Isotopy of the Dehn twist on K3#K3 after a single stabilization

Kronheimer-Mrowka recently proved that the Dehn twist along a 3-sphere in the neck of $K3\#K3$ is not smoothly isotopic to the identity. This provides a new example of self-diffeomorphisms on 4-manifolds that are isotopic to the identity in the topological category but not smoothly so. (The first such examples were given by Ruberman.) In this paper, we use the Pin(2)-equivariant Bauer-Furuta invariant to show that this Dehn twist is not smoothly isotopic to the identity even after a single stabilization (connected summing with the identity map on $S^{2}\times S^{2}$). This gives the first example of exotic phenomena on simply connected smooth 4-manifolds that do not disappear after a single stabilization.

Geometry & Topology msp Volume 27 (2023) Isotopy of the Dehn twist on K 3 # K 3 after a single stabilization Understanding smooth structures on 4-manifolds remains one of the most difficult topics in low-dimensional topology.In this dimension, many results that hold in the topological category do not hold in the smooth category.Such phenomena are called "exotic phenomena."To motivate our discussion, we list three major instances of exotic phenomena: By the groundbreaking work of Donaldson [16; 18] and Freedman [20] (and many subsequent works), there exist many pairs of simply connected closed smooth 4manifolds that are homeomorphic but not diffeomorphic.
By the combined work of Wall [36], Perron [31], Quinn [32] and Donaldson [16], there exist pairs of embedded 2-spheres in 4-manifolds with simply connected complement that are topologically isotopic to each other, but not smoothly so; see [3; 5] for explicit families of such examples.
Exotic phenomena appear in each of these three problems, which we call the "diffeomorphism existence problem", the "diffeomorphism isotopy problem" and the "surface isotopy problem".A fundamental principle, discovered by Wall [36; 37] in the 1960s, states that these exotic phenomena will eventually disappear after sufficient many stabilizations on the 4-manifolds.(Here stabilization means taking the connected sum with S 2 S 2 .)More precisely: Wall [37] proved that any pair of homotopy equivalent simply connected smooth 4-manifolds are stably diffeomorphic.Namely, they become diffeomorphic after sufficiently many stabilizations.
Gompf [22] and Kreck [25] further proved that any pair of homeomorphic orientable smooth 4-manifolds (not necessarily simply connected) are stable diffeomorphic.They also proved that nonorientable pairs can be made stably diffeomorphic by first doing a twisted stabilization (ie connected summing a twisted bundle S 2 z S 2 ).In fact, for any G with H 1 .GI Z=2/ ¤ 0, Kreck [24] constructed examples of homeomorphic nonorientable smooth 4-manifold pairs with fundamental group G which are not stably diffeomorphic.(Different constructions of such examples were given by Cappell and Shaneson [13] for G D Z=2 and Akbulut [2] for G D Z.) This implies that a twisted stabilization is indeed necessary in the nonorientable case.
By combining the results of Kreck [23] and Quinn [32], we know that homotopic diffeomorphisms of any simply connected smooth 4-manifold are smoothly isotopic after sufficient many stabilizations.Here stabilization means first isotoping the diffeomorphisms so that they all pointwise fix a small ball B, and then taking the connected sum with the identity map on S 2 S 2 along B.
The work of Wall [36], Perron [31] and Quinn [32] shows that any two homologous closed surfaces of the same genus embedded in a 4-manifold with simply connected complement become smoothly isotopic after sufficiently many external stabilizations.
Here external means that the connected sums with S 2 S 2 are taken away from the surfaces.

These results motivate the following natural question:
Geometry & Topology, Volume 27 (2023) Question 1.1 How many stabilizations are necessary in each of these three problems?
There has been speculation that one stabilization is actually enough in all three problems.This is based on several known results: It is shown by Baykur and Sunukjian [12] that exotic pairs of nonspin 4-manifolds produced by "standard methods" (logarithmic transforms, knot surgeries, and rational blow-downs) all become diffeomorphic after a single stabilization.
In the large families of examples (of embedded surfaces and self-diffeomorphisms) established in Akbulut [3] and Auckly, Kim, Melvin and Ruberman [5], exactly one stabilization is needed.
