Embedding calculus for surfaces

We prove convergence of Goodwillie-Weiss' embedding calculus for spaces of embeddings into a manifold of dimension at most two, so in particular for diffeomorphisms between surfaces. We also relate the Johnson filtration of the mapping class group of a surface to a certain filtration arising from embedding calculus.


Introduction
For smooth manifolds M and N and an embedding e ∂ : ∂M → ∂N , we write Emb ∂ (M, N ) for the space of embeddings that agree with e ∂ on ∂M , equipped with the smooth topology.Embedding calculus à la Goodwillie and Weiss provides a space T ∞ Emb ∂ (M, N ) and a map which approximates the space of embeddings through restrictions to subsets diffeomorphic to a finite collection of open discs and a collar.The space T ∞ Emb ∂ (M, N ) arises as a homotopy limit of a tower of maps whose homotopy fibres have an explicit description in terms of configuration spaces of M and N [Wei99,Wei11], so its homotopy type is sometimes easier to study than that of Emb ∂ (M, N ).The main result in this context is due to Goodwillie, Klein, and Weiss [GW99,GK15] and says that if the difference of the dimension of N and the relative handle dimension of the boundary inclusion ∂M ⊂ M is at least three, then embedding calculus converges in the sense that (1) is a weak homotopy equivalence.If this assumption is not met, little is known about for which choices of M and N embedding calculus converges (but see Remark 1.1 (ii) and (vi) below).
1.1.Convergence in low dimensions.In the first part of this work, we study (1) when the target N has dimension at most two.Our main result shows that embedding calculus always converges under this assumption, even though the assumption on the handle codimension is not satisfied.
Theorem A. For compact manifolds M and N with dim(N ) ≤ 2, the map is a weak homotopy equivalence for any embedding e ∂ : ∂M → ∂N .
Perhaps the most interesting (hence eponymous) instance of Theorem A is when M = N is a surface Σ and e ∂ = id ∂Σ .In this case Theorem A specialises to the following: Corollary B. For a compact surface Σ, possibly with boundary and non-orientable, the map is a weak homotopy equivalence.Remark 1.1.
(i) We prove Theorem A as a special case of a more general result that also treats embeddings spaces of triads (see Theorem 3.1).(ii) Theorem A is special to dimension at most 2: in [KK22], we show that this results fails for N = D 3 and for most high-dimensional compact manifolds N .In the language of that paper, Theorem A implies that the smooth Disc-structure space S Disc ∂ (N ) is contractible if dim(N ) ≤ 2. (iii) The proof of Theorem A does not rely on Goodwillie, Klein, and Weiss' convergence results.(iv) Theorem A is stronger than Corollary B, even if dim(M ) = dim(N ) = 2.It implies that T ∞ Emb ∂ (Σ, Σ ) = ∅ if Σ and Σ are connected compact surfaces that are not diffeomorphic.(v) Composition induces an E 1 -structure on T ∞ Emb ∂ (M, M ) with respect to which the map Emb ∂ (M, M ) → T ∞ Emb ∂ (M, M ) is an E 1 -map.For a compact manifold M , the E 1 -space Emb ∂ (M, M ) = Diff ∂ (M ) is grouplike, but it is not known whether the same holds for T ∞ Emb ∂ (M, M ).Theorem A implies that this is the case if dim(M ) ≤ 2. (vi) Theorem A provide a class of examples for which the map Emb ∂ (M, N ) → T ∞ Emb(M, N ) is a weak equivalence in handle codimension less than three.A few examples of this form were known before; see [KK20, Theorem C, Section 6.2.4].In contrast, there are some cases for which it is known that embedding calculus does not converge, such as for M = D 1 and N = D 3 by an argument due to Goodwillie.

Embedding calculus and the Johnson filtration. The Johnson filtration
of the mapping class group π 0 Diff ∂ (Σ) of an orientable surface Σ of genus g with one boundary component is the filtration by the kernels of the action of π 0 Diff ∂ (Σ) on the quotients of the fundamental group π 1 (Σ, * ) based at the point in the boundary, by the constituents of its lower central series.By work of Moriyama [Mor07], this filtration can be recovered from the action of π 0 Diff ∂ (Σ) on the compactly supported cohomology of the configuration spaces of the punctured surface Σ\{ * }.
It is reasonable to expect a relationship between the Johnson filtration and embedding calculus, as the latter may be viewed as the study of embeddings via their induced maps between the homotopy types of configuration spaces of thickened points in source and target.
The second part of this work serves to establish one such a relationship: we introduce a filtration arising from the cardinality filtration of embedding calculus in HZ-modules applied to the space of self-embeddings fixed on an interval in the boundary (see Section 4 for precise definitions), and we use [Mor07] to show that this filtration contained in the Johnson filtration:

Generalities on spaces of embeddings and embedding calculus
We begin by fixing some conventions on spaces of embeddings, followed by recalling various known properties of embedding calculus and complementing them with some new properties such as a lemma for lifting embeddings along covering spaces in the context of embedding calculus.
2.1.Spaces of embeddings and maps.All our manifolds will be smooth and may be noncompact, disconnected, or nonorientable.A manifold triad is a manifold M together with a decomposition of its boundary ∂M = ∂ 0 M ∪ ∂ 1 M into two codimension zero submanifolds that intersect at a set ∂(∂ 0 M ) = ∂(∂ 1 M ) of corners.Any of these sets may be empty or disconnected.If this decomposition is not specified, we implicitly take ∂ 0 M = ∂M and ∂ 1 M = ∅.
When studying embeddings between manifolds triads M and N , we always fix a boundary condition, i.e. an embedding e ∂0 : ∂ 0 M → ∂ 0 N , and only consider embeddings e : M → N that restrict to e ∂0 on ∂ 0 M and have near ∂ 0 M the form e ∂0 × id [0,1) : ∂ 0 M × [0, 1) → ∂ 0 N × [0, 1) with respect to collars of ∂ 0 M and ∂ 0 N .We denote the space of such embeddings in the weak C ∞ -topology by Emb ∂0 (M, N ).We replace the subscript ∂ 0 by ∂ to indicate if ∂ 0 M = ∂M , and drop the subscript if we want to emphasise if ∂ 0 M = ∅ holds.As a final piece of notation, given manifold triads M and L, we consider M L as manifold triad via ∂ 0 (M L) = ∂ 0 M ∂ 0 L.
Similarly, we also consider the space of bundle maps Bun ∂0 (T M, T N ).By this we mean the space of fibrewise injective linear maps T M → T N that restrict to the derivative d(e ∂0 ) on T ∂ 0 M , in the compact-open topology.Taking derivatives induces a map Emb ∂0 (M, N ) → Bun ∂0 (T M, T N ) which we may postcompose with forgetful map Bun ∂0 (T M, T N ) → Map ∂0 (M, N ) to the space of continuous maps extending e ∂0 , equipped with the compact-open topology.

Manifold calculus.
Given manifold triads M and N and a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N as above, Goodwillie-Weiss' embedding calculus [Wei99,GW99] gives a space T ∞ Emb ∂0 (M, N ) (or rather, a homotopy type) together with a map (3) Embedding calculus converges if the map (3) is a weak homotopy equivalence (shortened to weak equivalence throughout this work).This fits into the more general context of manifold calculus, and we shall need this generalisation at several places.

