f f ⊆ M for the set of double points of f . We record the self-intersections of f by the sum of signed group elements µ(f ) := p∈f f ε(p) · η(p) as follows.
• For p = f (x 1 ) = f (x 2 ) for x 1 = x 2 ∈ Σ, let γ p 1 and γ p 2 be paths in Σ from the basepoint to x 1 and x 2 respectively. • The sign ε(p) ∈ {±1} is defined as follows. Transport the local orientation of Σ at the basepoint to x 1 along γ p 1 , and along γ p 2 to x 2 . This induces a local orientation at p. Another local orientation is obtained by transporting the local orientation at the basepoint of M to p along the concatenated path v f * (f • γ p 2 ). We define ε(p) = 1 when the two local orientations match at p, and −1 otherwise.
There is a similar discussion about homotopy invariance of µ(f ) as for λ(f, g) earlier: homotopies f t that are generic immersions for all t preserve the formal sum of signed elements but finger moves and Whitney moves (of f with itself) create pairs −η(p) + η(p) so it is convenient to use abelian groups. This takes care of regular homotopies of f but there is an additional subtlety for cusp homotopies f t where there is exactly one time t 0 for which f t0 is not an immersion. These issues will be discussed carefully below.
For simply connected Σ, the self-intersection invariant µ(f ) is well defined in the quotient (as an abelian group) of Z[π 1 (M )] obtained by introducing the relators for all γ ∈ π 1 (M ). For general Σ, the quantity µ(f ) is well defined in the abelian group in other words a quotient of Γ f,f from Definition 2.11. Here, as above w M : π 1 (M ) → {±1} is the orientation character. We now change our notation slightly from the discussion of λ(f, g) in order to work in this further quotient. Let ∼ denote the equivalence relation on π 1 (M ) induced by the composition and let |π 1 (M )| be the quotient of π 1 (M ) obtained by identifying γ 1 and γ 2 whenever [γ 1 ] = ±[γ 2 ]. Then we obtain the following analogues of Lemmas 2.13 and 2.14.
Lemma 2.18. Let γ 1 , γ 2 ∈ π 1 (M ). One of the relations [γ 1 ] = ±[γ 2 ] holds if and only if γ 1 and γ 2 represent the same element in the quotient of the double coset by inversion. In other words, the identity map induces a bijection We write |γ| := pm(γ) for the quotient map pm : π 1 (M )/∼ |π 1 (M )|. Again pm has fibres of order 1 or 2 and we decompose |π 1 (M )| as a disjoint union B 1 B 2 according to this distinction as before. Choose a section s : |π 1 (M )| → π 1 (M )/∼ of pm. As before, for b ∈ B 2 we denote the elements of the fibre by pm Reading off the coefficients in this decomposition gives homomorphisms leading to a decomposition of Γ f as a direct sum of copies of Z and Z/2.
Proof. The proof is exactly analogously to that of Lemma 2.14.
As before the subgroups F A and V of Γ f do not depend on the choice of s, only the basis of F A does. As a consequence, the coefficient maps c s(b) only depend on s up to sign and satisfy c −s(b) = −c s(b) . Given a ∈ π 1 (M )/∼ we may again take s(pm(a)) := a to get c a and in particular c γ for γ ∈ π 1 (M ) independent of the choice of s at other points. This gives the following definition.
Definition 2.20. For γ ∈ π 1 (M ), we write µ(f ) γ := c γ (µ(f )). This quantity lies in Z (respectively, Z/2) when |γ| lies in B 2 (respectively B 1 ), or equivalently when [γ] has infinite order (respectively, order 2) in Γ f . The values does not depend on the choice of s and satisfies c γ1 = −c γ2 whenever We focus on µ(f ) 1 , which plays an important rôle in the distinction between the homotopy class and regular homotopy class of f , as we will discuss in the next subsection. In the usual case, where Σ is simply connected, µ(f ) 1 ∈ Z. However, in general µ(f ) 1 may lie in either Z or Z/2. The following lemma gives the precise conditions determining the home of µ(f ) 1 .
Suppose that w Σ is trivial on ker(f • ) and w M is trivial on Im(f • ). Then the first case (i) cannot Similarly, (ii) cannot happen: suppose δ is order two in π 1 (M ) and δ ∼ 1. Then by definition, , for some α, β ∈ π 1 (M ). In particular, δ ∈ Im(f • ), and so again w M (δ) = 1 by hypothesis, contradicting (ii). Therefore, [1] has infinite order as claimed.
Now suppose there is some and [1] has order two.
As with Proposition 2.16 the proof of the following proposition is virtually identical to the case of discs and spheres using Proposition 2.10 (see e.g. [PR21b]), and we leave it for the interested reader. The proof is again very similar to the case of discs and spheres [PR21b]. By this geometric characterisation, observe that it is meaningful to refer to generic immersions (not necessarily based) having trivial intersection and self-intersection numbers. The Whitney discs above may be assumed to be convenient in the following sense (see e.g. [FQ90, Section 1.4; PR21b]).
Definition 2.24. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1. A convenient collection of Whitney discs for F is a collection W of framed, generically immersed Whitney discs pairing all the intersections and self-intersections of F , with interiors transverse to F , and with disjointly embedded boundaries. A collection of arcs on F (Σ) is called a collection of Whitney arcs if it extends to a convenient collection of Whitney discs.
By pushing interior intersections towards the boundaries of the Whitney discs, we may further assume that the Whitney discs are pairwise disjoint and embedded, but we shall not make this restriction.
Recall that Whitney discs are required to pair intersection points of opposite sign, but as we saw in the previous subsection, the signs of intersection points depend on choices of paths transporting local orientations. The next two definitions are intended to clarify the notion of 'opposite signs' in the current general setting.
Definition 2.25. Let f : Σ → M and g : Σ → M be based maps that either intersect transversely, or f = g and f is a generic immersion. We say that two points p, q ∈ f g are paired by arcs if we equip them with the extra data of a simple arc A in Σ from f −1 (p) to f −1 (q) and a simple arc A in Σ from g −1 (q) to g −1 (p). In case f = g we require that each point in f −1 (p) and in f −1 (q) is the endpoint of precisely one arc, i.e. the arcs lie in distinct sheets at both p and q.
With the extra data of the arcs A and A , we can make sense of whether two intersection points that are paired by arcs have opposite sign.
Definition 2.26. Let f : Σ → M and g : Σ → M be based maps that either intersect transversely, or f = g and f is a generic immersion. Two intersection points p, q ∈ f g paired by simple arcs A ⊆ Σ and A ⊆ Σ have opposite sign if the following holds. Fix a local orientation on each of Σ and Σ at f −1 (p). This choice induces a local orientation on M at p. Transport the local orientation of Σ to q along the path A on Σ, and the local orientation on Σ to q along the path A on Σ . This gives a local orientation of M at the point q. Compare this with the local orientation on M at q induced by transporting the local orientation at p to q along f (A). If these orientations disagree then the paired points are said to have opposite sign.
If the orientations agree, the paired points p and q are said to have the same sign. Observe that this depends on the choice of arcs A and A .
2.5. Homotopy versus regular homotopy of generic immersions. Let f : Σ M be a generic immersion. Local orientations of M and Σ determine a local orientation of νf . Hence, given a framing of f | ∂Σ , one can define a relative Euler class of the normal bundle νf in H 2 (Σ, ∂Σ; Z w1(νf ) ). If f * (w 1 (M )) = w 1 (νf ) + w 1 (T Σ) = 0 then the local orientation of Σ determines a Poincaré duality isomorphism from this twisted cohomology group to Z, and we denote the resulting integer by e(νf ). Note that e(νf ) does not depend on the local orientation of Σ but only on the local orientation of M . If f * (w 1 (M )) = 0 then there is still a mod 2 normal Euler number, which we also denote by e(νf ) ∈ Z/2.
Next we give an extension of [PRT20, Theorem 1.2] from the simply connected to the general setting, restricting ourselves to the case of connected Σ for convenience. By the following theorem, in some cases, for example if M = R 4 and Σ is closed and nonorientable, then e(νf ) ∈ Z is an additional invariant of regular homotopy classes of immersions. It changes by ±2 during a cusp homotopy and hence there are infinitely many regular homotopy classes of immersions, whereas there is a unique homotopy class of continuous maps.
In Theorem 2.27, in the case that Σ has nontrivial boundary, we fix a framing on the embedding f | ∂Σ , in order to define the relative Euler number e(ν f ), for f any generic immersion homotopic to f . Theorem 2.27. Let Σ be a compact, connected surface and let M be a 4-manifold. Then the inclusion of the subspace of generic immersions Imm(Σ, M ) in the space of all continuous maps induces a map where [Σ, M ] ∂ denotes the set of homotopy classes of continuous maps that restrict on ∂Σ to embeddings disjoint from the image of the interior of Σ.
(2) The fibres of i are related by cusp homotopies. More precisely, suppose that f and g are homotopic generic immersions. Then we can add cusps to f and g, to obtain f and g respectively, such that f and g are regularly homotopic.
Proof. By Proposition 2.10 (1), the map is surjective. That is, every homotopy class contains a generic immersion. This proves (1).
For (2), note that if f and g are homotopic generic immersions, then by Proposition 2.10 (1) there exists a generic homotopy H between them. We can modify H such that there are real numbers t 1 < t 2 ∈ [0, 1] such that the singularities of H in [0, t 1 ] only consist of cusp homotopies that create double points, the singularities in [t 1 , t 2 ] only consist of finger moves and Whitney moves, and those in [t 2 , 1] only consist of cusp homotopies that remove double points. The statement then follows by taking f := H t1 and g := H t2 .
To achieve this modification, note that we can bring all the creating cusp singularities forward, so that they occur earlier, and we can delay all the removing cusps. To arrange for a creating cusp to be rearranged earlier than a finger or Whitney move, choose an arc in the image of H starting from H t (Σ), for some t ∈ (0, t 1 ), and ending at the cusp, which intersects each level in a point and is disjoint from all Whitney arcs and double points. The homotopy can then be altered in a neighbourhood of this arc so that the cusp singularity occurs at time t. Delaying a removing cusp is the same procedure but with the direction of time reversed. This completes the proof of (2).
The proof of (3) splits naturally into two cases.
As noted in Section 2, the sign of an intersection point is not always well-defined. Nevertheless, in the case that f * (w 1 (M )) = 0 the sign of a cusp homotopy is well-defined. The key point is that a cusp not only specifies a double point p but also an arc between the preimages of p. In the case that f * (w 1 (M )) = 0, using this path, the sign of the double point p is well-defined, independent of the choice of path transporting the local orientation at the basepoint to the double point. Thus in this setting we define the sign of a cusp to be the sign of the double point it creates or removes. We will use the terminology of creating cusps for cusps that create a double point and removing cusps for those that remove a double point.
