aa r X i v : . [ m a t h . A T ] J a n EMBEDDING CALCULUS FOR SURFACES
MANUEL KRANNICH AND ALEXANDER KUPERS
Abstract.
We prove convergence of Goodwillie–Weiss’ embedding calculus for spaces of diffeo-morphisms of surfaces and for spaces of arcs in surfaces. This exhibits the first nontrivial classof examples for which embedding calculus converges in handle codimension less than three andresults in particular in a description of the mapping class group of a closed surface in terms ofderived maps of modules over the framed E -operad. For smooth manifolds M and N and an embedding e ∂ : ∂M ֒ → ∂N , we denote by Emb ∂ ( M, N )the space of embeddings e : M ֒ → N that agree with e ∂ on ∂M in the smooth topology. Embeddingcalculus `a la Goodwillie and Weiss provides a space T ∞ Emb ∂ ( M, N ) and a mapEmb ∂ ( M, N ) −→ T ∞ Emb ∂ ( M, N ) , (1)which can be thought as an approximation to the space of embeddings through restrictions tosubsets diffeomorphic to a finite collection of open discs. The space T ∞ Emb ∂ ( M, N ) arises asa homotopy limit of a tower of fibrations whose homotopy fibres have an explicit description interms of the configuration spaces of M and N [Wei99, Wei11], so its homotopy type is sometimeseasier 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 the target andthe handle dimension of the source is at least three, then embedding calculus converges : the map(1) is a weak homotopy equivalence. If this assumption is not satisfied, little is known about thequality of the approximation (1) provided by embedding calculus.In this work, we study the map (1) for the first class of nontrivial examples in which theassumption on the handle codimension is never met: when the target N is a compact surface. Diffeomorphisms of surfaces.
Our main result shows that embedding calculus converges forcodimension zero embeddings of compact surfaces.
Theorem A.
For compact surfaces Σ and Σ ′ , possibly with boundary and non-orientable, andan embedding e ∂ : ∂ Σ ֒ → ∂ Σ ′ , the map Emb ∂ (Σ , Σ ′ ) −→ T ∞ Emb ∂ (Σ , Σ ′ ) is a weak homotopy equivalence. Specialising this result to Σ = Σ ′ and e ∂ = id ∂ Σ results in the following corollary. Corollary B.
For a compact surface Σ , possibly with boundary and non-orientable, the map Diff ∂ (Σ) −→ T ∞ Emb ∂ (Σ , Σ) is a weak homotopy equivalence.Remark. (i) In dimension 2, Corollary B answers a question of Randal-Williams [Kra19, Question 2.3].(ii) Theorem A and Corollary B are a low-dimensional phenomenon: as part of [KK], we provethat these results fail for most high-dimensional compact manifolds M , for instance for allspin manifolds of dimension d ≥
5. In the language of that paper, Theorem A shows thatthe smooth
Disc -structure space S Disc ∂ (Σ) of a compact surface is contractible. (iii) Theorem A is stronger than Corollary B. For instance, it implies that T ∞ Emb ∂ (Σ , Σ ′ ) = ∅ if Σ and Σ ′ are connected and not diffeomorphic.(iv) Composition induces a canonical A ∞ -structure on T ∞ Emb ∂ ( M, M ) with respect to whichthe map Emb ∂ ( M, M ) → T ∞ Emb ∂ ( M, M ) is A ∞ . For a compact manifold M , the A ∞ -space Emb ∂ ( M, M ) = Diff ∂ ( M ) is grouplike, but it is not known whether the same holdsfor T ∞ Emb ∂ ( M, M ). Corollary B implies that this is the case for compact surfaces.(v) Through the connection between embedding calculus and modules over the framed little d -discs operad (see e.g. [BdBW13, Tur13] and the next section), Theorem A has as consequencethat Ω Aut h ( E ) / O(2) is contractible (see Corollary 3.3).(vi) The analogues of Theorem A and Corollary B remain valid in dimension 0 and 1; see Remark1.4 and Section 4.1.
Relation to the framed little d -discs operad. One of the strengths of embedding calculusstems from the fact that there are several equivalent ways to describe the homotopy type of T ∞ Emb ∂ ( M, N ) and the map (1), some of which are related to the framed little discs operads .To illustrate this in the simplified case where M = N has no boundary, recall that the collec-tion of embedding spaces { Emb( ⊔ k R d , R d ) } k ≥ assembles to an operad E fr d which is well-knownto be weakly equivalent to the operad of framed little d -discs. Precomposition equips the collec-tion E M := { Emb( ⊔ k R d , M ) } k ≥ with the structure of a right-module over E fr d . Localising thecategory of such right-modules at the levelwise weak homotopy equivalences gives rise to derivedmapping spaces Map hE fr d ( − , − ) between such right-modules. Precomposition induces an action ofthe monoid of self-embeddings Emb( M, M ) on E M , so we obtain a mapEmb( M, M ) −→ Map hE fr d ( E M , E M ) . (2)By work of Boavida de Brito–Weiss [BdBW13, Section 6] (see also [Tur13]), the target of thismap is weakly equivalent to T ∞ Emb(
M, M ) if d = dim( M ) and with respect to this equivalence(1) agrees with (2). Specialising to surfaces and taking components, Corollary B thus implies thefollowing operadic description of the mapping class group. Corollary C.