2.2.1.
Manifold calculus in terms of presheaves.Among the various models for the map (3) and manifold calculus in general, that of Boavida de Brito and Weiss in terms of presheaves [BdBW13] is most convenient for our purposes.We refer to Section 8 of their work for a proof of the equivalence between this model and the classical model of [Wei99].
To recall their model (in a slightly more general setting, see Remark 2.5), we fix a (d−1)-manifold K possibly with boundary, thought of ∂ 0 M for manifold triads M .We write Disc K for the topologically enriched category whose objects are smooth d-dimensional manifold triads that are diffeomorphic (as , and whose morphisms are given by spaces of embeddings of triads as described in Section 2.1.If K is clear from the context, we abbreviate Disc K by Disc ∂0 . We write PSh(Disc ∂0 ) for the topologically enriched category of space-valued enriched presheaves on Disc ∂0 , and we consider it as a category with weak equivalences by declaring a morphism of presheaves to be a weak equivalence if it is a weak equivalence on all its values.Localising at these weak equivalences (for instance as described in [DK80]) gives rise to a topologically enriched category PSh(Disc ∂0 ) loc together with an enriched functor Denoting by Man ∂0 the topologically enriched category with objects all manifold triads M with an identification ∂ 0 M ∼ = K and morphism spaces the spaces of embeddings of triads, a presheaf If F is the restriction of a presheaf F ∈ PSh(Man ∂0 ), then we have a composition of maps of presheaves on Man ∂0 where the first map is given by the enriched Yoneda lemma and the second is induced by the restriction along Disc ∂0 ⊂ Man ∂0 and the functor (4).Note that this is a weak equivalence whenever M ∈ Disc ∂0 , that is, manifold calculus converges on manifolds diffeomorphic to the disjoint union of a collar on ∂ 0 M and a finite number of open discs.
Example 2.1 (Embedding calculus).For triads M and N and a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N , we have a presheaf Emb ∂0 (−, N ) of embeddings of triads extending e ∂0 .Choosing K = ∂ 0 M , the map (8) gives rise to a model for the embedding calculus map (3), Remark 2.2.There are several alternative points of view on the maps (5) and ( 6), for instance in terms of modules over variants of the little discs operad (see [BdBW13,Section 6] or [Tur13]).

A smaller model.
In some situations, it is convenient to replace Disc ∂0 by a smaller equivalent category.There is a chain of enriched functors The right arrow is the inclusion of the full subcategory Disc sk ∂0 ⊂ Disc ∂0 on the objects has the same objects as Disc sk ∂0 and space of morphisms pairs (s, e) of a parameter s ∈ (0, 1] and an embedding of triads ) is multiplication by s.Composition is given by composing embeddings and multiplying parameters, and the functor to Disc sk ∂0 forgets the parameters.Both functors in (7) are Dwyer-Kan equivalences, the first by a variant of the proof of the contractibility of the space of collars and the second by definition, so we may equivalently define T ∞ F (−) using any of the three categories (7).

Two properties of manifold calculus. The following two properties of the functor
together with the natural transformation id PSh( It is often convenient to use a stronger version of descent, namely with respect to complete Weiss ∞-covers U, which are Weiss ∞-covers that contain a Weiss ∞-cover of any finite intersection of elements in U. Regarding U as a poset ordered by inclusion, the map induced by restriction is a weak equivalence by [KK20, Lemma 6.7]. Remark 2.3.At several points in the remainder of this work, we will construct maps between spaces of the form T ∞ Emb ∂0 (M, N ) by using the descent property from Section 2.2.3 (b).Strictly speaking, these will only be weak maps i.e. zig-zags of maps whose wrong-way maps are weak equivalences.This will be good enough for all purposes.More formally, a weak map X → Y gives an actual morphism from X and Y in the localisation of the category of spaces at the weak equivalences, and all our statements involving weak maps can be viewed as taking place in this localisation.In particular, when we say that a square involving weak maps commutes up to canonical homotopy then we mean that the square can be enhanced in a preferred way to a homotopy commutative square in this localisation.
2.3.Properties of embedding calculus.We explain various features of embedding calculus which illustrate that T ∞ Emb ∂0 (M, N ) has formally similar properties to Emb ∂0 (M, N ) even in situations where embedding calculus need not converge.(a) Postcomposition with embeddings: Given triads M , N , and K, with boundary conditions that is associative in the evident sense and compatible with the composition maps for embeddings spaces, both up to higher coherent homotopy.
In the model of Section 2.2.1, these maps are given by applying the map induced by postcomposition in the second factor, followed by composition in PSh(Disc ∂0M ) loc .Note that the codomain of (9) does in general not agree with T ∞ Emb ∂0 (N, K).(b) Naturality and isotopy invariance: In the situation of (a), if we assume dim(M ) = dim(N ), then there are composition maps that are associative in the evident sense and compatible with (9) and the composition for embeddings, up to higher coherent homotopy.Combining this with (a), we see that like spaces of embeddings, T ∞ Emb ∂0 (−, −) is isotopy-invariant in source and target: if M ⊂ M is a sub-triad with ∂ 0 M ⊂ ∂ 0 M such that there is an embedding of triads M → M which is inverse to the inclusion up to isotopy of triads, then the maps induced by restriction and inclusion are weak equivalences.Here L is any other triad with a boundary condition e ∂0 : ∂ 0 L → ∂ 0 M .In the model described in Section 2.2.1, the composition map (10) can implemented as follows: the codimension 0 embedding e ∂0M : ∂ 0 M → ∂ 0 N induces enriched functor (e ∂0M ) * : Disc and (e ∂0M ) * : PSh(Disc On morphisms, (e ∂0M ) * keeps the parameter s fixed and sends an embedding e to the embedding given by id ∂0N ×(s•(−)) on ∂ 0 M ×[0, 1) and by (e ∂0M ×[0, 1) id n×R d )•e| n×R d on n × R d .The functor (e ∂0M ) * is given by precomposition with (e ∂0M ) * .The restriction maps are weak equivalences by the contractibility of spaces of collars, and similarly for Emb ∂0 (−, K), so we have weak equivalences in PSh(Disc Using the model ) , the composition (10) is given by applying (e ∂0M ) * to the second factor, composition in the category PSh(Disc which is compatible with composition maps from (10) up to higher coherent homotopy.This follows from the discussion in Section 2.2.3(b) by observing that the target in the natural transformation Emb ∂0 (−, N ) → Bun ∂0 (−, T N ) is a homotopy J 1 -sheaf, so the map Bun ∂0 (−, T N ) → T ∞ Bun ∂0 (−, T N ) is a weak equivalence of presheaves.(e) Extension by the identity: Suppose that we have another triad Q with an identification of ∂ 0 Q with a codimension zero submanifold of ∂ 0 M .Then we can form, up to smoothing corners, the triad strictly speaking this requires the addition of collars to the definitions to guarantee the glued map is smooth but we forego the addition of this contractible space of data), which can be shown to fit into a diagram commutative up to preferred homotopy.The existence of the dashed map in (12) is proved by is a weak equivalence.Then, fixing a triad embedding e : P → N disjoint from ∂N \e ∂0M (∂ 0 P ), there is a map of fibration sequences whose right square results from (10) and whose left square is an instance of the diagram (12).The homotopy fibres are taken over the embedding e and its image in T ∞ Emb ∂0 (P, N ), and ) with boundary condition induced by e and e ∂0M .For the upper row, this is a form of the usual parametrised isotopy extension theorem.For the lower row, this is a mild generalisation of a result of Knudsen and Kupers [KK20, Theorem 6.1, Remarks 6.4 and 6.5].Note that every triad embedding P → N is disjoint from ∂N \e ∂0 (∂ 0 P ) up to isotopy of triad embeddings, so if we would like to draw conclusions about all homotopy fibres of the right horizontal maps, it suffices to restrict to embeddings of this form.We record the following immediate corollary of Properties (c) and (f) which will allow us to restrict to triads with ∂ 0 M = ∅ when proving convergence results.