Since f * (w 1 (M )) = 0, e(ν f ) is defined in Z for any generic immersion f homotopic to f . Since regularly homotopic generic immersions have equal Euler numbers, the bijection given is well-defined on equivalence classes in the domain of i. Note that a cusp homotopy changes e(ν f ) by 2 or −2, depending on the sign of the cusp and whether it is a creating or a removing cusp. So every element of Z can be realised as the image of the given map i −1 (f ) → Z by some generic immersion in the homotopy class.
To complete the proof when f * (w 1 (M )) = 0, it remains to show that we can take a generic homotopy between generic immersions with equal Euler numbers, and can modify the homotopy in a way that cancels cusps, until we are left with a regular homotopy.
First, note that when we have a removing cusp, and later in the homotopy we have a creating cusp with the same sign, then we can cancel these two cusps along a level-preserving path in the homotopy as indicated in Figure 2.1. However this is not sufficient. We also have to show that we can also cancel cusps in the following two situations. (a) Two creating cusps of opposite sign, or two removing cusps of opposite sign. (b) A creating cusp paired with a later removing cusp, both of the same sign. If, as in (a), we have two creating cusps of opposite sign, then we can modify the homotopy as follows. At the start of homotopy, add a homotopy from the starting surface to itself that performs a trivial extra finger move together with two removing cusps for the double points created by the finger move. Now these new cusps can be cancelled with the previous two cusps by the above argument. In total this replaces the two cusps of opposite sign with a finger move. Doing the same from the other side of the homotopy, we can also cancel two removing cusps of opposite sign.
Similarly, as in (b), we can trade a creating cusp for a removing cusp, as follows. Change the homotopy, again by adding a homotopy from the starting surface to itself. To construct this homotopy, perform a trivial finger move and then add two removing cusps for the double points created by the finger move. Now we may cancel the original creating cusp with one of the new removing cusps. This entire operation replaces a cusp with a cusp of opposite sign and direction. As before we can repeat the operation at the end of the homotopy to replace the removing cusp with a creating one, also with the opposite sign. Thus when we have a creating cusp with a later removing cusp of the same sign, we can replace both by cusps of opposite sign and direction. Since now the removing cusp happens before the creating cusp, the two can be cancelled and we are done with case (b). This completes the proof of (3) in the case that f * (w 1 (M )) = 0.
Note that a cusp homotopy changes µ( f ) 1 ∈ Z/2 by one. So both values of Z/2 can be realised within the homotopy class. To show injectivity in this case, we have to show that we can cancel cusps in a homotopy in arbitrary pairs. First use the trading argument above to get all the removing cusps before the creating cusps in the homotopy. Then for any pair of cusps, one removing and one creating, choose some level-preserving path in the homotopy between the first and the second cusp, and restrict to a small disc containing the path. If they have the same sign with respect to this disc, cancel the two cusps as before.
If they have opposite signs, change the choice of the arc to arrange that the union of the new arc and the old arc maps nontrivially under w 1 (M ). Such an arc exists since f * (w 1 (M )) is nontrivial and Σ is connected. With this new choice, the signs of the cusps in the disc become the same and we can again cancel the cusps. This completes the proof of both halves of (3).
Finally for (4) note that if f * (w 1 (M )) = 0 and w 1 (Σ)| ker(f•) = 0, then recall that by Lemma 2.21 that µ( f ) 1 is well-defined in Z. By the discussion above the statement of the theorem, e(ν f ) is also well-defined in Z. In this case the formula holds by the proof of the corresponding fact for discs and spheres (see e.g. [PR21b, Proposition 11.8]). Any cusp homotopy leaves λ( f , f ) 1 unchanged, while it changes µ( f ) 1 by ±1. By the formula, it changes e(ν f ) by ∓2. Thus if f and f are generic immersions homotopic to f , then e(ν f ) = e(ν f ) if and only if µ( f ) 1 = µ( f ) 1 . Hence (4) follows from (3).

The Kervaire-Milnor invariant
The Kervaire-Milnor invariant for generically immersed collections of discs or spheres in a 4-manifold first appeared in [FQ90,Definition 10.8A] where they named the invariant after [KM61]. We will expand the definition to arbitrary generically immersed compact surfaces and show that it is a secondary obstruction to a generic immersion F : Σ M of a compact surface in a 4-manifold being regularly homotopic relative to the boundary to an embedding. By Lemma 2.23, it is defined only when all the intersections and self-intersections of F may be paired by a convenient collection of Whitney discs W.
The definition of the Kervaire-Milnor invariant below is based on a geometric criterion involving a collection of Whitney discs. Our main goal will be to provide a method to compute this invariant in practice.
and only if after finitely many finger moves on the interior of F , taking F to some F , there is a convenient collection of (framed) Whitney discs, with interiors disjoint from F , pairing all the intersections and self-intersections of F .

Regular homotopy invariance of km(F ) is the following straightforward proposition.
Proposition 3.2. Let Σ and M be as in Convention 1.1. If F 1 and F 2 are regularly homotopic generic immersions of Σ in M , and km(F 1 ) is defined, then km(F 2 ) is defined and km(F 1 ) = km(F 2 ) ∈ Z/2.
Proof. First, km(F i ) is defined if and only if all intersection numbers of F i vanish. Since these are regular homotopy invariants, km(F 1 ) is defined if and only if km(F 2 ) is. To show that km(F 1 ) = km(F 2 ), by symmetry it suffices to show that km(F 1 ) = 0 implies km(F 2 ) = 0. So suppose that km(F 1 ) = 0, and let F 1 be obtained from F 1 by finger moves such that the intersections of F 1 can be paired up by Whitney discs {W i } as in Definition 3.1. Since F 1 and F 2 are regularly homotopic, there is a generic immersion F 3 such that F 3 can be obtained from both F 1 and F 2 by finger moves and ambient isotopies. Since F 3 is obtained from F 1 by finger moves and ambient isotopies and the finger moves can be assumed to be disjoint from {W i }, all the double points of F 3 can also be paired up by Whitney discs with interiors disjoint from F 3 , as in Definition 3.1. Since F 3 is obtained from F 2 by finger moves, taking F 2 := F 3 it follows that km(F 2 ) = 0.
If F is homotopic to F and has trivial self-intersection numbers, then F is regularly homotopic to F by Theorem 2.27. Therefore F and F have equal Kervaire-Milnor invariant. However, a non-regular homotopy may change the self-intersection number of a generic immersion, and the Kervaire-Milnor invariant might no longer be defined.
We will analyse the situation where w 1 (Σ)| ker(fi)• is nontrivial for some i in Section 5 after setting up more notation.
Definition 3.1 is optimised for the proof of Theorem 1.2, as we will see shortly, but is difficult to use in practice. In particular, while one may fortuitously detect specific finger moves and Whitney discs to show km(F ) = 0, without a combinatorial description it appears, for a given F , to be hard to prove that the required finger moves from F to some F , together with Whitney discs for F , do not exist. We will provide precisely such a combinatorial reformulation of the Kervaire-Milnor invariant, similar in spirit to the definition in [FQ90, Section 10.8A] and [Sto94]. In the proof of Theorem 1.3 we will show that this agrees with Definition 3.1.

The proof of the surface embedding theorem
The surface embedding theorem (Theorem 1.2) can be deduced using the proof of [FQ90, Theorem 10.5(1)], combined with an observation in [PRT20, Theorem A, Lemma 6.5] for the condition on the homotopy class of G, using our definition of the Kervaire-Milnor invariant (Definition 3.1). Since the surface embedding theorem does not follow directly from the statement of [FQ90, Theorem 10.5(1)], as previously discussed, and also since the latter requires a correction by Stong [Sto94], it can be hard for the uninitiated to piece together a correct proof. Therefore we provide one in this section.
Further, the statement of [FQ90, Theorem 10.5(1)] is itself quite complicated, and our version focused on the surface embedding problem may be useful for those looking to apply the technology of Freedman-Quinn without delving into the details. 4.1. Ingredients. The statement of the surface embedding theorem uses the notions of algebraically and geometrically dual spheres. We recall the definitions.
gj for all i, j, for some choice of basings for F and G.
(2) A collection G = {g i : S 2 M } of generic immersions is geometrically dual to F if F G is a generic immersion and the geometric count of intersections satisfies |f i g j | = δ ij for all i, j.
We will need the following lemma, the idea behind which is due to Casson [Cas86; Fre82, Section 3]. The formulation we give here is from [PRT20, Lemma 5.1].
Lemma 4.2 (Geometric Casson lemma). Let F and G be transversely intersecting generic immersions of compact surfaces in a connected 4-manifold M . Assume that the intersection points {p, q} ⊆ F G are paired by a Whitney disc W . Then there is a regular homotopy from That is, the two paired intersections have been removed. The regular homotopy may create many new self-intersections of F and G; however, these are algebraically cancelling.
The proof of the surface embedding theorem also relies on Freedman's disc embedding theorem, whose statement we recall.  [BKK + 21]). Let M be a connected topological 4-manifold with good fundamental group, and let be a generic immersion of finitely many discs. Assume that F has framed algebraically dual spheres G = {[g i ]} ⊆ π 2 (M ) such that λ(g i , g j ) = 0 = µ(g i ) for all i = j. Then there is a flat embedding F : ( D 2 , S 1 ) → (M, ∂M ), which is equipped with geometrically dual spheres G = {g i }, such that F and F have the same framed boundary and [g i ] = [g i ] ∈ π 2 (M ) for all i.
We will also freely use standard constructions such as symmetric contraction, boundary twisting, and interior twisting (i.e. adding local cusps). See [FQ90, Chapters 1-2; PR21a] for further details.

4.2.
Proof of the surface embedding theorem. We recall the statement for the convenience of the reader. Apply the geometric Casson Lemma 4.2 to upgrade G = {g i } from algebraically to geometrically dual spheres G = {g i }, changing F to F by a regular homotopy in the process. The intersection and self-intersection numbers, and the Kervaire-Milnor invariant, vanish for F . So they also vanish for F , since all three quantities are preserved under regular homotopy relative to the boundary by Propositions 2.16, 2.22, and 3.2.
Then by the definition of the Kervaire-Milnor invariant (Definition 3.1), after further finger moves changing F to some F , we can find a convenient collection of Whitney discs W = {W } for F whose interiors are disjoint from F . Moreover F and G are still geometrically dual, since the finger moves may be assumed to miss G .