Auckly, Kim, Ruberman, Melvin and Schwartz [6] proved that any two homologous surfaces F 1 and F 2 of the same genus embedded in a smooth 4-manifold X with simply connected complements are smoothly isotopic after a single stabilization if they are not characteristic (ie OEF i is not dual to the Stiefel-Whitney class w 2 .X /).This shows that in the noncharacteristic case, one stabilization is indeed enough in the surface isotopy problem.(When the surfaces are characteristic, they proved a similar result involving a single twisted stabilization.) We prove the following theorem.
Theorem 1.2 (main theorem) Let ı be the Dehn twist along a separating 3-sphere in the neck of the connected sum K3 # K3.Then ı is not smoothly isotopic to the identity map even after a single stabilization.
To the author's knowledge, Theorem 1.2 provides the first example that exotic phenomena on simply connected smooth 4-manifolds do not disappear after a single stabilization with respect to S 2 S 2 .In particular, it implies that one stabilization is in general not enough in the diffeomorphism isotopy problem.
Note that Kronheimer and Mrowka [26] proved that ı itself is not smoothly isotopic to the identity, using the nonequivariant Bauer-Furuta invariant for spin families.Our result is based on the Kronheimer-Mrowka theorem and makes use of the Pin.2/equivariant version of the Bauer-Furuta invariant.This invariant was defined in Bauer and Furuta [11] (for a single manifold) and in Szymik [35] and Xu [38] (for families).It has been extensively studied in many papers, including Baraglia [7] and Baraglia and Konno [9], and it is the central tool in Furuta's proof of the 10  8 -theorem [21].The idea of using gauge-theoretic invariants for families to study the isotopy problem first appears in Ruberman [33].The idea of using the Pin.2/-equivariantBauer-Furuta invariant to further study Dehn twists on 4-manifolds was suggested by Kronheimer and Mrowka in [26].
We outline the proof of Theorem 1.2:By taking the mapping torus of ı, we form a smooth bundle N with fiber K3 # K3 and base S 1 .Then it suffices to show that the bundle z N , formed by fiberwise connected sum between N and .S 2 S 2 / S 1 , is not a product bundle.This is proved by showing that the Pin.This gives information on BF Pin.2/ .N / and its S 1 -reduction We can explicitly compute the homotopy group fS RC2H ; S 6 z R g S 1 as Z ˚Z=2.Based on this computation, information from (1) and the fact that BF S 1 .N / gives a vanishing family Seiberg-Witten invariant, we can prove that BF S 1 .N / D 0. This further implies that the nonequivariant Bauer-Furuta invariant BF feg .N / vanishes, which contradicts Kronheimer and Mrowka's result that BF feg .N / equals the nonzero element Á 3 2 3 .
Note that e z R becomes trivial when reducing to the subgroup S 1 Pin.2/.As a consequence, the S 1 -equivariant Bauer-Furuta invariant vanishes after a single stabilization (just like the classical Seiberg-Witten invariants and Donaldson's polynomial invariants).This explains why the Pin.2/-equivariance is essential in our proof.
We end this introductory section by remarking that it is still open whether one stabilization is enough to make any pairs of simply connected homeomorphic 4-manifolds diffeomorphic.(See Akbulut, Mrowka and Ruan [4], Donaldson [17] and Fintushel and Stern [19] for a possible approach using the 2-torsion instanton invariants.)It's also unknown whether two homotopic characteristic surfaces with simply connected complements become smoothly isotopic after a single stabilization.The proof of Theorem 1.2 suggests that the Bauer-Furuta invariant could be useful in attacking these problems.As a first step, one needs to establish new examples of spin 4-manifolds with sufficiently interesting higher-dimensional Pin.2/-equivariant Bauer-Furuta invariants.Note that in a recent paper by the author and Mukherjee [29], we use Theorem 1.2 to establish the first pair of orientable exotic surfaces (in a punctured K3 surface) which are not smoothly isotopic even after one stabilization.
The paper is organized as follows: In Section 2, we give a brief review of some basic Pin.2/-equivariant stable homotopy theory and recall the definition of the equivariant Bauer-Furuta invariant.We also use this section to set up notation and to adapt some standard results to our setting.The actual proof of Theorem 1.2 is given in Section 3. Experts may directly skip to Section 3 and occasionally refer back to Section 2 for notation and results.

Acknowledgements
The author is partially supported by NSF grant DMS-1949209.The author would like to thank Tye Lidman and Danny Ruberman for very enlightening conversations, Mark Powell for pointing out Kreck's work [24], and Selman Akbulut for explaining his work in [2; 3].