For a closed compact surface Σ , the map (2) induces an isomorphism π Diff(Σ) ∼ = −→ π Map hE fr2 ( E Σ , E Σ ) . Remark. (i) Injectivity of the map in Corollary C also follows from the Dehn–Nielsen–Baer theorem (seeRemark 1.3). The new part of Corollary C is that this map is also surjective.(ii) In Section 1.2.1, we recall a slightly different model for T ∞ Emb ∂ ( M, N ) from [BdBW13] interms of presheaves which combined with Corollary B gives a version of Corollary C thatalso applies if the surface Σ has boundary.
Spaces of arcs in surfaces and strategy of proof.
Our strategy for proving Theorem Ais to imitate for T ∞ Emb(Σ , Σ) Gramain’s argument [Gra73] of the contractibility of the pathcomponents of Diff ∂ (Σ) in positive genus as outlined in Hatcher’s exposition of the Madsen–Weiss theorem [Hat14], as well as Cerf’s proof of Smale’s theorem that Diff ∂ ( D ) is contractible[Cer63]. This relies on an isotopy extension theorem for T ∞ Emb( − , − ) due to Knudsen and thesecond-named author [KK20], other (partly new) properties of embedding calculus (Section 1),and a study of spaces of embedded arcs in a surface in this context (see Section 2). The latterinvolves proving the following convergence result, which might be of independent interest. Theorem D.
For a compact surface Σ and two distinct points in its boundary { , } ⊂ ∂ Σ , Emb ∂ ([0 , , Σ) −→ T ∞ Emb ∂ ([0 , , Σ) is a weak homotopy equivalence. MBEDDING CALCULUS FOR SURFACES 3
Remark.
To our knowledge, Theorems A and D provide the first class of nontrivial examples inwhich embedding calculus is known to converge in handle codimension less than three (however,see [KK20, Theorem C, Section 6.2.4] for a few particular examples). It is worth pointing outthat our results do not rely on the convergence results of Goodwillie, Klein, and Weiss.
Acknowledgements.
The first-named author was supported by the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programme (grantagreement No. 756444).1.
Spaces of embeddings and embedding calculus
We begin by fixing some conventions on spaces of embeddings, recall various known propertiesof embedding calculus, and complement the latter with several new properties such as a lemmafor lifting embeddings along covering spaces in the context of embedding calculus. The latter willbe essential to the proof of Theorem D on spaces of arcs in surfaces.1.1.
Spaces of embeddings and maps.
All our manifolds will be smooth and may be noncom-pact and disconnected. A manifold triad is a manifold M together with a decomposition of itsboundary ∂M = ∂ M ∪ ∂ M into two codimension zero submanifolds that intersect at a corner ∂ ( ∂ M ) = ∂ ( ∂ M ). Any of these sets may be empty or disconnected. If this decomposition isnot specified, we implicitly take ∂ M = ∂M and ∂ M = ∅ .When studying embeddings between manifolds triads M and N , we always fix a boundarycondition , i.e. an embedding e ∂ : ∂ M ֒ → ∂ N , and only consider embeddings e : M ֒ → N thatrestrict to e ∂ on ∂ M and have near ∂ M the form e ∂ × id [0 , : ∂ M × [0 , ֒ → ∂ N × [0 ,
1) withrespect to choices of collars of ∂ M and ∂ N . We denote the space of such embeddings in theweak C ∞ -topology by Emb ∂ ( M, N ). We replace the subscript ∂ by ∂ to indicate if ∂ M = ∂M and drop it if we want to emphasise that ∂ M = ∅ holds. As a final piece of notation, if we havefixed a manifold triad M , and L is a manifold without boundary, we consider M ⊔ L as manifoldtriad via ∂ ( M ⊔ L ) = ∂ M .Similarly, we also consider the space of bundle maps Bun ∂ ( T M, T N ). By this we meanthe space of fibrewise injective linear maps
T M → T N that restrict to the derivative d ( e ∂ )on T ∂ M , in the compact open topology. Taking derivatives induces a map Emb ∂ ( M, N ) → Bun ∂ ( T M, T N ) which we may postcompose with forgetful map Bun ∂ ( T M, T N ) → Map ∂ ( M, N )to the space of continuous maps extending e ∂ in the compact open topology.1.2. Manifold calculus.
Given manifold triads M and N and a boundary condition e ∂ : ∂ M ֒ → ∂ N as described above, Goodwillie–Weiss’ embedding calculus [Wei99, GW99] gives a space T ∞ Emb ∂ ( M, N ) (or rather, a homotopy type) together with a mapEmb ∂ ( M, N ) −→ T ∞ Emb ∂ ( M, N ) . (3)Embedding calculus converges if the map (3) is a weak homotopy equivalence. This fits into themore general context of manifold calculus , and we will need this generalization at several places.1.2.1. Manifold calculus in terms of presheaves.