We shall apply the disc embedding theorem (Theorem 4.3) to the collection of generically immersed discs W in the 4-manifold M \ νF , so we verify that the hypotheses are satisfied. The Whitney discs W have framed algebraically dual spheres as follows. The Clifford tori at the double points of F are geometrically dual to W. Symmetrically contract half of these tori, one per Whitney disc, using meridional discs for F tubed into the geometrically dual spheres G = {g i }. The resulting spheres are only algebraically dual to W since the elements of W and G may intersect arbitrarily; however, they have vanishing intersection and self-intersection numbers since they were produced by symmetric contraction. They are also framed, as we argue briefly now. If a sphere g i in G is not framed, then the symmetric contraction uses incorrectly framed caps. However in the symmetric contraction process each cap is used twice, with opposite orientations, and so any framing discrepancies cancel out. Since F has geometrically dual spheres, the fundamental group π 1 (M \ νF ) ∼ = π 1 (M ) and is thus good. This verifies the hypotheses of the disc embedding theorem (Theorem 4.3) for W, as desired.
Apply the disc embedding theorem to the Whitney discs W in M \νF to obtain disjointly embedded, flat, framed Whitney discs {W } for the intersections and self-intersections of F , with interiors still disjoint from F , along with a collection of geometrically dual spheres for the {W } in M \ νF .
Tube  This corollary follows from the surface embedding theorem and the fact that a generic immersion F : Σ M is π 1 -negligible if and only if F admits geometrically dual spheres, which can be seen as follows. For the forwards direction, the meridional circles are null-homotopic in M , so by π 1negligibility they are null-homotopic in M \ F (Σ). The union of null-homotopies with meridional discs gives geometrically dual spheres. For the reverse direction, first note that by general position the homomorphism π 1 (M \ F (Σ)) −→ π 1 (M ) is surjective. The kernel is normally generated by a collection consisting of one meridional circle for each connected component of Σ. Since geometrically dual spheres provide null homotopies for these meridians, the assertion follows. By the geometric Casson lemma (Lemma 4.2), the map F in the statement of the surface embedding theorem (Theorem 1.2) is regularly homotopic to a π 1 -negligible map, due to the existence of the algebraically dual spheres G. Indeed, this is the first step of the proof of the surface embedding theorem. Note that Theorem 1.2 also controls the homotopy class of the dual spheres, and so is slightly stronger than Corollary 4.4.

Band characteristic maps and the combinatorial formula
In this section we define b-characteristic surfaces and give the combinatorial definition of the Kervaire-Milnor invariant. We postpone many of the proofs to Section 6. We hope this will help the reader to assimilate the overall structure more easily.
A sphere g : S 2 M in a topological 4-manifold M is said to be twisted if the Euler number of the normal bundle is odd. We say g is framed if the normal bundle is trivial. To coincide with the usual meaning of framed, one can also implicitly choose a trivialisation, although we will not make use of such a choice. Observe that if the normal bundle of g has even Euler number then it is homotopic to a generically immersed sphere with trivial normal bundle, via adding local cusps.
Definition 5.1. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1, with components {f i }. Let Σ := Σ i denote the collection of components of Σ whose images under F do not admit framed, algebraically dual spheres with respect to F , i.e. Σ i ∈ Σ if and only if there is no framed immersed sphere g i with λ(f j , g i ) = δ ij for all j. Then we use to denote the restriction of F to Σ . Note that if an f i does not admit an algebraically dual sphere at all, then it belongs to F .
Recall that x ∈ H 2 (M, ∂M ; Z/2) is said to be characteristic if x · a ≡ a · a mod 2 for every a ∈ As a consequence of Lemma 5.3, we shall henceforth only discuss maps F of surfaces for which F is s-characteristic.
Recall that a convenient collection of Whitney discs for the intersections within F consists of framed, generically immersed Whitney discs with interiors transverse to F , and with disjointly embedded boundaries. In other words, t(F, W) is the mod 2 count of transverse intersections between F and the interiors of the Whitney discs pairing intersections within F.
We will usually apply this definition, as in Theorem 1.3, to (F, W) = (F , W ).
Lemma 5.5. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1, with components {f i }. Suppose that F admits algebraically dual spheres, and that all the intersections and self-intersections of F are paired by a convenient collection W of Whitney discs. Let W ⊆ W denote the sub-collection of Whitney discs for the intersections within F , where F is as in Definition 5.1. If t(F , W ) = 0, then km(F ) = 0.
We wish to find practically verifiable conditions on F that guarantee that t(F , W ) is independent of the collection of Whitney discs W . More precisely, the value of t(F , W ) should be independent of the pairing of double points, the Whitney arcs joining the paired double points (which includes the choice of sheets at each double point) and finally the Whitney discs. In the case that each Σ i is simply connected, t(F , W ) agrees with [FQ90, Definition 10.8A]. However, Freedman-Quinn claim in their Lemma 10.8B that, for simply connected Σ, the quantity t(F , W ) only depends on F , and not on the Whitney discs, as long as F is s-characteristic. This is not true in general, as pointed out and corrected by Stong [Sto94]. Further, again with π 1 (Σ i ) = 1 for all i, Stong established that the value of t does not depend on the choice of W using the notion of r-characteristic discs and spheres. Here is our generalisation of his notion. Remark 5.7. A map c : RP 2 → S 2 of odd degree (e.g. a collapse map) composed with elements of π 2 (M ) can be used to show that r-characteristic maps are s-characteristic. Indeed, given F and a ∈ π 2 (M ) we obtain a • c : RP 2 → M , and we have 0 ≡ F · (a • c) + (a • c) · (a • c) ≡ F · a + a · a mod 2.
Stong's key observation [Sto94] was that in some instances involving elements of order two in π 1 (M ), one can change the choice of sheets at two double points of F , and hence the Whitney arcs and corresponding Whitney disc, with a resulting change in the value of t(F , W ) by one. The restriction to r-characteristic maps removes this source of indeterminacy. To summarise, Stong showed the following theorem.
Theorem 5.8 (Stong [Sto94]). Let F : (Σ, ∂Σ) → (M, ∂M ) be a map of discs or spheres in a 4-manifold with components {f i }. Suppose that λ(f i , f j ) = 0 for all i = j, that µ(f i ) = 0 for all i, and that F admits algebraically dual spheres. If F is not r-characteristic then km(F ) = 0, and if F is r-characteristic then km(F ) = t(F , W ) for any choice of Whitney discs W for the intersections within F . Remark 5.9. Combining Theorem 2.27 and Theorem 5.8 with Theorem 1.2 gives the complete answer to the embedding problem for spheres and discs with algebraically dual spheres and with good fundamental group of the ambient space, due to Freedman, Quinn, and Stong. Theorem 2.27 allows one to fix the regular homotopy class of generic immersions within the homotopy class to be that with µ(−) 1 = 0, Theorem 5.8 computes the Kervaire-Milnor invariant, and then one applies Theorem 1.2 to conclude whether or not there is a regular homotopy to an embedding.
Our contribution in the present paper extends this solution to the case that the components of Σ are not all simply connected. In this case there is a further source of indeterminacy coming from the choice of Whitney arcs on Σ. For this reason we need a stronger restriction on F . M , to simplify the notation we will not mention the mapping cylinder and refer to B(F ) ⊆ H 2 (M, Σ; Z/2). We use the notation for the connecting homomorphism from the long exact sequence of the pair. A class represented by a compact surface S is mapped to its boundary ∂S under the map ∂. Then ∂B(F ) ⊆ H 1 (Σ; Z/2) consists of (the homology classes of) those closed 1-manifolds immersed in Σ whose images under F bound bands in M satisfying (5.1).
In the case that λ Σ | ∂B(F ) is trivial, we define a further invariant Θ on elements of B(F ). We will need the following notions.
(1) For C a closed 1-manifold generically immersed in Σ, the self-intersection number µ Σ (C) ∈ Z/2 of C counts the number of double points, which we assume without loss of generality to be disjoint from the double points of F . As usual, this is not invariant under homotopies of C in Σ, only under regular homotopies.
(2) Let S be a compact surface with boundary C and with a generic immersion of pairs (S, C) (M, Σ). Suppose that w 1 (Σ) is trivial on each component of C, e.g. if Σ is orientable. Then the normal bundle of C in Σ is trivial and we can pick a nowhere vanishing section (if S is closed, this is an empty choice). Extend this section to the normal bundle of S in M (such a normal bundle exists by Corollary 2.7) such that the extension is transverse to the zero section. Then we define the Euler number e(S) to be the number of zeros of this section modulo 2. Observe that this is analogous to how one measures the twisting of a Whitney disc with respect to the Whitney framing. For S a closed surface this coincides with the usual definition of the Euler number.
Here is the definition of Θ in the case that w 1 (Σ) is trivial on every component of C. For closed S and Σ, we have Θ A (S) ≡ |Int S F | + e(S) mod 2, and thus Θ A (S) vanishes for all closed S if and only if F is characteristic in the traditional sense. In the proof of Theorem 1.3, we will only use the definition of Θ A for bands. But the case of general surfaces will be useful for our proof that, in the cases relevant to us, Θ A does not depend on the choice of A (see Lemma 5.15 below).
If a component of ∂S is orientation-reversing in Σ, then its normal bundle in Σ is nontrivial and hence we may not use it to choose a nowhere vanishing section of the normal bundle of S on its boundary as before to define the Euler number. However, bands with such boundaries may exist in the ambient 4-manifold and must be considered. In case w 1 (Σ) is nontrivial on precisely one component of ∂S, e.g. when ∂S is connected, we know λ Σ (∂S, ∂S) = 1. Then, by Lemma 5.11, if a band with such boundary exists, then km(F ) = 0, and there is no need to define Θ. In particular, note that this means that we need not consider the case of a Möbius band whose boundary consists of a curve on which w 1 (Σ) is nontrivial. There is one final case of relevant bands left to consider, which we do next. Remark 5.14. An alternative definition of Θ A (B, D) would use the arc D to add a tube to F (Σ), in such a way that the tube intersects the band B in two parallel copies of D. More precisely, we perform an ambient surgery on F (Σ). This requires choosing a 2-dimensional sub-bundle of the normal bundle of D in M -the tube itself consists of the circle bundle for this sub-bundle, considered within a tubular neighbourhood of D. We build the required sub-bundle by first choosing a section lying in the normal bundle of D in B, denoted ν B D . The second section can be chosen freely. Let F denote the immersion constructed by the tubing procedure. Observe that the domain of F , denoted Σ , is obtained from the abstract surface Σ by adding a 1-handle. Depending on the choice of the second section above, this may be an orientation-reversing or orientation-preserving 1-handle, but this will not matter for us.
Adding the tube changes the band B to a disc ∆, by removing a thin strip neighbourhood of D. The disc ∆ has boundary lying on Σ . Observe that ∂∆ is orientation-preserving on Σ , since it is the result of banding together two orientation-reversing curves. Since no new intersection points were added, the collection A is a collection of Whitney arcs for the intersection points of F . As a result, we may define where the latter is computed as in Definition 5.12. In order to see that this agrees with Definition 5.13, we need only check that the definition of the Euler number terms agree, assuming we choose the tube to be thin enough to miss any double points. For the Euler number term, compare Figures 5.1(b) and (c). In particular note that the freedom involved in tubing, namely of the second section of the 2dimensional sub-bundle which determines the tube, equals the freedom involved in choosing the section of γ * ν M B on the part of ∂ B that maps to D. Note that we did not prove that e( B) is independent of the choice of D. This will follow from the upcoming proof of (ii) in Lemma 5.15 below, which states that Θ A (B, D) depends only on the homology class of B. The following lemma, whose proof is again deferred to Section 6, shows that Θ is well defined in all required cases. An initial, important observation is that b-characteristic maps are r-characteristic.