is a weak equivalence if and only if for all embeddings
Proof.This is an instance of the fact that for a commutative square whose right arrow is a weak equivalence, the map E → E is a weak equivalence if and only if the map hofib(E → B) → hofib(E → B ) is a weak equivalence for all choices of basepoints.We apply this to the commutative square induced by restriction whose right-hand map is a weak equivalence by the convergence on discs (Property (c)).By isotopy extension (Property (f)), the map on homotopy fibres over an embedding e : D d → int(N ) agrees with the second map in the statement, so the claim follows.
We continue with a pair of remarks about these properties: Remark 2.5.Boavida de Brito and Weiss [BdBW13, Section 9] restrict their attention to the case ∂ 0 M = ∂M , but this turns out to be no less general: given a manifold triad M , the manifold triad Remark 2.6.As a consequence of property (d) above, to show that the map of Corollary B on path-components is injective, which is true for all compact surfaces and can be seen as follows.
First, one reduces to the case of connected surfaces.For this, it suffices to show that closed connected surfaces are homotopy equivalent if and only if they are diffeomorphic, which is a consequence of the fact that closed surfaces are classified by orientability and the Euler characteristic, and both of these are preserved by homotopy equivalences relative to the boundary.In the connected case, the claimed injectivity is proved for instance in [Bol09, Theorem 4.6], with the exception of Σ = S 2 and Σ = RP 2 .These two cases can settled using the fibre sequence resulting from restricting to an embedded 2-disc and the fact that the mapping class groups of a disc and a Möbius strip are trivial (see [Sma59, Theorem B], [Eps66, Theorem 3.4]).
In fact, the forgetful map ( 13) is often an isomorphism: for closed orientable surfaces of positive genus this is an instance of the Dehn-Nielsen-Baer theorem [FM12, Theorem 8.1], but there is also an argument for most surfaces with boundary [Bol09, Theorem 1.1 (1)].
The proof of Theorem A relies on some additional properties of embedding calculus which we establish in the ensuing subsections.These properties are not very surprising, but seem to have not appeared in the literature before.

Thickened embeddings.
The first property concerns the behaviour of embedding calculus upon replacing the domain M by a thickening, that is, a vector bundle V over M .
Fix manifold triads M and N and a k-dimensional vector bundle p : Lemma 2.7.There exists a dashed map in (14) such that the diagram commutes up to preferred homotopy and so that the two subsquares are homotopy cartesian.
Proof.Let O be the poset of open subsets U ⊂ M containing a collar on ∂ 0 M .Taking derivatives as well as restricting embeddings and bundle maps induces a commutative diagram of space-valued presheaves on O, where the bottom equivalence results from the discussion in Section 2.3 (d).Since homotopy pullbacks of presheaves are computed objectwise, this is a homotopy-cartesian square of presheaves.We define a new presheaf F (−) on O as the homotopy pullback The result will follow by evaluation at M ∈ O once we provide an identification compatible with the maps to Bun ∂0 (T V, T N ) and from Emb ∂0 (V, N ).It follows from Section 2.2.3(b) and Section 2.3 (c), that it suffices to verify that (a) F satisfies descent for the complete J ∞ -cover U ⊂ O given by those open subsets U ⊂ M equal to a collar on ∂ 0 M and a finite collection of open discs, and (b) the map Emb ∂0 (p −1 (−), N ) → F (−) is a weak equivalence when evaluated on U ∈ U.For (a), we observe that all entries but F (−) in the homotopy pullback diagram (15) defining F (−) satisfy descent with respect to J ∞ -covers, so F (−) does as well.For (b), we observe that on U ∈ U, the right vertical map of ( 15) is a weak equivalence so it suffices to verify that Emb ∂0 (p −1 (U ), N ) → Bun ∂0 (T p −1 (U ), T N ) is a weak equivalence.This is indeed the case because p −1 (U ) is a disjoint union of a collar on ∂ 0 V and a finite collection of open discs.
We derive from Lemma 2.7 two lemmas that will allow us to interpolate between convergence questions for Emb ∂0 (M, N ) and for Emb ∂0 (V, N ).
Lemma 2.8.Let M and N be manifold triads, p : V → M be a vector bundle considered as a triad by ∂ 0 V = p −1 (∂ 0 M ), and e ∂0 : ∂ 0 V → ∂ 0 N be a boundary condition.Then the map Proof.This follows from the upper homotopy cartesian square in (14) provided by Lemma 2.7.Lemma 2.9.Let M be a d-dimensional manifold triad, N be a (d + k)-dimensional manifold triad, and e ∂0 : ∂ 0 M → ∂ 0 N be a boundary condition.Then the map , we choose a metric on T N , let V be the vector bundle over M whose fibre over m ∈ M is the orthogonal complement to β (T m M ) in T β (m) N , and extend the boundary condition e ∂0 : ∂ 0 M → ∂ 0 N to ∂ 0 V by exponentiation.Writing Emb ∂0 (V, N ) β and T ∞ Emb ∂0 (V, N ) β for the unions of the path components mapping to β in (14), Lemma 2.7 yields a homotopy pullback whose left vertical map a weak equivalence by assumption.By construction, β lifts to a bundle map in Bun ∂0 (T V, T N ) under the bottom horizontal map in (14), so it follows from Lemma 2.7 that such that there exists a lift [ α] ∈ π 0 Map ∂0 (M, N ).We shall assume that ∂ 0 M → M is 0-connected, so that this lift is unique.We write for the unions of the path components that map to [α] ∈ π 0 Map ∂0 (M, N ) via the maps in (11).We similarly define subspaces Emb Lemma 2.10.In this situation, there exists a dashed map making the diagram Here the top map is given by sending an embedding ) be the presheaf on Disc ∂0 of those embeddings that remain an embedding after composition with π.This fits in a pullback diagram of presheaves on Disc ∂0M whose vertical maps are given by inclusion.This is homotopy cartesian in the projective model structure on PSh(Disc ∂0 ), since (π ) is a objectwise fibration by the lifting property of covering maps.Evaluating at M and using that T ∞ (−) preserves homotopy limits by Section 2.2.3 (a), we arrive at a commutative cube with front and back faces homotopy cartesian, and bottom diagonal maps weak equivalences as Map ∂0 (−, N ) and Map ∂0 (−, N ) are homotopy J 1 -sheaves (see Section 2.2.3 (b)).By the uniqueness of lifts (this uses that ∂ 0 M → M is 0-connected), the bottom horizontal maps become weak equivalences when we restrict domain and target to the path components of [ α] and [α] respectively.Doing so and using the homotopy pullback property, the top of the cube provides a commutative square with horizontal weak equivalences.The top map is even a homeomorphism, by the uniqueness of lifts.