Pin(2)-equivariant homotopy theory
In this section, we collect some standard results (mostly from [1; 28; 30; 34]) on G-equivariant stable homotopy theory in the case Instead of stating the most general form of these results, we will only focus on the special cases that are actually needed in our argument.We refer to [1; 34] for an introduction to equivariant stable homotopy theory (in the case of finite groups) and to [28; 30] for a more general treatment.
Since all objects we study here are finite G-CW complexes, for simplicity, we will work with the G-equivariant Spanier-Whitehead category [1] (instead of the homotopy category of G-spectra).Of course, there are a lot of drawbacks (eg one cannot always take limits/colimits), but it is enough for our purpose.

Basic facts and definitions
Let U be a countably infinite-dimensional Grepresentation space equipped with a G-invariant inner product, which we call a "universe".We assume that U contains the concrete representation Here R is the trivial representation, z R is the 1-dimensional representation on which S 1 acts trivially and j acts as 1, and H is acted upon by G via left multiplication in the quaternions.
To apply the results in [30] directly without checking additional conditions, we further assume that U is "complete".This means that U contains infinitely many copies of all isomorphism classes of irreducible G-representations. 1   We will use H to denote either the group G or its subgroups S 1 or feg.By restricting the G-action on U , we can also use U as a complete H -universe.We use R H to denote the set of all finite-dimensional H -representations contained in U .We will treat R G as a subset of R S 1 and R feg by restricting the G-action.
For any V 2 R H , we use S V to denote the 1-point compactification of V (called the representation sphere) and use S.V / to denote the unit sphere.We set 1 as the basepoint of S V and we use S.V / C to denote the union of S.V / with a disjoint basepoint.
Let X , Y and Z be based finite H -CW complexes; see for example [15, Chapter I] for a definition.We use the notation OEX; Y H to denote the set of homotopy classes of based H -maps from X to Y (ie maps that preserve the basepoint and are equivariant under H ).
Given any V; W 2 R H with V W , let V ?be the orthogonal complement of V in W . Then smashing with the identity map on S V ?provides a map One can check that these maps make the collection into a direct system.We define fX; Y g H as the direct limit of this system.As in the nonequivariant case, the set fX; Y g H is actually an abelian group.A based H -map will be called a stable H -map from X to Y .An element in the group fX; Y g H will be called a stable homotopy class of H -maps.
1 Since all G-CW complexes we consider can have only G, S 1 or feg as their isotropy group, all arguments we make actually will still hold for the incomplete universe , which is more relevant to the geometric setting.
Geometry & Topology, Volume 27 (2023) Fact 2.1 Given any based H -map f W X !Y , we form the mapping cone Cf and let i W Y !Cf be the natural inclusion.Then for any Z, the functor f ; Zg H is a generalized cohomology theory [30, page 157].As a result, there is a long exact sequence Fact 2.2 Suppose the H -action on X is free away from the basepoint.Then there is a natural map from the equivariant homotopy group to the nonequivariant homotopy group of the quotient space.This map is constructed as follows: Since the H -action on X is free away from the basepoint, any OEf 2 fX; Y g H can be represented by an The map f induces a nonequivariant map between the quotient space, Then we define q H .OEf / as OEf =H .One can check that this does not depend on the choice of f and V .
Fact 2.3 [1, Theorem 5.3; 28, Theorem 4.5, page 78] Suppose the H -action on X is free away from the basepoint and the H -action on Y is trivial.Then the map q H is an isomorphism.
For the rest of the section, we assume X and Y are based finite G-CW complexes.The next few facts concern various relations between the G-equivariant homotopy groups and the S 1 -equivariant homotopy groups.
Fact 2.4 [1, Theorem 5.1; 28, Theorem 4.7, page 79] There is a natural isomorphism constructed as follows: Take any OEf 2 fX; Y g S 1 represented by an S 1 -map By enlarging V if necessary, we may assume V 2 R G .Then we consider the G-map for any x 2 S V ^X .We let Ã.OEf / D OEf 0 .This map Ã turns out to be an isomorphism.