Lemma 5.17. Every b-characteristic map F : Σ → M is r-characteristic. Moreover the two notions agree for unions of discs or spheres.
Proof. Let R : RP 2 → M be a generic immersion which is transverse to F and so that R * w 1 (M ) = 0. Take a small disc on F (Σ) away from R F , and tube into the image of R. This creates a Möbius band B with boundary on Σ. Here ∂B is homotopically trivial in Σ, so w 1 (Σ), ∂B = 0. The core C of B corresponds to RP 1 within the original immersed RP 2 . Therefore, since R * w 1 (M ) = 0, we have that w 1 (M ), C = 0. So B ∈ B(F ). Note that Θ : B(F ) → Z/2 is well-defined by Lemma 5.15 (iv) since F is b-characteristic. Further Θ(B) = Θ(R) = F · R + R · R ∈ Z/2 by Lemma 5.15 (i) since [B] = [R] ∈ H 2 (M, Σ; Z/2). But this vanishes since F is b-characteristic. Hence F is r-characteristic.
For the second sentence, suppose that Σ is a union of discs or spheres and is r-characteristic. Let B ∈ B(F ) be a band. Since Σ is simply connected, the boundary of B is null homotopic in F (Σ). Therefore B can be closed up using a codimension zero submanifold of Σ to either a sphere or an RP 2 immersed in M . The resulting closed surface R again satisfies Θ(B) = F · R + R · R ∈ Z/2 by Lemma 5.15 (i). Here λ Σ | ∂B(F ) = 0 since Σ is simply connected and so Θ : B(F ) → Z/2 is again well-defined by Lemma 5.15 (iv). Once again, since null homotopic circles on Σ must be orientationpreserving, w 1 (Σ), ∂B = 0 and so (5.1) implies that R * w 1 (M ) = 0. So Θ(B) = 0 since F is r-characteristic. Thus F is b-characteristic. It follows that the notions of b-characteristic and rcharacteristic coincide as claimed.
Recall that if F is b-characteristic, then Theorem 1.3 states that km(F ) = t(F , W ), so we have a combinatorial description of km(F ). Moreover, since Θ only depends on the homology class of a band in H 2 (M, Σ; Z/2), we can in principle determine whether or not F is b-characteristic by computing Θ on finitely many homology classes. Having said that, as mentioned in the introduction, in practice deciding precisely which homology classes can be represented by maps of bands may be difficult.
The next lemma shows that any map which is regularly homotopic to a b-characteristic map is itself b-characteristic.
Lemma 5.18. Let F : Σ → M be a generic immersion as in Convention 1.1, with components {f i }.
Suppose that λ(f i , f j ) = 0 for all i = j and that µ(f i ) = 0 for all i. Assume that H is regularly homotopic to F and that F is b-characteristic. Then H is b-characteristic.
Proof. First observe that λ Σ | ∂B(F ) = λ Σ | ∂B(H) . In particular, boundaries of bands are not allowed to pass through double points, so are unaffected by finger and Whitney moves on F . Thus the answer to the question of which classes of H 2 (M, Σ; Z/2) can be represented by bands only depends on the homotopy class of the immersion F . It remains to show that Θ does not change under a regular homotopy of F . By definition, it suffices to show that Θ is unaffected by ambient isotopy, finger moves, and Whitney moves. This is immediate for all but the Whitney moves. Since Θ only depends on the homology class of the band by Lemma 5.15, we can assume that the boundary of the band B is disjoint from the boundary of the Whitney disc W along which the Whitney move is performed. Then the Whitney move leaves all terms in the definition of Θ except | Int B F | unchanged. Since the Whitney move uses two copies of the Whitney disc the change in | Int B F | is twice | Int B W |. As Θ takes values in Z/2, Θ(B) is unchanged as claimed.
In contrast to this, homotopic maps need not have the same b-characteristic status. For example, let Σ be the Klein bottle. Then an embedding f : Σ → R 4 must have normal Euler number e(νf ) ∈ {−4, 0, 4} by [Mas69]. It can be directly verified that exactly the embeddings with e(νf ) = 0 are b-characteristic and hence this notion is not invariant under homotopy. To see this, think of Σ ∼ = RP 2 #RP 2 , with a corresponding isomorphism H 1 (Σ; Z/2) ∼ = Z/2⊕Z/2. There is a standard embedding of RP 2 with normal Euler number ±2, and there are essentially three ways to take connected sums of these embeddings, realising the three options e(νf ) ∈ {−4, 0, 4}. With e(νf ) fixed, these embeddings are unique up to regular homotopy by Theorem 2.27. By Lemma 5.18, it suffices to check whether these standard embeddings are b-characteristic. To do so, one has to compute that λ Σ | ∂B(f ) = 0; this is straightforward. The b-characteristic status of the embedding is then decided by Θ(B), where B is a band with boundary (1, 1) ∈ H 1 (Σ; Z/2). One can compute that Θ(B) = 0 for this band if and only if e(νf ) = 0, that is if the embedding of Σ arises from the connected sum of the two different standard embeddings of RP 2 i.e. those with opposite normal Euler numbers.
We finish this section by describing the construction mentioned in Section 1.4, which we use to compare homotopy and regular homotopy of maps. Briefly, if we are interested in finding an embedding in a given homotopy class, rather than a regular homotopy class, we may use the construction to replace a given map by a homotopic map for which the invariant t is trivial. In particular, the construction is applicable in the cases from Theorem 2.27 in which there are infinitely many regular homotopy classes with µ(−) 1 = 0 in a given homotopy class. Let N ⊆ Σ i be a Möbius band with f i | N π 1 -trivial. In a small disc in N introduce four double points with the same sign by cusp homotopies and call the resulting immersion f i . Let F denote the map given by ( i =j f i ) f i . Then there is a convenient collection of Whitney discs W for all the double points of F such that t(F , W ) ≡ t(F, W) + 1 mod 2.
Proof. We will pair up the two new pairs of double points with new Whitney discs. Pick any pair of the four new double points and pair them by arcs in the small disc, as in Definition 2.25, such that the resulting circle is null-homotopic in M . For one of the arcs perform a connected sum in the interior with the core α of N . With this new pair of arcs, the double points have opposite sign, and by our choice of N the resulting Whitney circle bounds a Whitney disc W 1 in M . Up to boundary twisting, we assume that W 1 is framed, and by pushing off we ensure there is no intersection between the boundary of W 1 and the boundaries of the elements of W. Do the same for the remaining two new double points in f i , namely paired by a Whitney disc W 2 , which by definition is a parallel copy of W 1 . Since λ N (α, α) = 1, the boundaries of W 1 and W 2 intersect an odd number of times. To turn W ∪ {W 1 , W 2 } into a convenient collection of Whitney discs we have to remove any intersections between their boundaries. For this push any such intersection off the end of one of the Whitney arcs. This will in turn change the number of intersections between the interior of the Whitney discs and f i by one mod 2. Let W be the resulting collection of Whitney discs. We have Construction 5.19 together with Theorems 1.2 and 1.3 implies Theorem 1.9. The construction also has the following consequences. Proof. Since f * (w 1 (M )) is trivial, regular homotopy classes of generic immersions homotopic to f are detected by the Euler number of the normal bundle by Theorem 2.27. Further, since e(νf )−e(νf ) = ±8 we may add four cusps (of the same sign) to f to obtain a map f which is regularly homotopic to f . The map f is b-characteristic by assumption, while the map f is b-characteristic since f is, by Lemma 5.18. By Lemma 2.21, we know that µ(f ) 1 ∈ Z/2, so by construction µ(f ) = µ(f ) = µ(f ) = 0. So the quantities t(f ) and t(f ) are defined, and further t(f ) = t(f ). Apply Construction 5.19 to see that t(f ) ≡ t(f ) + 1 mod 2.
Applying Proposition 5.20 to immersions of RP 2 into R 4 we obtain the following. Proof. Recall that there exist embeddings g ± : RP 2 → R 4 with Euler number ±2. First we prove that e(νf ) ≡ 2 mod 4. By Lemma 2.21, we know that µ(f ) 1 ∈ Z/2. Then µ(g + ) = 0, and so µ(g + ) 1 = 0. Since f is homotopic to g + and µ(f ) = 0, it follows from Theorem 2.27 that e(νf ) ≡ e(νg + ) ≡ 2 mod 4. (The same argument would have applied with g − .) Note that any generic immersion of RP 2 into R 4 , and in particular the map f , is b-characteristic since H 2 (R 4 , RP 2 ; Z/2) ∼ = Z/2 and the nontrivial element does not satisfy condition (5.1) in Definition 5.10. Now we apply Proposition 5.20. Note that e(νf ) differs from one of ±2 by a multiple of 8. An easy inductive argument shows that t(f ) = 0 precisely if e(νf ) differs from ±2 by a multiple of 16.
Corollary 5.21 obstructs generic immersions of RP 2 in R 4 with e(f ) = ±2 mod 16 from being regularly homotopic to an embedding and thus partially recovers the result due to Massey [Mas69] that every embedding of RP 2 in R 4 must have Euler number ±2. Massey stated the result for smooth embeddings, since he used the G-signature theorem. But the G-signature theorem was later extended to the topological category by [Wal99,Chapter 14B], so Massey's result also holds for locally flat embeddings of RP 2 in R 4 .

Proofs of the statements from Section 5
In this section, we provide the proofs we skipped in Section 5. First we focus on Lemmas 5.3, 5.5, and 5.11, each of which gives conditions for the Kervaire-Milnor invariant of a generically immersed collection of surfaces F to be trivial. By Definition 3.1, a necessary condition for km(F ) to be trivial is that the intersection and self-intersection points are paired by a convenient collection of Whitney discs with interiors lying in the complement of F . We recall the statement of Lemma 5.3. Proof. Let G = {g i } be an algebraically dual collection of spheres for F = {f i }. By the definition of F , we can assume that precisely the elements of F ⊆ F have twisted dual spheres in G. By applying the geometric Casson lemma (Lemma 4.2) and Propositions 2.16, 2.22, and 3.2, we may assume without loss of generality that F and G are geometrically dual.