Using the inclusion of presheaves Emb
whose top composition is given by sending an embedding to its unique lift extending ẽ∂ , so we obtain a map Remark 2.11.If α has no lift, then there is no component of Emb ∂0 (M, N ) mapping to [α] under composition with π.In this case, the above argument shows that there is also no component of T ∞ Emb ∂0 (M, N ) mapping to [α] under the map of Section 2.3 (d) and composition with π.
2.6.Adding a collar to the source.The third property concerns the behaviour of embedding calculus when adding a disjoint collar to the domain.
We fix triads M and N and a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N .Given a compact (dim(M )−1)manifold K, we replace M by the triad M K × [0, 1) with ∂ 0 (M K × [0, 1)) = ∂ 0 M K × {0} and fix an extension e ∂0 : ∂ 0 (M K × [0, 1)) → ∂N of e ∂0 as boundary condition.By contractibility of the space of collars, the restriction map is a weak equivalence.Embedding calculus has this property as well: Lemma 2.12.In this situation, both horizontal maps in the diagram induced by restriction are weak equivalences.
Proof.Let U be the open cover of M K × [0, 1) given by subsets of the form where V ⊂ M is the union of a open subset diffeomorphic to a collar on ∂ 0 M and a finite disjoint union of open discs.This is a complete Weiss ∞-cover of M K × [0, 1), and whose bottom horizontal map is a weak equivalences by Section 2.2.3(b) and whose right vertical map is a weak equivalence by Section 2.3 (c).Similarly we have a square which receives a map from the former square by restriction, so it suffices to show that the maps are weak equivalence.This follows from the contractibility of spaces of collars.
Combined with Lemma 2.4 this yields the following lemma, which is often useful to justify the hypothesis needed to apply isotopy extension for embedding calculus (see Section 2.3 (f)).
Lemma 2.13.Let M and N be d-dimensional triads, and e ∂0 : is a weak equivalences.By Lemma 2.12 we may then forget the collars (R d \int(D d )) on ∂D d from the source, so the result follows.
2.7.Taking disjoint unions.The fourth and final general property of embedding calculus we shall discuss concerns taking disjoint unions in source and target.Its full strength is not needed to prove the main results of this paper-only Corollary 2.15 is-but we believe it to be of independent interest.Let M , M , N , and N be triads with dim(M ) = dim(M ) and dim(N ) = dim(N ).Given boundary conditions e ∂0 : ∂ 0 M → ∂ 0 N and e ∂0 : ∂ 0 M → ∂ 0 N , we consider the boundary condition e ∂0 e ∂0 : ∂ 0 (M M ) → ∂ 0 (N N ).Disjoint union of embeddings induces which is a weak equivalence (in fact, a homeomorphism) if both inclusions ∂ 0 M → M and ∂ 0 M → M are 0-connected.Embedding calculus has this property as well: Lemma 2.14.In this situation, there is a dashed weak equivalence that makes commute up to preferred homotopy.
Proof.As in the proof of Lemma 2.12, the property of embedding calculus we shall use is descent for complete Weiss ∞-covers (see Section 2.2.3 (b)).
We take U M to be the open cover of M given by open subsets U ⊂ M that are diffeomorphic to a collar on ∂ 0 M and a finite disjoint union of open discs, and similarly for U M .We take U M M to be the open cover of M M given by unions of an element of U M and an element of U M .The covers U M , U M , and U M M are all complete Weiss ∞-covers.
We consider U M M as a poset ordered by inclusion and let Emb ∂0 (−, N N ) be the presheaf on U M M that sends U U with U ∈ U M and U ∈ U M to the subspace Emb ∂0 (U U , N N ) ⊂ Emb ∂0 (U U , N N ) which map U into N and U into N .Defining Map ∂0 (−, N N ) similarly, we have a homotopy pullback diagram of presheaves on and this remains a homotopy pullback when taking homotopy limits over U M M .
To identify the term holim we note that there are isomorphisms ) of presheaves, so the Fubini theorem for homotopy limits implies that this homotopy limit is given by holim Combining descent with the fact that embedding calculus converges on U ∈ U M and U ∈ U M by Section 2.3 (c), we conclude that holim The same analysis holds for Map ∂0 (−, M M ) and since this is a homotopy J 1 -sheaf (see Section 2.2.3 (b)), we conclude that holim By the same argument (using descent, convergence on U U ∈ U M M , and that Map ∂0 (−, N N ) is a homotopy J 1 -sheaf), we have weak equivalences so altogether we obtain a homotopy pullback diagram of the form The condition that ∂ 0 M → M and ∂ 0 M → M are 0-connected implies that the bottom map is a weak equivalence, so the top map is a weak equivalence as well.The proof is finished by tracing through the weak equivalences to see that this makes the square in the statement homotopy commute.
Taking M = ∅, which is the only case used in this paper, Lemma 2.14 says: Corollary 2.15.In this situation, in the diagram induced by the inclusion N → N N both horizontal maps are weak equivalences.
is a weak equivalence for all boundary conditions e ∂0 : ∂ 0 M → ∂ 0 N .
Convergence for dim(M ) = 0 holds as a result of Section 2.3 (c).For M = ∅ and dim(M ) > dim(N ), we consider the composition Emb ∂0 (M, N ) → T ∞ Emb ∂0 (M, N ) → Bun ∂0 (T M, T N ) from Section 2.3 (d).If dim(M ) > dim(N ) then the final space in this composition is empty, so the same holds for the first and the second space.This implies convergence.
Step (2.1).Convergence holds for (M, N ) if M is an arc or a strip, and dim(N ) = 2 We divide this step into two substeps: the case where the boundary condition e ∂0 : ∂ 0 M → ∂ 0 N hits two distinct boundary components of N , and the case where the boundary condition hits a single boundary component.The arguments are inspired by Gramain's work [Gra73] and Hatcher's exposition thereof in [Hat14].