Next, we recall the two operations about changing groups, namely the restriction map and the transfer map The restriction map is defined by simply ignoring the j -action.To define the transfer map, we consider the Pontryagin-Thom map that crushes all points outside a normal neighborhood of S. z R/ in S z R .(Here we identify the Thom space of the normal bundle of S. z R/ as S z R ^.S. z R/ C /.) Then the transfer map is defined as the composition (7) To describe the composition of transfer and restriction, we define the conjugation map (8) as follows: Take any element OEf 2 fX; Y g S 1 represented by an S 1 -map f W S V ^X !S V ^Y .By enlarging V if necessary, we may assume V 2 R G .Then c j .OEf / is represented by the composition Note that when the S 1 -action on X is free away from the basepoint, the maps c j and the map q S 1 defined in (3) are compatible.That means (9) q S 1 .cj .˛//D j ı q S 1 .˛/ı j 1 for all ˛2 fX; Y g S 1 : Here j and j 1 are treated as elements in fY =S 1 ; Y =S 1 g feg and fX=S 1 ; X=S 1 g feg , respectively.
Next is a special case of the double coset formula [ (This element is called the Euler class of z R.) Then the kernel of the map equals the image of the transfer map (6).
Proof There is a cofiber sequence . Smashing this sequence with X and applying the functor f ; S z R ^Y g G , we get the exact sequence So we see that the image of p equals the kernel of the map (12).The lemma follows from the definition of Tr G S 1 ; see (7).

The characteristic homomorphism We now define the characteristic homo
following [11], where a, b and c are nonnegative integers with d a C 2. This homomorphism is of interest to us because the (family) Seiberg-Witten invariant can be obtained by applying t on the Bauer-Furuta invariant.Note that although z R is trivial as an S 1 -representation, we still distinguish it with R in order to keep track of the j -action.
To define t, we take the smash product of the cofiber sequence with the sphere S aR and get the cofiber sequence This induces the long exact sequence [14,Section 8.4] states that the stable homotopy class of an S 1 -equivariant stable map from S aR or S .aC1/R to S d z R is determined by its mapping degree on the S 1 -fixed point sets.Since this mapping degree is always 0 for dimension reasons, Therefore, we get an isomorphism Note that the S 1 -action on S .aC1/R^S.bH/ C is free away from the basepoint, with quotient space S .aC1/R^CP 2b 1

C
. By composing 1 with the isomorphism q S 1 given in (3), we get the isomorphism ( 16) Definition 2.7 Suppose d a is an odd number less than or equal to 4b 1.Then we define the characteristic homomorphism by setting t.˛/ as the image of 1 under the induced map on the reduced cohomology Here we use the standard orientations on S d z R , S .aC1/Rand CP 1 2 .da 1/ to identify the homology groups as Z.If either d a is even or d a > 4b 1, we simply define t as the zero map.
To discuss the behavior of t under the conjugation map c j defined in (8) We end this section with the following result, which is essentially the algebraic version of the vanishing result for the Seiberg-Witten invariant of connected sums.
Lemma 2.10 Given any ˛1 2 fS Proof The product ˛1˛2 belongs to the group Therefore, t.˛1˛2/ can be nonzero only if d 1 C d 2 a 1 a 2 is odd.Without loss of generality, we may assume d 1 a 1 is odd and d 2 a 2 is even.Since d i > a i for i D 1; 2, the group fS a i R ; S d i z R g S 1 vanishes.By the long exact sequence (14), we see that ˛i equals the image of some element Here we identify S b i H =S 0 with S R ^S.b i H/ C by treating S R as the one-point compactification of .0;C1/ and sending v 2 H b i nf0g to .jvj;v=jvj/ 2 .0;1/ S.b i H/.
Moreover, checking the explicit construction of the map q S 1 given in Fact 2.2, we see that q S 1 is also natural under the smash product and composition.Therefore, .˛1˛2/D q S 1 ..˛1˛2//D q S 1 .ˇ1ˇ2/ı q S 1 ./; and q S 1 .ˇ1ˇ2/equals the composition Because d 2 a 2 is even, the cohomology z H d 2 .S .a 2 C1/R ^.S.b 2 H/ C /=S 1 // equals 0. So q S 1 .ˇ2/induces the trivial map on the reduced cohomology.This implies that .˛2˛2/induces the trivial map on z H d 1 Cd 2 ./. Hence, t.˛1˛2/ D 0.

The Pin(2)-equivariant Bauer-Furuta invariant for spin families
In this section, we briefly summarize the definition and some important properties of the Bauer-Furuta invariant for spin families.This invariant was originally defined in [11] for a single 4-manifold.The family version was first defined in [35; 38] and later extensively studied in [7; 9].Because we want to construct the Bauer-Furuta invariant as a concrete element in the G-equivariant stable homotopy group of spheres, some care must be taken in the construction.