Let W = {W } be a convenient collection of Whitney discs pairing the intersections within F . By hypothesis F is not s-characteristic, so there exists some element a ∈ π 2 (M ) for which a · a ≡ F · a mod 2. For each f i , if f i · a is odd, tube one intersection of a with f i into the dual sphere g i . We obtain a sphere a for which the Z/2-valued intersection numbers f i · a vanish for all i. The mod 2 self-intersection number changes according to the parity of the Euler number of the normal bundles of the immersed sphere being tubed into, and therefore we have a · a ≡ a · a + F · a ≡ 1 mod 2.
Let h be an immersed sphere in M representing the class a . Suppose there is some W whose interior intersects some f i an odd number of times. Tube the interior into the geometrically dual sphere g i along f i once. If g i is not twisted, the unsigned count of intersections between W and f i is now even and the framing has changed by an even number. If g i is twisted, the unsigned count of intersections between W and f i is now even, but we have changed the framing of W by an odd number. Tube the interior of W into h. This can create many new intersections of W with F , but these all come in pairs. Since h · h ≡ 1 mod 2, we have changed the framing of W by a further odd number. The total change in framing is thus even.
Apply the procedure in the previous paragraph to each element of W. Each element of the resulting collection of Whitney discs has an even number of intersections with each f i , and the framing has changed by some even number. Now for each intersection of a W with an f i , tube into the corresponding geometrically dual sphere g i . A given sphere g i may or may not be twisted, but since each Whitney disc is tubed into a given g i an even number of times, the framing of each Whitney disc changes by an even number.
Finally correct the framing of all the Whitney discs by adding local cusps in the interior. This does not create any intersections with F . Call the resulting convenient collection of Whitney discs W := {W }. By construction, the interiors of W lie in M \ F , and thus km(F ) = 0.
The following transfer move will be useful for arranging that algebraically cancelling intersection points occur on the same Whitney disc. Let W 1 and W 2 be elements of W with Int W 1 H = ∅ = Int W 2 H. Then we may perform finger moves on H, so that the resulting generic immersion H admits six new double points, paired by framed, embedded Whitney discs {V, U 1 , U 2 }, each of which has two intersections with H , and such that the boundaries are disjoint and embedded. Moreover, the collection W := W ∪ {V, U 1 , U 2 } is a convenient collection of Whitney discs for H and we have |Int W 1 H | = |Int W 1 H| − 1 and |Int W 2 H | = |Int W 2 H| − 1.
Proof. Suppose that W 1 pairs intersections of h a and h b while W 2 pairs intersections of h c and h d , where repetition within a, b, c, d is allowed. Perform a finger move between h a and h c , creating two new double points paired by a corresponding framed, embedded Whitney disc V . Note that the interior of V is disjoint from the image of H. The operation depicted in Figure 6.1 gives a regular homotopy of Observe that V appears on both panels. As shown in the left panel, an intersection between W 1 and h e has been pushed down into h a and then one of the resulting intersection points has been pushed across to V . In the right panel, we see this new intersection between V and h e . We have performed a similar move in the right panel -an intersection between W 2 and h f has been pushed down to h c and one of the resulting intersection points is pushed over to V . These last three moves form a regular homotopy of H. Now we prove Lemma 5.5, whose statement we recall.
Lemma 5.5. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1, with components {f i }. Suppose that F admits algebraically dual spheres, and that all the intersections and self-intersections of F are paired by a convenient collection W of Whitney discs. Let W ⊆ W denote the sub-collection of Whitney discs for the intersections within F , where F is as in Definition 5.1. If t(F , W ) = 0, then km(F ) = 0.
By applying the geometric Casson lemma (Lemma 4.2) and Propositions 2.16, 2.22, and 3.2, we may assume without loss of generality, after a regular homotopy of F and G, that F and G are geometrically dual.
We modify the collection of Whitney discs, as follows, so that each has an even number of intersections with F . Since the count in (6.1) is zero, the number of Whitney discs with odd intersection with F is even, so we may pair them up (arbitrarily). For each such pair, apply Construction 6.1. This changes F by finger moves to some F , whose intersections and self-intersections are paired by a convenient collection of Whitney discs W := {W }, such that each element of W has an even number of intersections with (F ) . Note that the new Whitney discs created by the application of Construction 6.1 have been added to the collection.
For each intersection of some W with (F ) , tube W into the corresponding geometrically dual sphere. Note that each sphere being tubed into is necessarily twisted, but since we tube an even number of times, the total change in the framing of W is even. Do this for each element of W . The resulting family of Whitney discs may still intersect F , but not (F ) . For each such intersection with F , again tube into the appropriate geometrically dual sphere. Now the spheres are not twisted, so the framing of the Whitney discs changes by an even number. Correct the framing of all the Whitney discs by adding local cusps in the interior. We have now produced the desired convenient collection of Whitney discs for the intersections within F , whose interiors lie in the complement of F . This shows that km(F ) = 0, and therefore km(F ) = 0, as desired.
Before giving the proof of Lemma 5.11, we explain the key new construction in this paper, which we already mentioned in Section 1.2. Briefly, given a band B with boundary lying on an immersed surface, a finger move along a fibre of the band produces two new double points paired by a Whitney disc arising from B. Consider B ∈ B(F ) as a D 1 -bundle. Then we can do a finger move along a fibre of B, with endpoints missing A, as depicted in Figure 1.1. We call this fibre the finger arc, and denote it by D. We assume that D misses all double points of B and all intersections between Int B and F .
A finger move depends on a choice of 2-dimensional sub-bundle of the normal bundle to the finger arc (the proof of Lemma 6.3 will give further details). We use a sub-bundle that lies in the tangent bundle T F at one end of the arc, contains ν B D along D, and intersects T F in the line bundle ν B D at the other end of the arc. Here ν B D is the normal bundle of D in B. Call the immersion resulting for the above finger move F . We will check in the next paragraph that the remainder of B, i.e. the complement in B of a tubular neighbourhood of D, gives a Whitney disc for the new pair of double points. Make the boundary embedded and disjoint from A, by boundary push off operations, and then boundary twist if necessary, to obtain a framed Whitney disc W B for the new double points. Then W := W ∪ {W B } is a convenient collection of Whitney discs pairing the double points of F . Note that both F and W B depend on the choice of finger arc D and the 2-dimensional sub-bundle of its normal bundle mentioned above. Now, as promised, we check that W B is a Whitney disc. The finger move creates a trivial Whitney disc and we refer to the corresponding Whitney arcs as the trivial arcs. The double points are also paired by the arcs A 1 , A 2 ⊆ ∂B where A 1 ∪ A 2 = ∂W B . The existence of the disc W B implies that the group elements of the double points agree with respect to the arcs A 1 and A 2 . It remains only to to see that the double points have opposite signs with respect to the arcs A 1 and A 2 (see Definition 2.26). The case that both M and Σ are orientable is straightforward. The general case follows from the w 1 condition in the definition of a band, (5.1), as we now check.
First we consider the case that B is an annulus. Let ∂ 1 B and ∂ 2 B denote the two components of ∂B. Then A 1 and A 2 differ from the trivial arcs joining the new double points by ∂ 1 B and ∂ 2 B respectively. By Definition 2.26 the double points have opposite sign precisely when w 1 (Σ), ∂B + w 1 (M ), ∂ 1 B = 0, which matches (5.1) since ∂ 1 B is homotopic in M to the core of B. Now suppose that B is a Möbius band. Then the union of A 1 and A 2 and the trivial pair of arcs is the circle ∂B ⊆ Σ. Moreover, the union of the image of A 1 and either one of the trivial arcs forms a circle in M homotopic to the core C of B. As before, the double points have opposite sign precisely when w 1 (Σ), ∂B + w 1 (M ), C = 0, which again matches (5.1).
The following lemma explains how Construction 6.2 changes the value of t. In the proof we will also carefully explain how to make suitable choices of 2-dimensional sub-bundles, as required for the finger move in Construction 6.2. (ii) Suppose that F and W are obtained from F and W by a single application of Construction 6.2 with respect to a band B ∈ B(F ) and an arc D ⊆ B, where B is an annulus with w 1 (Σ) nontrivial on both boundary components and D ⊆ B connects the two boundary components. Let A denote the Whitney arcs corresponding to W. Then For the proof it will be advantageous to refrain from applying boundary twists and removing intersections involving ∂W B , and to instead use the following alternative definition of t(F, W), using a slightly weaker restriction on collections of Whitney discs, as in [Sto94], cf. [FQ90, Section 10.8A].
Definition 6.4. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1. A weak collection of Whitney discs for F is a collection W of Whitney discs pairing all the intersections and self-intersections of F , with generically immersed interiors transverse to F , and with Whitney arcs whose interiors are generically immersed in F (Σ) minus the double points of F .
In particular, we allow the boundaries of Whitney discs to be generically immersed on Σ and to intersect one another transversely. We also allow the Whitney discs to be twisted, i.e. for the framing of the normal bundle restricted to the boundary to disagree with the Whitney framing. Each of the discs in a weak collection of Whitney discs admits a normal bundle. The proof is the same as for a generic immersion of pairs, but with a preliminary step than one has to first fix the normal bundle in neighbourhoods of the two double points being paired.
Definition 6.5. Given a weak collection of Whitney discs W := {W } for the double points of a generic immersion F as in Convention 1.1, fix an ordering on the indexing set for W and define where e(W ) is the relative Euler number of the normal bundle with respect to the Whitney framing on ∂W .
Note that if W is a convenient collection of Whitney discs for F , then t alt (F, W) = t(F, W). The following lemma shows that t alt can be used as a proxy for t in general.
Lemma 6.6. Given a weak collection of Whitney discs W := {W } for the double points of a generic immersion F as in Convention 1.1, there exists a convenient collection of Whitney discs W such that t(F, W ) = t alt (F, W).
Proof. For each Whitney disc W with e(W ) = 0, add boundary twists to obtain W with e(W ) = 0. We have and also µ Σ (∂W ) = µ Σ (∂W ), and k> |∂W ∂W k | = k> |∂W ∂W k |. Next, remove intersections between Whitney arcs as well as self-intersections of Whitney arcs, as follows. For each self intersection of ∂W and, for k > , for each intersection of an arc ∂W k with ∂W , push the intersection off the end of one of the Whitney arcs of ∂W , to obtain the convenient collection W := {W }, where each W is the result of applying the above procedure to W . This yields Sum over to obtain that t(F, W ) = t alt (F, W) ∈ Z/2 as claimed.