Substep: The boundary condition hits two distinct boundary components of N .By Lemma 2.9 and isotopy invariance (see Section 2.3 (b)), it suffices to consider the case M = I × [0, 1] of a strip.To do so, we glue a disc D to the boundary component of N hit by {1}, and consider L = (I × [0, 1] ∪ D).
Smoothing corners and an application of isotopy extension justified by the convergence on discs (see Section 2.3 (c) and (f)) yields a map of fibre sequence with fibres taken over the standard inclusion.Since L is isotopy equivalent to I × [0, 1) relative to I × {0}, the middle vertical map is a weak equivalence by isotopy invariance and the convergence on collars (see Section 2.3 (b) and (c)), so the left vertical map is a weak equivalence as well.
is a weak equivalence by the previous substep.We next investigate the set of path-components.To do so, we will use that the dashed map in is surjective: if an embedding I → P is homotopic to a map I → N , then it is isotopic to an embedding I → N within the homotopy class of I → N .To see this, use the bigon criterion [FM12, Sections 1.2.4,1.2.7] to isotope I → P so that its geometric intersection number with the cocore β of the 1-handle is equal to the algebraic intersection number, which is 0 since it is homotopic to a map I → N .With this in mind, a diagram chase in the factorisation shows that the maps 1 and 2 have the same image.
Let us now fix a class [α] ∈ π 0 Map ∂ (I, N ) in this image.As the map 1 is injective because two embedded arcs are isotopic relative to the endpoints if and only if they are homotopic relative to the α R To do so, we will construct a homotopy-commutative diagram whose horizontal compositions are homotopic to the identity.This will finish the proof, since it exhibits T ∞ Emb ∂ (I, N ) α as a retract of the contractible space T ∞ Emb ∂ (I, P ) α Emb ∂ (I, P ) α .The left square in (18) is obtained by restricting the path-components of the homotopy commutative square (17).The right square arises as the composition of two squares which we explain now.The surface P is an appropriate covering space of P : the construction of P gives a decomposition π 1 (P ) ∼ = π 1 (N ) * Z and P is the cover corresponding to the subgroup π 1 (N ).Explicitly, the cover P can be constructed by cutting P along β to obtain a surface R (see Figure 2) and gluing two copies of the universal cover R of this surface to the two dashed intervals in the boundary resulting from β.Note that R contains a preferred lift α of α and hence so does P .We denote the endpoints of α and α in the various surfaces generically by {0, 1}.The cover P has the property that the map N → P lifts uniquely to P so that {0, 1} is fixed.Moreover, using that the interior of R is diffeomorphic to R 2 , there is an embedding e : P → N fixing {0, 1} such that the composition N → P → N is isotopic to the identity relative to {0, 1}.Viewing P as being glued together by three parts-N , the two half-strips resulting from the cut 1-handle, and the two copies of R attached to these two half-strips-this embedding e : P → N is given by the identity on N ⊂ P apart from a neighbourhood of the two arcs in the boundary to which the half-strips are attached, and by pushing the half-strips and the copies of R attached to them into this neighbourhood.
The right square is induced by post-composition with e, so homotopy commutes in view of Section 2.3 (a).The homotopy commutative left square is obtained by invoking the lifting lemma Lemma 3.3 for the covering map P → P .The top composition in ( 18) is homotopic to the identity by construction, but it remains to justify this for the bottom composition.Justifying this requires the details of the proof of Lemma 2.10, in particular the presheaf Emb π ∂ (−, P ) defined there.Viewing N as a submanifold of P as explained above, the projection π : P → N is isotopic to the identity when restricted to N , so we have a dashed inclusion map of presheaves on Disc ∂I that makes the triangle in the following diagram commute up homotopy .
Moreover, since N ⊂ P → N is isotopic to the identity, the composition Emb ∂ (−, N ) → Emb ∂ (−, N ) along the bottom is homotopic to the identity.Applying T ∞ , evaluating at I, and restricting to path-components, we obtain a homotopy commutative diagram ) α is homotopic to the identity.The composition along the top involving a wrong-way weak equivalence agrees by construction with the bottom composition of (18), so it is homotopic to the identity as claimed (recall Remark 2.3).
By Lemma 2.9, it suffices to prove the claim for the cylinder and the Möbius strip.We will do so for the Möbius strip M = Mo; the argument for the cylinder is analogous.We pick a disc D 2 ⊂ int(Mo).By Lemma 2.4 it suffices to prove that is a weak equivalence for all embeddings e : D 2 → int(Σ).To this end, we pick a subtriad I × [0, 1] ⊂ Mo\int(D 2 ) as in Figure 3 and attempt to show that the vertical restriction maps in the diagram are weak equivalences.Isotopy extension exhibits the homotopy fibre of the left vertical map up to smoothing corner and isotopy equivalence as Emb ∂ (I × [0, 1) I × [0, 1), N \int(e(D 2 ))) which is contractible by the contractibility of spaces of collars.To see that the right vertical map is an equivalence, one combines this observation with descent with respect to a Weiss ∞-cover of open discs and collars on ∂D 2 similarly to the proof of Lemma 2.12.As the bottom horizontal map is a weak equivalence by Step (2.1), the top horizontal map is a weak equivalence as well.
Here we used Lemma 2.12 and isotopy invariance to replace M (I × [0, 1]\int(P )) in the domain with M .The left vertical map is a weak equivalence by the induction hypothesis, so the middle vertical map is a weak equivalence one too.