Spin structures on the circle family of 4-manifolds
Let N be a smooth fiber bundle whose fiber is a closed spin 4-manifold M and whose base is another closed manifold B. For simplicity, we will make the following assumption throughout the paper: Assumption 2.11 The bundle N satisfies: (i) M is simply connected.
(ii) The signature .M / is at most 0.
(iii) Let M x be the fiber over the point x 2 B. Then the action of 1 .B; x/ on H 2 .M x I Z/ (given by the holonomy of the bundle) is trivial.
We equip N with a Riemannian metric and let Fr v .N / be the frame bundle of the vertical tangent bundle of N .This is an SO.
We are mainly interested in the case that B is a circle or a point.By Assumption 2.11, N has a unique spin structure when B is a point and has two spin structures when B is a circle.We give an explicit description of these two spin structures as follows: Let M W P M !Fr.M / be the covering map given by the unique spin structure on M .Then the bundle N is obtained by gluing the two boundary components of M OE0; 1 via a diffeomorphism f W M !M .The diffeomorphism induces a map f W Fr.M / !Fr.M /, which has two lifts f ˙ W P M !P M .These lifts differ from each other by the deck transformation W P M !P M .We use f ˙ to glue the two boundary components of P M I and form two spin structures on N .Definition 2.14 When N D M S 1 , the maps f ˙ are just the identity map and the deck transformation .We call the associated spin structures over N the product spin structure and the twisted spin structure, respectively.Let s be the unique spin structure on M .Then we use Q s to denote the former and use Q s to denote the latter.
For general M , the product family and the twisted family are not isomorphic.For example, Kronheimer and Mrowka [26] is a quaternionic linear operator.We form the operator D over N by putting D.M x / together.Now we consider four Hilbert bundles V C , V , U C and U over B. The fibers of V ˙are suitable Sobolev completions of .S ẋ /, and the fibers of U C and U are completions of 1 .M x / and 2 C .M x / ˚ 0 .M x /=R, respectively.We let G D Pin.2/ act on V ˙by left multiplication in the quaternions, and we let G act on U ˙by setting the S 1 -action to be trivial and setting the j -action as multiplication by 1.To apply the finite-dimensional approximation technique on the map SW, we carefully choose finite-dimensional subspaces of V ˙and U ˙as follows: First, we apply Kuiper's theorem [27] to get canonical trivialization of the bundles

The family
Here L 2 ./ denotes the completion with respect to the L 2 -norm.Choose m; n 0 and let U C U C and V V be the subbundles corresponding to the bundles B H n and B z R m under the isomorphism (18).Let H C 2 be the subbundle of U consisting of all self-dual harmonic 2-forms on M x .We set (Note that .dC ; d / is injective by our assumption that b 1 .M / D 0.) We choose m large enough so that V is fiberwise transverse to D and we set As explained in [11], when m and n are large enough, SW C .W C 1 / \ S.W ;? / D ∅; where S.W ;? / denotes the unit sphere in the orthogonal complement of W in U ˚V .Therefore, by composing SW C with a specific G-equivariant deformation retraction (18) gives canonical trivializations of the bundles V and U C .By Assumption 2.11, 1 .B/ acts trivially on H 2 .M x /.Therefore, as explained in [26], a homology orientation of M determines a canonical trivialization of H C 2 .At this point, we have obtained canonical trivializations of U ˙and V .Using these trivializations, we get the composition map (19) RCnH ; where pj denotes projection to the first factor.
From now on, we specialize to the case that B is a circle or point.Note that V C is a quaternionic bundle of dimension n 1  16 .M / and the group Sp n 1 16 .M / has trivial i for i Ä 2. So the bundle V C has a trivialization (canonical up to homotopy).This trivialization allows us to fix an identification and rewrite the map (19) as a G-map which represents an element in OE sw 2 fS ..M /=16/H ^BC ; S b C .M / z R g G .By checking the concrete construction of sw in [11], one establishes: Fact 2.17 Consider the map S m z R ^BC !S .mCbC .M // z R given by restricting sw to the S 1 -fixed point sets.This map can be explicitly described as the composition Definition 2.18 Suppose B is a point.Then M D N and S ..M /=16/H ^BC D S ..M /=16/H .In this case, we define the G-equivariant Bauer-Furuta invariant as We will neglect the spin structure s in our notation when it is obvious from the context.