Proof of Lemma 6.3. By Lemma 6.6, it will suffice to show that in case (i), Every self-intersection of ∂B and each intersection ∂B A will contribute one intersection of ∂W B and between ∂W B and A respectively, while each intersection Int B F will contribute one intersection between Int W B and F . Thus it remains to show that the framing e(B) appearing in the definition of Θ A (B) agrees with the framing e(W B ). For this it will be helpful to pick the finger arc and the 2-dimensional sub-bundle for its normal bundle needed for the finger move more carefully, which we do next. Let ∂ i B denote the connected components of ∂B ⊆ Σ. Consider the following decomposition of the tangent bundle of M restricted to ∂ i B: As shown in Figure 6.2, choose a section s of the normal bundle of B that is nonvanishing on the finger arc. In both boundary components ∂ i B, we assume that this section lies in ν Σ ∂iB . Now rotate the section near the top boundary component, as shown in the middle picture of Figure 6.2, to obtain a section s , so that on the top component s lies in (ν M Σ | ∂iB ∩ ν M B | ∂iB ). For the finger move, by definition, we use the 2-dimensional sub-bundle of ν M D determined by s and T (∂ i B), as shown in the right-most figure of Figure 6.2. Now consider the Whitney disc W B obtained from B after performing the finger move along D using the above 2-dimensional sub-bundle of its normal bundle. By definition, e(W B ) equals the number of zeros of s | W B , since on the boundary Whitney arcs it is normal to one sheet and tangent to the other. On the other hand, the number of zeros of s | W B equals the number of zeros of s, since s was obtained from s by a rotation, and neither section vanishes near the finger arc. Finally by definition e(B) counts the zeros of s. Therefore we see that e(B) = e(W B ).
Case 2. The band B is an annulus such that w 1 (Σ) restricted to both components of ∂B is nontrivial, as in (ii).   (Here we also implicitly identify ν M B with its image in M .) Now do the finger move using D × S 1 × {0} according to this parametrisation, where S 1 is the unit circle in the R 2 factor, along with a finger tip. The choices above imply that s | ∂W B is a Whitney framing, where W B denotes the new Whitney disc created according to Construction 6.2. Specifically, let F denote the result of the finger move. By our choice of the 2-dimensional sub-bundle for the finger move above, the section s is normal to F along half of ∂W B , and tangent along the other half; see Figure 6.5.
As previously mentioned, we need to check that the relative Euler number e(B) in Θ A (B) agrees with the twisting number e(W B ) in t alt . The relative Euler number e(B) is given by the number of zeros of the section s on the interior of B. As mentioned before, this coincides with the number of zeros of the section s . Since s | ∂W B is a Whitney framing, this further coincides with the twisting number e(W B ) as desired, since we assumed there are no zeros of s within the strip ([−1, −1 + ε] ∪ [1 − ε, 1])×[−1, 1] ⊆ B used for the finger move.
Next we prove Lemma 5.11. Here is the statement for the convenience of the reader. If t(F , W ) = 1 we proceed as follows. By hypothesis, λ Σ is nontrivial on ∂B(F ), meaning that there are bands B 1 and B 2 with boundary C 1 and C 2 on F (Σ ) minus double points such that λ Σ (C 1 , C 2 ) = 0 ∈ Z/2. Here it is possible that B 2 is a parallel push-off of B 1 . Using either B 1 or B 2 and Construction 6.2, perform a finger move and obtain a new framed Whitney disc. If this changes t(F , W ) using one of B 1 or B 2 , then km(F ) = 0 by Lemma 5.5. Hence we can assume the individual moves for B 1 and B 2 do not change t(F , W ). Now using Lemma 6.3 twice, for B 1 and B 2 simultaneously, the change in t(F , W ) is as before except that there is an odd number of intersections between the boundary arcs for the new Whitney discs coming from B 1 and B 2 . Removing these by pushing one Whitney arc off the end of the other introduces an odd number of intersections between the Whitney discs and F , and thus changes t(F , W ) by one. Once again we deduce that km(F ) = 0 by Lemma 5.5.
For the proof of Lemma 5.15, we will need the next four lemmas, Lemmas 6.7 to 6.10. Recall that the N 0 is isomorphic to Z/2 while the N 1 vanishes. It follows that in the Atiyah-Hirzebruch spectral sequence with E 2 -term H p (M, Σ; N q ) and converging to N p+q (M, Σ), there is no nontrivial differential going out of H 2 (M, Σ; N 0 ) ∼ = H 2 (M, Σ; Z/2); such a differential would have codomain H 0 (M, Σ; N 1 ) = 0. Thus the edge homomorphism N 2 (M, Σ) → H 2 (M, Σ; Z/2) is onto, as desired.
Lemma 6.8. Let F : (Σ, ∂Σ) (M, ∂M ) be as in Convention 1.1, with components {f i }. Suppose that λ(f i , f j ) = 0 for all i = j and that µ(f i ) = 0 for all i. Let A be a choice of Whitney arcs pairing the double points of F . Then the function Θ A is quadratic with respect to the Z/2-valued intersection form λ Σ . That is, let S and S be compact surfaces with boundaries C and C respectively, with generic immersions of pairs (S, C) (M, Σ) and (S , C ) (M, Σ) such that C and C intersect A and each other transversely, and are such that w 1 (Σ) is trivial on every component of C and C . Then we have Proof. The term e(S) in Definition 5.12 is defined component-wise and the terms |C A| and |Int S F | are linear in C and S, respectively. Hence the only term that is not linear in S is µ F (C). This term is also quadratic in the sense that which proves the lemma. The idea of the proof is to remove all intersections of C by locally adding a twisted disc D as indicated in Figure 6.6. More precisely, we add these discs D such that the interiors are disjoint from the interior of F and the boundary is disjoint from A. Then pick a section γ D of ν M D such that, along the aligned (i.e. parallel) parts of the boundaries, γ D and γ S are opposite. Glue D to S along the aligned parts of the boundaries and push this part of the boundary off F as indicated in Figure 6.7. Each of these local twisted discs has mod 2 Euler number 1, as can be seen from the nontrivial linking in Figure 6.8. Thus the resulting surface S has embedded boundary and the mod 2 Euler number of S differs from that of S by the number of intersections of C modulo two, i.e. µ Σ (C). Since we have neither changed the Proof. Without loss of generality, assume that Σ is connected. Considering the entire collection F = {Σ i }, every component of C would appear as the boundary of precisely two of the Σ i , since otherwise C would be nontrivial in H 1 (Σ; Z/2). To see this note that C can contain homologically essential curves in H 1 (Σ; Z/2), provided they cancel. However none of these can be orientation-reversing curves, since C is embedded.
The idea of the proof is to take "half" of the elements of F. Let x ∈ Σ | C be an arbitrary basepoint away from C. For each Σ i , define p(Σ i ) ∈ Z/2 as follows. Pick a point y ∈ Int Σ i and a path w in Σ from x to y which is transverse to C. Define p(Σ i ) as the mod 2 intersection number of w and C.
We show that p(Σ i ) is independent of the choices of w and y. If w is another path from x to y, then the concatenation w −1 · w is a loop in Σ and we have So p(Σ i ) does not depend on the choice of w. Also, since each Σ i is connected, p(Σ i ) does not depend on y. To see this let y ∈ Int Σ i , and choose a path z from y to y that lies in Int Σ i . Let w be a path from x to y , which is further transverse to C. Then |w C| = |(w · z) C| = |w C|.
The first equation uses that z ⊆ Σ i and the second uses independence of the choice of w. Hence p(Σ i ) ∈ Z/2 is well-defined as desired.
Now let F consist of all the components Σ i for which p(Σ i ) = 0. This subset is the one we seek, since for a fixed component C j of C, the two components of F containing a cut-open copy of C j have different values of p. This completes the proof of the lemma.
We are now ready for the proof of Lemma 5.15. Recall that we were able to assume that C is embedded in F (Σ) away from the double points and that λ M ([N ], [N ]) = e(νN ) mod 2. We claim that this in turn agrees with e(S) + Fi∈F e(F i ). Here we define e(F i ) as follows. We used F to define a nowhere vanishing section of ν M S | C . Since ν M S | C is two dimensional, we can pick a linearly independent nonvanishing section. This can be equivalently used for the definition of e(S). But this new section now can also be used to define e(F i ). Combining these vector fields that are transverse to the zero section defines a vector field on the normal bundle of This completes the proof of (i).
Before proving (ii) and (iii), we introduce a general construction. Let B be an annulus as in (ii) with an embedded arc D in B connecting its two boundary components. As in Remark 5.14, add a tube to F (Σ) along the arc D. Let d, d denote the two discs removed from F when the tube is added. Adding the tube changes F to some F , an immersion of a surface Σ , and changes B to a disc ∆. As before, observe that ∂∆ is an orientation-preserving curve in Σ and A is now a collection of Whitney arcs pairing the double points of F . By construction, we see that Θ A (B, D) = Θ A (∆).
Moreover, suppose there is either some immersed compact surface S in M as in (i) or some immersed annulus B as in (ii), with an embedded arc D on B connecting its two boundary components, where possibly B = B . We may choose the tube in the above construction thin enough so that Θ A (S) and Θ A (B , D ) remain unchanged. In particular, this means we assume, after a small local isotopy, that the discs d and d do not intersect the boundaries of S and B , so both represent classes in H 2 (M, Σ ; Z/2). We have the following claim.  We continue now to prove (ii). Let B and B be immersed annuli in M as in the statement of (ii). Choose arcs D in B and D in B connecting the boundary components of each, and assume that [B] = [B ]. By the construction from the proof of Claim 6.11 applied twice, once to B and once to B , we find discs ∆ and ∆ , coming from B and B respectively, such that Θ A (B, D) = Θ A (∆) and Θ A (B , D ) = Θ A (∆ ). From Claim 6.11, applied twice with the rôles of B and B reversed, we see that the classes [ , the proof of (ii) is completed by appealing to (i), which says that Θ A (∆) = Θ A (∆ ), since both ∆ and ∆ have w 1 (Σ ) trivial on the boundary. In the second case, [∆] = [∆ ] + [d], we also appeal to (i), but now for the pair of surfaces ∆ and ∆ ∪ d. So we have that Θ A (∆) = Θ A (∆ ∪ d). It follows directly from the definition that Θ A (∆ ∪ d) = Θ A (∆ ). This completes the proof of (ii). In particular, we have proved that Θ A (B, D) does not depend on the choice of arc D.
The proof of (iii) is similar. Suppose we have an immersed annulus B in M as in the statement of (ii), as well as an immersed compact surface S in M as in the statement of (i). For each pair of double points, we can pick Whitney discs W 1 and W 2 with boundary in A and A respectively. By adding small strips to the union of W 1 and W 2 in the neighbourhood of the double points, we can see that the difference of A and A is the boundary of some collection of bands B . For more details about this construction, see the upcoming proof of Theorem 1.3. Then we have which vanishes by assumption. Figure 6.10. Left: the Whitney discs W 1 , W 2 , and V 1 , pairing up double points as (p 1 , p 2 ), (q 1 , p 2 ) and (q 1 , q 2 ), respectively. Right: the Whitney disc V 2 pairing up (p 1 , q 2 ) is obtained as a union of W 1 , W 2 , and V 1 , by adding small bands at the points p 2 and q 1 to resolve the singularities, and pushing the interiors of the bands into the complement of F . Compare with [Sto94, Figure 2].
Case 2. The collections of Whitney arcs A and A correspond to a different pairing up of the double points of F .