Step (2.5). Convergence for (M, N
By Corollary 2.15 we may assume that N is connected.We first prove the case where the target N is not diffeomorphic to D 2 .In this case Emb ∂ (D 2 , N ) = ∅, so we need to show T ∞ Emb ∂ (D 2 , N ) = ∅.If this were to fail, then the target of the map ) from Section 2.3 (d) must be nonempty, so N would be a connected surface with a boundary component whose inclusion is null-homotopic.We claim this is impossible unless and we may choose this decomposition so that the homotopy class of the boundary inclusion represents the free product of the homotopy classes of boundary inclusions of those components at which we perform the boundary connected sums.By the classification of connected compact surfaces, it then suffices to observe that all boundary inclusions are non-trivial in the fundamental group of the surfaces Σ 0,2 , Σ 1,1 , and Mo.For Σ 0,2 , each inclusion represents a generator of π 1 (Σ 0,2 ) ∼ = Z, for Σ 1,1 the boundary inclusion represents xyx −1 y −1 ∈ π 1 (Σ 1,1 ) ∼ = x, y , and for the Möbius strip it represents twice a generator in π 1 (Mo) ∼ = Z.
Here H the union of J and H0, and H 0 ⊂ H0 is the component to the right of J.

It remains to show that Emb
2 ) a weak equivalence for which we follow the proof of what is sometimes called the Cerf Lemma [Cer63, Proposition 5].We consider the triad , an application of isotopy extension (see Section 2.3 (f)) justified by Step (2.1) in the case M = J ∼ = I × [0, 1] and Lemma 2.13 gives a map of fibre sequences with connected weakly equivalent bases and homotopy fibres over the standard inclusion .
As H is a closed collar on ∂ 0 H, the middle vertical map is a weak equivalence by isotopy invariance and the convergence on collars (see Section 2.3 (b) and (c)).By Lemma 2.12 we may discard the collar H 0 ∩((−∞, 0]×R) from the source of the left vertical map, and obtain that for ) is a weak equivalence.Invoking Corollary 2.15 to neglect D 2 0 \H 0 from the target and identifying H 0 with a disc upon smoothing corners, we conclude that Emb Step (2.6).Convergence for (M, N ) if M is an orientable surface of genus 0 with n ≥ 1 boundary components and ∂ 0 M = ∂M , and dim(N ) = 2.Note that by gluing (n−1) discs to M we obtain a disc D 2 .We also glue n−1 discs to the corresponding boundary components of N to obtain a triad N with a canonical embedding e : n−1 × D 2 → N .Then isotopy extension and the convergence on discs (see Section 2.3 (f) and (c)) yields fibre sequences The middle vertical map a weak equivalence by Step (2.5), so the left map is one as well.
Step (2.7).Convergence for As a result of Lemma 2.4, we may assume that ∂M = ∅, so M is a boundary connected sum for n ≥ 1 and possibly empty finite sets T 1 and T 2 .Thus we may find an embedding such that M \int(P ) ∼ = Σ 0,n with n = n+2|T 1 |+|T 2 |; see Figure 5 for an example.For any embedding e : M → N extending the boundary condition, an application of isotopy extension (see Section 2.3 (f)), justified by Step (2.3) and Lemma 2.13, gives a map of fibre sequences whose left vertical map a weak equivalence by Step (2.7).Varying the embedding e : M → N , we conclude that the middle vertical map is also a weak equivalence.
Step (2.8).Convergence for Choose a triad embedding and collars on components of ∂ 0 (M \int(P )); see Figure 6 for an example.By Step (2.3) and Lemma 2.13, we may apply isotopy extension as in Step (2.3) to the restriction map Emb ∂0 (M, N ) → Emb ∂0 (P, N ) and its T ∞ -version.From Step (2.7) and Lemma 2.12 we see that the map between fibres is a weak equivalence, from which we conclude the claim.
Step (2.9).Convergence for This is a induction on the number n of components of M .The initial case n = 1 is the previous one, and for the induction step we write M = M M with M connected.The induction hypothesis applied to M together with Lemma 2.13 ensures that we may apply isotopy extension (see Section 2.3 (f)) to the restriction Emb ∂0 (M, N ) → Emb ∂0 (M , N ) and its T ∞ -version from which the claim follows by noting that the map on fibres is a weak equivalence by applyinh the induction hypothesis to M .
Step (3).Convergence for (M, N ) if dim(M ) = dim(N ) = 1 This can be proved similarly to Step (2) but is easier.We outline the argument.
First one proves the case M = D 1 with ∂ 0 (D 1 ) = {−1, 1} by a strategy analogously to Step (2.5): one first uses Corollary 2.15 to reduce to N = D 1 as in the case for surfaces.Then one takes and D 1 0 = D 1 \int(J) and develops a map of fibre sequences similar to Step (2.5).Using Lemma 2.12 and Corollary 2.15, the map on fibres agrees with ), so it is a weak equivalence.
Next one shows the case of a general connected triad M : the case M = S 1 follows directly by an application of isotopy extension (see Section 2.3 (f)) together with the case M = D 1 above, and the cases Finally, the case of a possibly disconnected triad M can be settled as in Step (2.8).).Using this fact, we may conclude from Theorem 3.1 that

Automorphisms of the E
Combining Theorems 1.2, 1.4, and 6.4 of [BdBW18], we have where Aut h (E d )/O(d) is the homotopy fibre of the map BO(d) → BAut h (E d ) resulting from the standard action of O(d) on the little discs operad by derived operad automorphisms, so we deduce: Remark 3.5.Horel [Hor17, Theorem 8.5] proved that Aut h (E 2 )/O(2) * with different methods.His proof crucially uses the spaces of k-arity operations in the operad E 2 are K(π, 1) for all k.This fact can also be used to give an alternative proof of Ω 2 Aut h (E 2 ) * (and thus of Corollary 3.4): the derived mapping space Map h (O, P ) between operads O and P can be computed as a homotopy limit of a diagram whose values are products of spaces of operations in O and P ; this follows e.g. by using the alternative model of operads in terms of dendroidal Segal spaces.Applied to O = P = E 2 , one sees that Map h (E 2 , E 2 ) is a homotopy limit of K(π, 1), so it is contractible after looping twice.

Embedding calculus and the Johnson filtration
This section serves to introduce the filtration (2) of the mapping class group π 0 Diff ∂ (Σ g,1 ), and to prove in Theorem 4.2 that it is contained in the Johnson filtration.
4.1.The cardinality filtration.Returning to the general setting of manifold calculus of Section 2.2.1 with a fixed (d − 1)-manifold K, possibly with boundary, we consider the filtration of the topologically enriched category Disc ∂0 by its full subcategories Disc ∂0,≤k on triads that are diffeomorphic to K × [0, 1) T × R d for finite sets T of bounded cardinality ≤ k.Localising the categories PSh(Disc ∂0,≤k ) at the objectwise weak equivalences as we did for k = ∞ in Section 2.2.1, given a presheaf F ∈ PSh(Disc ∂0 ) we obtain presheaves on Man ∂0 by which are related by maps of presheaves induced by restriction along the inclusions (19).If F is the restriction of a presheaf on Man ∂0 , we can precompose this tower with the canonical map F (M ) → T ∞ F (M ) from (5).
4.1.1.Sheaf-theoretic point of view.The tower (20) can also be seen from the point of view of J k -sheaves as described in Section 2.2.3 (b): by [BdBW13, Theorem 1.2] the functor together with the natural transformation id PSh(Man ∂ 0 ) ⇒ T k is a model for the homotopy J ksheafification.From this point of view the maps (20) are induced by the universal property of homotopy sheafification, using the fact that any J k+1 -sheaf is in particular a J k -sheaf.
In particular, in the case of embedding calculus, i.e. for presheaves F (−) = Emb ∂0 (−, N ) for triads M and N and a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N (see Example 2.1), this implies that there is a factorisation of the map from the discussion in Section 2.3 (d) of the form 4.2.HZ-embedding calculus.Much of the above goes through for presheaves valued in categories other than spaces.We have use for one such generalisation, which we discuss now.It involves the topologically enriched category Sp of spectra and the topologically enriched category HZ-mod of module spectra over the Eilenberg-Mac Lane spectrum HZ, both modelled for example using symmetric spectra in spaces as in [MMSS01].We denote by PSh HZ (Disc ∂0 ) the category of HZ-module spectrum-valued enriched presheaves on Disc ∂0 , and its localisation at the objectwise stable equivalences by PSh HZ (Disc ∂0 ) loc for 1 ≤ k ≤ ∞, giving rise to an extension of the tower (20) to a map of towers which is compatible with the restrictions maps.
Theorem 4.2.For a compact orientable surface Σ with a single boundary component, the subgroup We will deduce Theorem 4.2 from Moriyama's work [Mor07].The key step for this deduction is not special to surfaces and applies to a general d-dimensional manifold triad M , so we will formulate it in this generality.To do so, we fix a presheaf F ∈ PSh(Disc ∂0,≤k ), restrict it to Disc ∂0,≤k−1 and homotopy left Kan extending it back along the inclusion ι k : Disc ∂0,≤k−1 ⊂ Disc ∂0,≤k to obtain a presheaf hLan ι k F with a natural map hLan ι k F → F .Evaluating it at where k := {1, . . ., k} we get a map of Σ