Example 2.19 BF G .S 4 / is an element in fS 0 ; S 0 g G represented by a G-map from the S m z RCnH to itself.By the equivariant Hopf theorem [15, Chapter II.4], such a stable homotopy class is determined by its restriction to the S 1 -fixed points.Hence, by Fact 2.17, we see that BF G .S 4 / D 1.
RCnH .Such a map is also determined by its restriction on the S 1 -fixed points.By Fact 2.17 again, we see that BF G .S 2 S 2 / D e z R .Here e z R is the Euler class defined in (11).
When B is a circle, we identify it with the unit sphere S.2R/ in S 2R .Consider the cofiber sequence The map p, which is just the Pontryagin-Thom map for the inclusion S.2R/ ,! S 2R , can be treated as a stable map from S R to B C .This stable map induces the map that sends ˛to ˛ı .idS ..M /=16/H ^p/.
Geometry & Topology, Volume 27 (2023) Definition 2.21 When B D S.2R/ we define the G-equivariant Bauer-Furuta invariant In either case, we define both the S 1 -equivariant and nonequivariant Bauer-Furuta invariants as the restriction of the G-equivariant Bauer-Furuta invariant: feg .BF G .N; s//: In [26], Kronheimer and Mrowka gave an alternative definition of BF feg .N; s/: Take a generic section r of the bundle W that is transverse to the map sw.Then the preimage sw 1 .r/ is a manifold.When B is a point, the canonical trivializations of the bundles W ˙determine a stable framing on sw 1 .r/.When B is S.2R/, we fix a stable framing on B that bounds a framed disk.Then together with the trivializations of W ˙, this determines a stable framing on sw 1 .r/.In [26], the family Bauer-Furuta invariant is defined as the framed cobordism class of sw 1 .r/.
Recall that the framed cobordism classes of smooth n-manifolds are classified by elements in the n th stable homotopy group of spheres.The following lemma states that our definition of BF feg is essentially identical to Kronheimer and Mrowka's definition.

Lemma 2.22
The framed cobordism class of sw 1 .r/ is classified by the nonequivariant Bauer-Furuta invariant BF feg .N; s/.
Proof By Sard's theorem, we can take r to be a constant section that sends the whole B to a generic point r 0 2 S .mCbC .M // z RCnH .Then sw 1 .r/ D sw 1 .r0 / and it is also the preimage of the point f0g r 0 2 S RC.mCb C .M // z RCnH under the composition (22) .id Because r 0 is a regular value of sw and any point in f0g B C is a regular value of p, we see that f0g r 0 is indeed a regular value of the map (22).Recall that an element in the stable group of spheres defines a stably framed manifold by taking the preimage of a regular value and taking the induced framing.The proof is finished by observing that the stable framing on B that bounds a framed disk (the one we used to fix the framing on sw 1 .r/) is exactly the framing induced by the inclusion B ,! S 2R .

Some properties of the Bauer-Furuta invariant
In this subsection, we summarize some important properties of the Bauer-Furuta invariant.We start with a vanishing result.Recall from Definition 2.14 that on the trivial bundle N D M S 1 there are two spin structures: the product spin structure Q s and the twisted spin structure Q s .
Lemma 2.23 The Bauer-Furuta invariants BF G , BF S 1 and BF feg of the product spin structure Q s are all vanishing.
Proof The cofiber sequence (21) induces a long exact sequence where q is induced by the map q W B C ! S 0 that preserves the basepoint and sends B to the other point.By its definition, the map sw for .M S 1 ; Q s/ is just a pullback of the corresponding map for .M; s/ via the map q.So OE sw 2 Image.q/, which implies The invariants BF S 1 and BF feg vanish because BF G vanishes.
Remark It would be interesting to prove a generalization of Proposition 2.24 for BF G .M S 1 ; Q s / and BF S 1 .M S 1 ; Q s /.
Next, we give a connected sum formula for the family Bauer-Furuta invariants.This formula was originally proved by Bauer [10] for a single 4-manifold.