From A we can construct Whitney arcs A so that A and A correspond to the same pairing up of double points, as in Figure 6.10. Here are the details. We will define the family A iteratively, starting with A. Let p 1 , p 2 , q 1 , q 2 be double points of F . Suppose that arcs in A pair up p 1 and p 2 , as well as q 1 and q 2 , while arcs in A pair up p 2 and q 1 . Pick Whitney discs W 1 and W 2 with boundary in A. Let V 1 be a Whitney disc for the points p 2 and q 1 with boundary away from A. Then, as indicated in Figure 6.10, we may choose Whitney arcs, away from the other elements of A, so that p 2 and q 1 are also paired by a Whitney disc V 2 , obtained as a union of W 1 , W 2 , and V 1 . Modify the family A by removing ∂W 1 and ∂W 2 , and adding in ∂V 1 and ∂V 2 . Comparing this new family with A , we see that we have reduced the number of mismatches in the pairing up of double points of F . Iterate this process and call the result A .
Looking more closely at the construction in the previous paragraph, observe that at each step, the family of arcs changes by adding in two parallel copies of the boundary of a Whitney disc V 1 . Since intersection points are counted modulo 2, Θ A and Θ A are equal. By Case 1 we know that Θ A and Θ A are equal when restricted to B(F ). Thus, Θ A and Θ A are equal when restricted to B(F ), as needed.
7. Proof of Theorems 1.3 and 1.6 First we prove Theorem 1.6 from the introduction, which shows that for b-characteristic surfaces, t(F , W ) ∈ Z/2 is well-defined, i.e. independent of the Whitney discs W . Note that the theorem has no assumption about the existence of algebraically dual spheres.  Proof. The definition of F (Definition 5.1) did not require algebraically dual spheres. Similarly Lemma 5.15, according to which if λ Σ | ∂B(F ) = 0 then Θ : B(F ) → Z/2 is a well-defined map, also did not require algebraic duals.
We will show below that t(F , W ) is independent of the choice of W , assuming that F is bcharacteristic, i.e. assuming that λ Σ | ∂B(F ) = 0 and that Θ is the zero map. This will prove (i). For the first part of (ii) note that the existence of algebraically dual spheres does not change under regular homotopy. So if H is regularly homotopic to F then F is regularly homotopic to H , and therefore H is b-characteristic by Lemma 5.18. To establish (ii) it remains to observe that the count t does not change under finger moves and (framed) Whitney moves in the interior. (iii) is a consequence of (i) and (ii) since t(F , W ) clearly vanishes if W is a collection of Whitney discs with interiors disjoint from F (Σ), as in the definition of km(F ). Now we prove (i). Since F is b-characteristic, by definition λ Σ | ∂B(F ) is trivial and Θ is trivial on B(F ). As indicated above, the function Θ, as well as which classes of H 2 (M, Σ ; Z/2) can be represented by bands, only depends on the immersion F up to regular homotopy. We need to show that t(F , W ) does not depend on the choice of pairing of the double points, the choice of Whitney arcs, nor the choice of Whitney discs; see Figure 7.1. Let W be a given initial choice of convenient collection of Whitney discs for the double points of F . Let A be the corresponding collection of Whitney arcs for the double points of F .
The remainder of the proof is similar to Stong's [Sto94] and [FQ90, Section 10.8A]. We will work with weak collections of framed Whitney discs and the alternative count t alt ∈ Z/2, as in Definitions 6.4 and 6.5. So the boundaries of our collections of Whitney discs might not be disjointly embedded, but the Whitney discs will be framed (as can always be arranged by boundary twisting). We will show that t alt (F , W ) does not depend on the choice of weak collection of Whitney discs W , and then use that t alt (F , W ) = t(F , W ) for W a convenient collection (Lemma 6.6).
Claim 7.1. Given a weak collection of Whitney discs W corresponding to some choice of pairing up of double points of F , then for any other choice of pairing, there exists a weak collection of Whitney discs V for that choice, so that t alt (V ) = t alt (W ).
Proof. Let p 1 , p 2 , q 1 , q 2 be double points of F . Suppose that in the initial choice of data, p 1 and p 2 are paired by a Whitney disc W 1 ∈ W , and q 1 and q 2 by a Whitney disc W 2 ∈ W . Suppose we instead pair up p 1 and q 2 by some Whitney disc V 1 . Then, as indicated in Figure 6.10, p 2 and q 1 are also paired by a Whitney disc V 2 , obtained as a union of W 1 , W 2 , and V 1 . Then (W \{W 1 , W 2 })∪{V 1 , V 2 } is a weak collection of framed Whitney discs. The contribution of V 1 and V 2 to t alt (F , (W \{W 1 , W 2 })∪{V 1 , V 2 }) counts the intersections of F with each disc W 1 and W 2 once, while it counts the intersections of F with the disc V 1 twice. Each intersection of ∂V 1 with A\(∂W 1 ∪∂W 2 ) can be paired with an intersection of ∂V 2 with A \ (∂W 1 ∪ ∂W 2 ). Each intersection of ∂V 1 with ∂W 1 ∪ ∂W 2 gives rise to two contributions to t alt : an intersection of ∂V 2 with ∂V 1 and a self-intersection of ∂V 2 . Since intersections are counted mod 2 in the definition of t alt , we see that as needed. Iterate this process to complete the proof of Claim 7.1. Figure 7.1. Within Σ we see the preimages p ± 1 and p ± 2 , for the double points p 1 and p 2 of F respectively. Blue denotes the Whitney arcs for W 1 while red denotes the Whitney arcs for the new disc V 1 . On the left, the choice of sheets stays the same, while it changes on the right. Compare with [Sto94, Figure 3].
Continuing with the proof of Theorem 1.6, next we check that t alt is independent of the choice of Whitney discs. This includes potentially changing the Whitney arcs and the choice of sheets at each double point. Suppose we are given another weak collection of framed Whitney discs V for the double points of F . By applying Claim 7.1, we may assume that V corresponds to the same pairing of double points of F as W . Assume the collections are indexed so that W ∈ W and V ∈ V correspond to the same pair of double points. For each i, define the weak collection of Whitney discs Let A i denote the collection of Whitney arcs for U i . First we prove a special case.
Claim 7.2. Suppose the Whitney disc W i is framed, embedded, with interior disjoint from F and the Whitney discs U i−1 \ {W i }, and with ∂W i disjoint from A i , other than the endpoints. Then t alt (F , U i−1 ) = t alt (F , U i ) ∈ Z/2.
Proof. We refer the reader to Figure 7.2. As described in the figure, we wish to perform the Whitney move using W i pushing towards the Whitney arc a i for W i . Observe that the union of V i with a strip, corresponding to the unit outward pointing normal vector field of a i ⊆ ∂W i , is a band; this requires a small isotopy of V i to ensure that the chosen vector field of a i is compatible with V i , as shown in Figure 7.2. For this band, performing a finger move as in Construction 6.2 creates W i as the standard Whitney disc, and V i as the new Whitney disc arising from the band. Since F is b-characteristic, the disc V i , has trivial contribution to t alt (F , U i ) by Lemma 6.3. By hypothesis, so does W i to t alt (F , U i−1 ). Therefore t alt (F , U i−1 ) = t alt (F , U i ) ∈ Z/2 as asserted in the claim.
For the case that ∂W i = ∂V i , we first perform a small isotopy of each Whitney arc in ∂V i relative to the endpoints, so that the result intersects ∂W i only in the endpoints. Then we perform the same procedure as in the previous paragraph, including the requisite isotopy of V i , to ensure that the union of V i with the strip forms a band. Alternatively in this case, we could sidestep this argument and appeal to the argument of Stong from [Sto94] -the union of V i and W i is an immersed sphere in M , and the claim follows from the fact that F is s-characteristic (Lemma 5.17 and Remark 5.7). Now we prove the general case. Denote the double points paired by W i by p 1 and p 2 . By performing a finger move, split W i into new Whitney discs W i and U 1 creating two new double points q 1 and q 2 in the process, paired by a standard trivial Whitney disc U 2 . See Figure 7.3. By an appropriate choice of splitting arc, we ensure that the points p 1 and q 1 are paired by W i and the points q 2 and p 2 are paired by U 1 , where U 1 is trivial, i.e. it is framed, embedded, with interior disjoint from everything and boundary ∂U 1 disjoint from ∂V i except at p 2 . In the case that ∂W i = ∂V i , this also requires changing ∂V i by a small isotopy, close to ∂U 1 .
Let (F ) denote the result of performing this regular homotopy on F . Note that t alt (F , U i−1 ) = t alt ((F ) , (U i−1 \ {W i }) ∪ {W i , U 1 }) (7.1) by construction. Let V i denote the Whitney disc obtained as the union of V i , W i , and U 2 , as in Figure 6.10. Observe that the Whitney discs U 1 and V i pair the same double points, namely q 2 and Left: two sheets of the surface Σ and two Whitney discs W i and V i pairing the double points p 1 and p 2 . The disc W i is assumed to be framed, embedded, have interior disjoint from all other surfaces, and ∂W i disjoint from A i−1 \ ∂W i . One of its Whitney arcs a i is also labelled. The blue strip to the right of a i is an extension of the Whitney disc beyond its boundary, that is part of the data for the Whitney move. Right: the result of the Whitney move. The strip and the disc V i from the previous panel have formed a band (blue).
p 2 , where U 1 is trivial. So by Claim 7.2, By the proof of Claim 7.1 (see Figure 6.10), Since U 2 is trivial, we can use it to undo the Whitney move, and obtain The combination of (7.1), (7.2), (7.3), and (7.4) imply t alt (F , U i−1 ) = t alt (F , U i ). This completes the proof that t alt is independent of the choices of Whitney discs, and therefore completes the proof that t alt is well-defined. Finally by Lemma 6.6 we know that t alt (F , W ) = t(F , W ) for W a convenient collection, so t is well-defined for convenient collections W , as desired. Figure 7.3. Splitting a Whitney disc W i into two Whitney discs. One of the new Whitney discs, U 1 , pairing p 2 and q 2 , satisfies the hypotheses of Claim 7.2. The other Whitney disc W i intersects whatever W i intersected. The trivial Whitney disc U 2 pairing the new double points q 1 and q 2 is shown in grey.
Next we recall the statement of Theorem 1.3 for the convenience of the reader. Suppose that λ(f i , f j ) = 0 for all i = j, that µ(f i ) = 0 for all i, and that F has algebraically dual spheres. If F is not b-characteristic then km(F ) = 0. Hence if π 1 (M ) is good then (ii) of Theorem 1.2 holds. If F is b-characteristic then km(F ) = t(F , W ) ∈ Z/2 for every convenient collection of Whitney discs W pairing all the intersections and self-intersections of F .