and then taking homotopy quotients by the subgroup
In Proposition 4.5 below, we relate this map for F (−) = Emb ∂0 (−, M ) to a certain "boundary inclusion" of the ordered configuration spaces Emb(k, M ).For this, recall the Fulton-Mac Pherson compactification FM k (M ) of Emb(k, M ) (e.g. from [Sin04]) which comes with a natural inclusion Emb(k, M ) → FM k (M ) that is homotopy equivalence, and a "macroscopic location" map which, when varying M , defines a zig-zag of weak equivalences in the arrow category of Fun(Man ∂0 , S).
Before turning to the proof of Proposition 4.5, we explain how it implies Theorem 4.2.
Proof of Theorem 4.2.An element φ ∈ Diff ∂ (Σ) induces a commutative diagram Abbreviating E Σ := Emb ∂0 (−, Σ), this agrees by Proposition 4.5 with the square up to a zig-zag of weak equivalences of maps of squares.As (−) + ∧ HZ commutes with taking homotopy orbits and left Kan extensions, we conclude that the square (27) depends up to natural weak equivalences only on the endomorphism φ * : Emb ∂0 (−, Σ) + ∧ HZ → Emb ∂0 (−, Σ) + ∧ HZ in PSh HZ (Disc ≤k−1 ) and moreover, as homotopy orbits and homotopy left Kan extensions preserve weak equivalences, only on its image in PSh HZ (Disc ≤k−1 ) loc .Taking vertical cofibres in (27) and homotopy groups, we conclude that the map depends only on the image of φ under the map π 0 Diff ∂ (Σ) → π 0 T HZ k Emb ∂0 (Σ, Σ).In particular, if φ lies in the kernel T J HZ ∂ /2 (k) of this map, then (28) is the identity.Using excision as in [KRW20, Section 5.4.1] one see that the macroscopic location map µ : where the coproduct is taking over ordered collections c 0 , . . ., c p of objects in C and face maps are induced by the composition in C and the functoriality of F and G.We denote the geometric realisation of this semi-simplicial space by omitting the •-subscript.Since geometric realisations of levelwise weak equivalences of semi-simplicial spaces are weak equivalences, the object B(G, C, F ) is weakly homotopy invariant in triples (G, C, F ), in the appropriate sense.
Given an enriched functor ι : C → D and d ∈ D, the space B D(d, ι(−)), C, F agrees, naturally in d, with the homotopy left Kan extension hLan ι F (d) (see e.g.[Rie14, Example 9.2.11]; the cofibrancy conditions are not relevant for us as we consider the bar-construction as a semi-simplicial space and geometric realisations of semi-simplicial spaces preserve weak equivalences).Moreover, if F extends to a functor on D, then there is a natural augmentation map induced by composition and evaluation, which agrees upon geometric realisations with the canonical map hLan ι F (d) → F (d) (or rather, it provides a model thereof).In particular, using the notation introduced above, the left vertical map in the statement of Proposition 4.5 is given by the map induced by (29) and taking homotopy orbits Proof sketch.The strategy is to show that this map is a Serre microfibration and has weakly contractible fibres, which implies the statement by a lemma of Weiss [Wei05, Lemma 2.2].This is a standard argument, so we will explain the idea somewhat informally and avoid spelling out lengthy but routine technical details that are similar to e.g.[KKM21, Section 4].
To verify that the map is a Serre microfibration the task is to show that in a commutative diagram The pair [ e, t] must have the property that the image µ(x) of x under the macroscopic location map is contained in the interior of the deepest level (see Figure 7 for an example) and the equivalence relation is that if a coordinate of t ∈ ∆ p = {(t 0 , . . ., t p ) ∈ [0, 1] p+1 | t 0 + • • • + t p = 1} is 0 then we may forget it and the corresponding level of discs.
In these terms, the right vertical map in the diagram sends (x, [ e, t]) to x.The map D i → B(∂ 0 FM k (−), Disc ∂0,≤k , Emb ∂0 (−, M )) provides for each s ∈ D i a configuration x(s) ∈ ∂ 0 FM k (M ) together with nested embedded discs and weights [ (s), t(s)].The map D i × [0, 1] → ∂ 0 FM k (M ) defines a homotopy x t (s) with t ∈ [0, 1] starting at x(s).If t is small enough then this remains within the deepest level of the discs for x(s, 0), and by compactness of D i we find a single ε > 0 such that this is the case for all (s, t) with t ≤ ε.The dashed lift is then given by sending (s, t) to (x(s, t), [ e(s), t(s)]).
To see that the fibre over x ∈ ∂ 0 FM k (M ) is weakly contractible, i.e. any map from S i to the fibre extends over D i+1 , we observe that given an equivalence class [ e, t] represented by a family of nested embedded discs in M with weights, whose deepest level contains x, we find a smaller collection of ≤ (k − 1) discs around points in the macroscopic image µ(x) of x and contained in the deepest level.By compactness we can find a single such small collection which works for all images of s ∈ S i .Adding this collection and transferring all weight to this collection provides an extension to D i+1 .
Proof of Proposition 4.5.For brevity, we abbreviate We claim that the commutative diagram induced by restriction and inclusion, 3 is induced by inclusion, and 4 is another instance of (29).
As the diagram is natural in M and the leftmost vertical map agrees with the left vertical map in the statement by the discussion around (30), it remains show that 1 -4 are weak equivalences.
The map 1 factors as a composition whose first map is a weak equivalence since left Kan extensions commute with homotopy orbits.To show that the second map in this composition (and also the map 2 ) is a weak equivalence, we argue that the composition (31) consists of weak equivalences upon applying (−) O(d) k to the first two spaces.For the first map this follows by shrinking the collar, for the second map it holds because the derivative Emb(R d k , N ) → k × Fr(N ) is a weak equivalence for any manifold N where Fr(N ) is the frame bundle, and for the third map it is clear.
The map 3 is a weak equivalence because ∂ 0 FM k (−) ⊂ FM k (−) is a weak equivalence when evaluated on objects U of D ≤k−1 .Indeed, if U consists of a collar and ≤ k − 1 discs we have and ∂ 0 FM k (U ) is the union of such terms where one FM ni is replaced by ∂ 0 FM ni .By the pigeonhole principle we have n 0 ≥ 1 or n i ≥ 2 for some 1 ≤ i ≤ , so it suffices to observe that in these cases ∂ 0 FM n0 (∂M × [0, 1)) → FM n0 (M × [0, 1)) or ∂ 0 FM ni (R d ) → FM ni (R d ) are inclusions of deformation retracts, either by modifying configurations such that one has a macroscopic location in ∂ 0 M × {0} ⊂ ∂ 0 M × [0, 1) or such that all have macroscopic location at {0} ∈ R d .Finally, 4 is a weak equivalence by Lemma 4.7.
will be of use: (a) Homotopy limits: The mapping spaces resulting from the localisation (4) can be viewed equivalently as the derived mapping spaces formed with respect to the projective model structure on PSh(Disc ∂0 ) (see [BdBW13, Section 3.1]).That is, the functor (8) models the homotopy right Kan-extension along the inclusion Disc ∂0 ⊂ Man ∂0 [BdBW13, Section 4.2].The functor (8) thus preserves homotopy limits in the projective model structures, which are computed objectwise.(b) J ∞ -covers and descent: If F is the restriction of a presheaf F ∈ PSh(Man ∂0 ) then T ∞ F can be seen alternatively as the homotopy J ∞ -sheafification of F : for 1 ≤ k ≤ ∞ (we will only use the cases k = 1, ∞), an open cover U of a triad M is called a Weiss k-cover if every U ∈ U contains an open collar on ∂ 0 M and every finite subset of cardinality ≤ k of int(M ) is contained in some element of U.An enriched presheaf on Man ∂0 is a homotopy J k -sheaf if it satisfies descent for Weiss k-covers in sense of [BdBW13, Definition 2.2].Note that a homotopy J 1 -sheaf is a homotopy sheaf in the usual sense, and a homotopy J k -sheaf is also a homotopy J k -sheaf for any k ≥ k.By [BdBW13, Theorem 1.2], the functor Fixing a boundary condition e ∂0 : ∂ 0 V → ∂ 0 N , we obtain a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N by restriction along the zero-section M ⊂ V .From (11), we obtain the solid arrows in the diagram this implies that the left vertical map in the homotopy pullback is a weak equivalence.2.5.Lifting along covering maps.The second property is concerned with the problem of lifting embeddings of triads M → N along covering maps π : N → N .To state the result, we consider the cover N as a triad by setting ∂ 0 N := π −1 (∂ 0 N ) and ∂ 1 N := π −1 (∂ 1 N ), and fix a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N as well as a lift ẽ∂0 : ∂ 0 M → ∂ 0 N .We pick a homotopy class [α] ∈ π 0 Map ∂0 (M, N ) is a weak equivalence for any finite set T , if the maps Emb ∂0 (M, N ) → T ∞ Emb ∂0 (M, N ) are weak equivalences for all d-dimensional triads N and all boundary conditions e ∂0 : ∂ 0 M → ∂ 0 N .Proof.By induction over |T | it suffices to prove the case |T | = 1.In that case, it suffices by Lemma 2.4 to prove that for all embeddings e : D d → int(N ) the map