To set up the theorem we let .N i ; s i / for i D 1; 2 be two spin families over B D S.2R/ with fiber M i , both satisfying Assumption 2.11.To form the connected sum, we pick sections i W B ! N i .By Assumption 2.11(i), the section i is unique up to homotopy.We remove small standard 4-balls around these sections to form the family N i D 4 S 1 of 4-manifolds with boundary.Then we can form the fiberwise connected sum by identifying the collars of their boundaries.To fix such an identification, we need to choose a smooth family of orientation reversing isomorphisms Proof Denote by Q ˙the two choices of Q .Then they provide gluing maps which differ from each other by a Dehn twist on @.N 2 D 4 S 1 /.Under any boundary parametrization @.N 2 D 4 S 1 / Š S 3 S 1 , this Dehn twist can be written as Ã.v; x/ D .˛.x/v; x/ for .v;x/ 2 S 3 S 1 ; where ˛W S 1 !SO.4/ is an essential loop.Note that S 3 S 1 , regarded as the product S 3 -bundle over S 1 , has two family spin structures (the product spin structure and the twisted spin structure), which are related to each other by Ã.We see that exactly one of the two maps f ˙sends s 1 j @.N 1 D 4 S 1 / to s 2 j @.N 2 D 4 S 1 / .This finishes the proof.We also note that when Q D Q .s 1 ; s 2 /, the gluing map on the boundary has two lifts to the gluing map on the spin bundle, but they give isomorphic spin structures on the connected sum.
From the discussion above, there is a unique way to take the connected sum of two spin families .N i ; s i /.The resulting spin family .N 1 # Q .s 1 ;s 2 / N 2 ; s 1 # s 2 / will also be written as .N 1 ; s 1 / # .N 2 ; s 2 /.
To talk about the Bauer-Furuta invariant of a connected sum, we also need to specify a rule for homology orientation.Given homology orientations on M 1 and M 2 , we let the homology orientation on M 1 # M 2 be defined by putting the oriented basis for Proof The proof is essentially identical to the single 4-manifold case in [10]; see [26] for a sketch of the proof for the family version (in the nonequivariant setting).A central step is an excision argument that builds a homotopy between the approximated Seiberg-Witten maps sw (20) for the bundle 3 Proof of the main theorem

The key proposition
In this subsection, we prove the homotopy theoretic Proposition 3.2, which will be the key ingredient in the proof of our main theorem.
Recall that the group fS RC2H ; S 6 z R g S 1 admits a conjugation action c j ; see (8).The following lemma computes this group and this action: along the separating S 3 in the neck.We want to show that ı is not smoothly isotopic to the identity map even after a single stabilization.Without loss of generality, we may assume that the stabilization is done in the first copy of X 1 .Then we need to show that the map is not smoothly isotopic to the identity map.As in [26], we will prove this by forming the mapping torus N ı s WD ..X 0 # X 1 # X 1 / OE0; 1/=.x; 0/ .ıs .x/;1/ and showing that it is a nontrivial smooth bundle over S 1 .
By Lemma 2.23, the product spin structure over the trivial bundle has vanishing BF G .So, it suffices to show that both spin families associated to N ı s have nontrivial BF G .
To prove this, we consider the product family .X i S 1 ; Q s i / and the twisted family .X i S 1 ; Q s i /.By the discussion in [26, begining of Section 5], the mapping torus N ı can be formed as the fiberwise connected sum .X 1 S 1 / # '.Q s 1 ;Q s 1 / .X 1 S 1 /: Therefore, the bundle N ı s can formed as the fiberwise connected sum .X 0 S 1 / # '.Q s 0 ;Q s 1 / .X 1 S 1 / # '.Q s 1 ;Q s 1 / .X 1 S 1 / as well as the fiberwise connected sum .X 0 S 1 / # '.Q s 0 ;Q s 1 / .X 1 S 1 / # '.Q s 1 ;Q s 1 / .X 1 S 1 /: The two spin families associated to N ı s are We will show that BF G ..X 0 S 1 ; Q s 0 / # .X 1 S 1 ; Q s 1 / # .X 1 S 1 ; Q s 1 // ¤ 0; and the other family is similar.We use ˛to denote the element BF G ..X Seiberg-Witten equations give a fiber-preserving G-equivariant map SW W U C ˚VC !U ˚V : This Seiberg-Witten map can be written as l C c, where l is the fiberwise Fredholm operator l WD D ˚.d C ; d / and c is a certain 0 th order operator.Furthermore, by the boundedness property of the Seiberg-Witten equations [11, Proposition 3.1], SW extends to a map SW C W .U C ˚VC / 1 !.U ˚V / 1 between the one-point completions .U ˙˚V ˙/1 WD .U ˙˚V ˙/ [ f1g: Geometry & Topology, Volume 27 (2023) 2/-equivariant Bauer-Furuta invariant BF Pin.2/ .z N / is nonvanishing for both spin structures.Note that BF Pin.2/ .z N / equals the product of BF Pin.2/ .N / with the Euler class e z R (a stable homotopy class represented by the inclusion from S 0 D f0; 1g to the 1-dimensional representation sphere S 30, Chapter XVIII, Theorem 4.3].It can be verified directly by unwinding the definitions.