Proof. First we show that if F is not b-characteristic then km(F ) = 0. By Lemma 5.3 we reduce to the case that F is s-characteristic. Then by Lemma 5.11 we reduce to the case that λ Σ | ∂B(F ) is trivial, which implies that the function Θ is well defined on B(F ). Since we assume that F is not b-characteristic, there exists B ∈ B(F ) so that Θ(B) = 1, so we can apply Construction 6.2 First we claim that F is b-characteristic. To see this, we start by computing H 2 (W, S 1 × S 1 ; Z/2) using the long exact sequence of the pair with Z/2 coefficients: H 2 (S 1 × S 1 ) − H 2 (W ) 0 − − → H 2 (W, S 1 × S 1 ) −→ H 1 (S 1 × S 1 ) ∼ = Z/2 ⊕ Z/2 − H 1 (W ) ∼ = Z/2. Therefore H 2 (W, S 1 × S 1 ; Z/2) ∼ = Z/2 is generated by S × {p} where S ⊆ S 3 is a Seifert surface for the knot K(S 1 ) and p ∈ S 1 . The intersection form of S 1 × S 1 restricted to ∂S is trivial. Since Θ is well defined on homology classes we can compute it using S. But S has interior disjoint from the image of F , embedded boundary, and trivial relative Euler number, so Θ(S) = 0. Thus F is b-characteristic as claimed.
Observe that km(f 1 ) = 1 inside * CP 2 (see e.g. [FQ90, Section 10.8]). We can pick a convenient collection of Whitney discs for f 1 in * CP 2 . Since f 2 is an embedding, these constitute a convenient collection of Whitney discs for F . It follows that km(F ) = km(f 1 ) = 1. Note that the choice of knot K was irrelevant, since for any two choices the resulting immersions F are regularly homotopic and hence have equal Kervaire-Milnor invariant.
Example 8.5. In the previous example we constructed a generically immersed torus in * CP 2 #(S 1 ×S 3 ) with nontrivial Kervaire-Milnor invariant. In particular, this torus is not homotopic to an embedding (cf. Section 1.4). Now we show that in contrast to this every map from a closed surface Σ to S 1 × S 3 is homotopic to an embedding. Note that these classes do not have algebraically dual spheres since π 2 (S 1 × S 3 ) = 0. The surfaces in the regular homotopy class with µ(f ) 1 = 0 are either not bcharacteristic or t(f ) vanishes.
Since the projection S 1 × S 3 → S 1 is 3-connected, the induced map [Σ, S 1 × S 3 ] → [Σ, S 1 ] is bijective. In particular, the homotopy class of a map f : Σ → S 1 × S 3 is determined by the induced map on fundamental groups.
We first consider the case that Σ is connected. There exists a decomposition Σ = H#Σ , where H is either a sphere, a torus, or a Klein bottle, with respect to which f can be written as an internal connected sum T #f with f π 1 -trivial. In particular, f is homotopic to an embedding inside a ball D 4 ⊆ S 1 × S 3 . It remains to show that T is homotopic to an embedding, which will show that the connected sum is homotopic to an embedding.
If H is a sphere, we are done. If H is a torus, let i : S 1 × S 1 → S 3 be an embedding. For each k ∈ Z, define the embedding h k : S 1 ×S 1 → S 1 ×(S 1 ×S 1 ) by (s, t) → (s k , (s, t)). Let h k := (Id ×i)•h k . There exists some k and some identification of H with S 1 × S 1 such that T and h k induce the same map on fundamental groups and thus are homotopic. If H is a Klein bottle, let p : H → S 1 be a fibre bundle with fibre S 1 . For each k ∈ Z there exists an immersion i : H S 3 such that h k (x) := (p(x) k , i(x)) is an embedding H → S 1 × S 3 . As before, there exists some k such that T and h k are homotopic.
The argument easily generalises to disconnected surfaces since the above embeddings can be realised disjointly.
Next we prove Proposition 1.7 and Corollary 1.8, which we restate for the convenience of the reader.
Proof. The vanishing of the self-intersection number of F i is witnessed by a collection of Whitney discs W i in M i , for each i. The union W 1 W 2 , now considered in M 1 #M 2 , shows that the intersection and self-intersection numbers of F 1 F 2 , as well as for F 1 #F 2 , vanish in M 1 #M 2 . Since the union W 1 W 2 pairs all the intersections and self-intersections of F 1 F 2 (resp. F 1 #F 2 ), and since elements of W i cannot intersect F j (Σ j ) for all i = j, the claimed relationship t(F 1 F 2 ) = t(F 1 #F 2 ) = t(F 1 ) + t(F 2 ) holds as long as F 1 F 2 and F #F 2 are b-characteristic.
As a preliminary step, note that neither F i has a framed dual sphere in M i , since otherwise it would not be s-characteristic, and therefore, not b-characteristic by Lemma 5.17. As a result, Σ i = Σ i for i = 1, 2.
Next we consider the immersion F 1 F 2 : (Σ 1 Σ 2 , ∂Σ 1 ∂Σ 2 ) M 1 #M 2 . Let S ⊆ M 1 #M 2 denote a connected sum 3-sphere. Consider a band [B] ∈ H 2 (M 1 #M 2 , Σ 1 Σ 2 ). By (topological) transversality we can assume that B is immersed, the double points of B are disjoint from S, and the intersection B ∩ S corresponds to an embedded 1-manifold in the interior of the domain of B, since ∂B ⊆ Σ 1 Σ 2 ⊆ (M 1 #M 2 ) \ S. The image of this 1-manifold in S is embedded and bounds a collection of immersed (perhaps intersecting) discs in S. Surger B using two copies each of these discs to produce B 1 ⊆ M 1 and B 2 ⊆ M 2 , where each B i is an immersed collection of surfaces with ∂B i ⊆ Σ i .
We claim that each component of B i can be replaced by a band. Recall that since each M i and Σ i is oriented, there is no condition on Stiefel-Whitney classes for bands, and we need only arrange that each component is either a Möbius band or an annulus. By considering the Euler characteristic, we see that each component is homeomorphic to either a sphere, an RP 2 , a disc, a Möbius band, or an annulus. Then use the tubing trick from the proof of Lemma 5.17 to replace each sphere, RP 2 , or disc component by a band. More precisely, choose a small disc on Σ 1 or Σ 2 , as appropriate, away from all Whitney arcs and double points, and tube into the disc, sphere, or RP 2 .
For each i, the value of Θ(B i ) does not depend on whether the ambient manifold is M i or M 1 #M 2 , since B i does not intersect F j (Σ j ) for all i = j (see Definition 5.12). Since each F i is b-characteristic, Θ(B) = Θ(B 1 ) + Θ(B 2 ) = 0 + 0 = 0 ∈ Z/2. This completes the proof that F 1 F 2 is b-characteristic.
Now we consider the connected sum F 1 #F 2 . Let B ∈ H 2 (M 1 #M 2 , Σ 1 #Σ 2 ) be a band. As above, we assume that the intersection B ∩ S corresponds to an embedded 1-manifold in the domain of B. Unlike above, this may include embedded arcs with endpoints on the boundary. These endpoints are mapped to the intersection (F 1 #F 2 )(Σ 1 #Σ 2 )∩S. By connecting the endpoints with arcs on (F 1 #F 2 )(Σ 1 #Σ 2 )∩S, we again get a collection of closed circles in S, which bound an immersed collection of discs in S. Surger using these discs as before to produce collections B i ⊆ M i . Once again, each component of B i is homeomorphic to either a sphere, an RP 2 , a disc, a Möbius band, or an annulus. By the tubing trick from above applied to the sphere, RP 2 , and disc components, we may arrange that each component is a band. The argument of the previous paragraph now applies to show that F 1 #F 2 is b-characteristic.
Corollary 1.8. For any g, there exists a smooth, closed 4-manifold M g , a closed, oriented surface Σ g of genus g, and a smooth, b-characteristic, generic immersion F : Σ g M g with t(F ) = 0 and therefore km(F ) = 0.
Proof. By the same proof as in Example 8.4, for any knot K the product T := K ×Id : S 1 ×S 1 → S 3 ×S 1 is an embedded b-characteristic torus. A generic immersion S : S 2 → CP 2 representing three times a generator of H 2 (CP 2 ; Z) is b-characteristic with km(S) = 1. This was the original example of Kervaire and Milnor [KM61]. To see that km(S) = 1, represent S in the following way. Take a cuspidal cubic, which is a smooth embedding of a 2-sphere away from a single singular point. In a neighbourhood of the singular point we see a cone on the trefoil. Replace a neighbourhood of the singular point with an immersed disc ∆ in D 4 with boundary the trefoil, and two double points that are paired by a framed Whitney disc that intersects ∆ once. Alternatively, [Sto94, p. 1313, Lemma], which uses [FK78], computes t(S) as (σ(CP 2 ) − S · S)/8 = (1 − 9)/8 ≡ 1 mod 2. 1 By Proposition 1.7, for every k ∈ N S# k T : Σ g −→ CP 2 # k (S 3 × S 1 ) is a b-characteristic generic immersion of a closed surface of genus k with nontrivial t, and therefore km(S# k T ) = 0. In particular S# k T is not regularly homotopic to an embedding. Note these examples are smooth, but have no algebraically dual sphere. We could replace (CP 2 , S) with (CP 2 # 8 CP 2 , S ) where S is a generic immersion representing the class (3, 1, . . . , 1) ∈ Z 9 ∼ = H 2 (CP 2 # 8 CP 2 ), to obtain an example with an algebraically dual sphere and km(S ) = (−7 − 1)/8 ≡ 1 mod 2.
Remark 8.6. Let M denote the infinite connected sum CP 2 # ∞ (S 3 × S 1 ). The proof of the Corollary 1.8, along with the formula from Proposition 1.7, shows that for every g there exists a smooth generic immersion F : Σ g M with t(F ) = 0 and therefore km(F ) = 0. The following proposition shows that if there is such a compact 4-manifold M and such an F then the 4-manifolds must have nonabelian fundamental group.
Proposition 8.7. Let M be a compact 4-manifold such that π 1 (M ) is abelian with n generators. Let F : Σ M be a b-characteristic generic immersion where Σ is a closed, connected surface. Then the Euler characteristic satisfies χ(Σ) ≥ −2n.
Proof. Suppose that χ(Σ) < −2n. Note that Σ can be written as a connected sum of a genus g orientable surface Σ for some g > n with zero, one, or two copies of RP 2 . There exists a surjection Z n π 1 (M ). Then the induced map H 1 (Σ ) → H 1 (M ) ∼ = π 1 (M ) admits a lift H 1 (Σ ) → Z n , which has kernel of rank at least 2g − n > g. So there exist closed curves γ 1 , γ 2 in Σ \D 2 ⊆ Σ that are null-homotopic in M and λ Σ (γ 1 , γ 2 ) ≡ 1 mod 2. It follows that F is not b-characteristic.