Substep:Figure 1 .
Figure 1.The surface P .The original surface N is the region within the dotted circle.

Figure 3 .
Figure 3.The complement of an open disc in Möbius strip.The orange copy of I × [0, 1] differs up to isotopy equivalence from Mo\int(D 2 ) only in hatched region which is diffeomorphic to I × [0, 1) I × [0, 1).
1 -and E 2 -operad.The above arguments do not rely on the fact that Diff ∂ (D d ) = Emb ∂ (D d , D d ) is contractible for d ≤ 2 (this is folklore for d = 1 and due to Smale for d = 2 [Sma59] . The composition of the left-adjoints Σ ∞ + : Top → Sp and − ∧ HZ : Sp → HZ-mod induces the vertical arrows in the commutative diagram PSh(Disc ∂0 ) PSh(Disc ∂0 ) loc PSh HZ (Disc ∂0 ) PSh HZ (Disc ∂0 ) loc .F ∈ PSh(Disc ∂0 ) we define presheaves T HZ k F (M ) := Map PSh HZ (Disc ∂ 0 ,≤k ) loc (Emb ∂0 (−, M ) + ∧ HZ, F + ∧ HZ) ) whose vertical maps are induced by(22)  and whose horizontal maps are induced by restriction along (19).Note that forF (−) = Emb ∂0 (−, M ), composition induces an E 1 -structure on T k F (M ) = T k Emb ∂0 (M, M ) and T HZ k F (M ) = T HZ k Emb ∂0 (M, M ) which upgrades (23) to a diagram of E 1 -spaces.Remark 4.1.In [Wei04],Weiss considers manifold calculus applied to the space-valued presheaf Ω ∞ (Emb ∂0 (−, M ) + ∧ HZ).This agrees with the above HZ-embedding calculus since the adjunctions Σ ∞ + Ω ∞ and − ∧ HZ U , with U : HZmod → Sp the forgetful functor, induce adjunctions on presheaf categories, which in turn induce for F ∈ PSh(Man ∂0 ) and 1 ≤ k ≤ ∞ an identificationMap PSh(Disc ∂ 0 M,≤k ) loc (Emb ∂0 (−, M ), Ω ∞ (F + ∧ HZ))THZ  k F (M ) = Map PSh HZ (Disc ∂ 0 M,≤k ) loc (Emb ∂0 (−, M ) + ∧ HZ, F + ∧ HZ) the union of the subspace A k ⊂ M k where at least one point lies in ∂ 0 M and the fat diagonal∆ k := {(m 1 , . . ., m i ) ∈ M k |m i = m j for some i = j} ⊂ M k .The key step in the proof of Theorem 4.2 is to identify the map (26) for F (−) = Emb ∂0 (−, M ) with the boundary inclusion ∂ 0 FM k (M ) ⊂ FM k (M ) in the following sense: Proposition 4.5.There is zig-zag of compatible weak equivalences But the subgroup of mapping classes with this property is exactly J (k), by [Mor07, Theorem A, Proposition 3.3].Remark 4.6.It might be interesting to study the various filtrations of the mapping class group obtained by replacing HZ in the definition of T J HZ ∂ /2 (k) with HR for any ring R, such as Q or F p .As long as R has characteristic 0, the resulting filtration is contained in the Johnson filtration.This follows from the proof for Z we gave above, together with the fact from [Mor07, Proposition 3.3] that H * (Σ k , ∆ k ∪ A k ; Z) is trivial if * = k and free abelian for * = k.4.4.The proof of Proposition 4.5.It will be convenient for us to work with an explicit model for the homotopy left Kan extension as a bar construction, which we recall next.4.4.1.The enriched bar construction.Given enriched space-valued functors F and G on a topologically enriched category C where F is contravariant and G is covariant, the bar construction B • (G, C, F ) is the semi-simplicial space given by [p] −→ c0,...,cp

•
∂0M) loc , and using the weak equivalences of presheaves above.(c)Convergenceon disjoint unions of discs: Embedding calculus converges if the domain M is diffeomorphic (as a triad) to∂ 0 M ×[0, 1) T ×R d for a finite set T , where ∂ 0 ∂ 0 M ×[0, 1) T ×R d = ∂ 0 M × {0}.This follows from the corresponding fact for manifold calculus (see Section 2.2.1).By isotopy invariance, it remains true with T × R d replaced by T 1 × R d T 2 × D d for finite sets T i .(d) Comparison to bundle maps: The derivative map Emb ∂0 (M, N ) → Bun ∂0 (T M, T N ) fits into a natural commutative diagram (up canonical homotopy) of the form and N are of the same dimension and we are further given a boundary condition e ∂0 : ∂ 0 M → ∂ 0 N , we can form N ∪ Q in the same manner.Extending embeddings by the identity gives a map Emb ∂0 HZ ∞ Emb∂ /2 (Σ, Σ) is injective.(ii) If the genus of Σ is at least 3, then the inclusion T J HZ ∂ /2 (1) ⊂ J (1) is strict.Indeed, an element of the mapping class group lies in T J HZ ∂ /2 (1) if and only if induced the identity on the homology of frame bundle Fr(T Σ).By [Tra92, Theorem 2.2, Corollary 2.7], this is the case if and only if it lies in the Chillingworth subgroup of the Torelli subgroup J (1) [Chi72a, Chi72b].Theorem 4.2 and the final part of the previous remark suggest: which implies that the Johnson filtration is exhaustive, i.e. ∩ k J (k) = {id}.By Theorem 4.2, the same holds for {T J HZ ∂ /2 (k)} so in particular the map π 0 Diff ∂ (Σ) → π 0 T