z R g G be the element represented by the inclusion map z R : , we prove: Lemma 2.8 For any ˛2 fS aRCbH ; S d z R g S 1 , Proof By formula (9), .cj .˛//equals the composition of .˛/with the elements which are just .1/ d and a suspension of m, respectively.Corollary 2.9 When d a is odd , t.c j .˛//D .1/ Proof When restricted to CP 1 , the map m is just the antipodal map, and so has degree 1.Using the ring structure on H .CP 2b 1 /, we see that m has degree .1/ .c j .˛//D .1/ d m ı .˛/;where m 2 fCP 2b 1 C ; CP 2b 1 C g feg is the "mirror reflection map" defined as m.OEz 1 ; z 2 ; z 3 ; z 4 ; : : : ; z 2b 1 ; z 2b / D .OE N z 2 ; N z 1 ; N z 4 ; N z 3 ; : : : ; N z 2b ; N z 2b 1 / for z i 2 C: 1 2 .da 1/ on z H d .S .aC1/R^CP 2b 1 C /.The result follows from Lemma 2.8.
4/-bundle over N .Definition 2.12 A spin structure s on N is a double covering map W P !Fr v .N / that restricts to a nontrivial covering map Spin.4/ !SO.4/ on each fiber.Two spin structures .; P / and . 0; P 0 / are called isomorphic if there exists a homeomorphism P !P 0 that covers the identity map on Fr v .N /.
Definition 2.13 The pair .N; s/ is called a spin family.Two spin families .N 1 ; s 1 / and .N 2 ; s 2 / over the same base B are called "isomorphic" if there exists a bundle isomorphism Example 2.15 The product family .K3 S 1 ; Q s/ and the twisted family .K3 S 1 ; Q s / are not isomorphic, as can be proved by the nonequivariant Bauer-Furuta invariant.However, for the special case of S 2 S 2 , these two families are indeed isomorphic:Lemma 2.16 ..S 2 S 2 / S 1 ; Q s/ and ..S 2 S 2 / S 1 ; Q s / are isomorphic.ProofThere is an S 1 -action on S 2 with fixed points f0; 1g.We use W S 1 S 2 !S 2 to denote this action.As x varies from 0 to 2 , the induced map .idS 2 .x;// W T .0;0/.S 2 S 2 / !T .0;0/.S 2 S 2 / gives an essential loop in SO.4/.Using this fact, one can verify that the bundle automorphism f W .S 2 S 2 / S 1 !.S 2 S 2 / S 1 Definition of the Bauer-Furuta invariant As in the case of a single 4manifold, a spin structure s gives rise to two quaternion bundles S ˙over N .Denote by S ẋ the restriction of S ˙to the fiber M x .Then the spin Dirac operator established:Geometry & Topology, Volume 27 (2023) defined by f .y 1 ; y 2 ; x/ D .y 1 ; .x;y 2 /; x/ satisfies f .Q s/ D Q s .2.2.2 denote the resulting bundle over B, with fiber M 1 # M 2 .In general, the result N 1 # Q N 2 will depend on the choice of Q up to homotopy.Because 1 .SO.4// D Z=2, there are essentially two choices.Lemma 2.25 There exists exactly one choice of Q such that the spin structures s 1 and s 2 can be glued together to form a spin structure on N 1 # Q N 2 .We denote this choice by Q .s 1 ; s 2 / and denote the resulting spin structure by s 1 # s 2 .
in front of the oriented basis for H 2 C .M 2 /.The following theorem is a family version of Bauer's connected sum formula [10]: Proposition 2.26 Let .M S 1 ; Q s/ be the product family for some spin 4-manifold .M; s/.Then BF H ..N 1 ; s 1 / # .M S 1 ; Q s// D BF H .N 1 ; s 1 / ^BF H .M; s/ for H D G, S 1 or feg.