Among the various models for the map (3) andmanifold calculus in general, that of Boavida de Brito and Weiss in terms of presheaves [BdBW13]is particularly convenient for our purposes. We refer to Section 8 of their work for a proof of theequivalence between this model and the classical model of [Wei99].To recall their model (in a slightly more general setting, see Remark 1.2), we fix a ( d − K , which we think of ∂ M for a manifold triad M . We write Disc K for thetopologically enriched category whose objects are smooth d -dimensional manifold triads that arediffeomorphic (as triads) to K × [0 , ⊔ S × R d for a finite set S with ∂ ( K × [0 , ⊔ S × R d ) = K ×{ } , MANUEL KRANNICH AND ALEXANDER KUPERS and whose morphisms are given by spaces of embeddings of triads as described in Section 1.1. If K is clear from the context, we abbreviate Disc K by Disc ∂ .We write PSh ( Disc ∂ ) for the topologically enriched category of space-valued enriched presheaveson Disc ∂ and consider it as a category with weak equivalences by declaring a morphism ofpresheaves to be a weak equivalence if it is a weak homotopy equivalence on all its values. Local-ising at these weak equivalences (for instance as described in [DK80]) gives rise to a topologicallyenriched category PSh ( Disc ∂ ) loc together with an enriched functor PSh ( Disc ∂ ) −→ PSh ( Disc ∂ ) loc . (4)Denoting by Man ∂ the topologically enriched category with objects all manifold triads M withan identification ∂ M ∼ = K and morphism spaces the spaces of embeddings of triads, a presheaf F ∈ PSh ( Disc ∂ ) induces a new presheaf T ∞ F ∈ PSh ( Man ∂ ) by setting T ∞ F ( M ) := Map PSh ( Disc ∂ ) loc (cid:0) Emb ∂ ( − , M ) , F (cid:1) . If F is the restriction of a presheaf F ∈ PSh ( Man ∂ ), then we have a composition of maps ofpresheaves on Man ∂ F ( M ) ∼ = −→ Map
PSh ( Man ∂ ) (cid:0) Emb ∂ ( − , M ) , F (cid:1) −→ T ∞ F ( M ) (5)where the first map is given by the enriched Yoneda lemma and the second is induced by therestriction along Disc ∂ ⊂ Man ∂ and the functor (4). Note that this is a weak homotopy equiv-alence whenever M ∈ Disc ∂ , that is, manifold calculus converges on manifolds diffeomorphic tothe disjoint union of a collar on ∂ M and a finite number of open balls. Example . For triads M and N and a boundary condition e ∂ : ∂ M ֒ → ∂ N , we have a presheaf Emb ∂ ( − , N ) of embeddings of triads extending e ∂ . Choosing K = ∂ M ,the map (7) gives rise to a model for the embedding calculus map (3),Emb ∂ ( M, N ) −→ Map
PSh ( Disc ∂ ) loc (cid:0) Emb ∂ ( − , M ) , Emb ∂ ( − , N ) (cid:1) = T ∞ Emb ∂ ( M, N ) . A smaller model.
In some situations, it is convenient to replace
Disc ∂ by a smaller equiv-alent category. There is a chain of enriched functors Disc • ∂ −→ Disc sk ∂ −→ Disc ∂ . (6)The right arrow is the inclusion of the full subcategory Disc sk ∂ ⊂ Disc ∂ on the objects ∂ M × [0 , ⊔ n × R d for n = { , . . . , n } with n ≥
0. The category
Disc • ∂ has the same objects as Disc sk ∂ and space of morphisms pairs ( s, e ) of a parameter s ∈ (0 ,
1] and an embedding of triads e : ∂ M × [0 , ⊔ n × R d → ∂ M × [0 , ⊔ m × R d with e | ∂ M × [0 , = id ∂ M × s · ( − ), where s · ( − ) : [0 , → [0 ,
1) is multiplication by s . Compositionis given by composing embeddings and multiplying parameters and the functor to Disc sk ∂ forgetsthe parameters. Both functors in (6) are Dwyer–Kan equivalences, the first by a variant of theproof of the contractibility of the space of collars and the second by definition. As a result, theinduced functors on enriched presheaf-categories are Quillen equivalences (see e.g. [K¨o17] for aproof), so we may equivalently define T ∞ F ( − ) using any of these three categories.1.2.3. Two properties of manifold calculus.
The following two properties of the functor
PSh ( Disc ∂ ) ∋ F T ∞ F ∈ PSh ( Man ∂ ) (7)will be of use: MBEDDING CALCULUS FOR SURFACES 5 (a)
Homotopy limits:
The mapping spaces resulting from the localisation (4) can be viewed equiv-alently as the derived mapping spaces formed with respect to the projective model structureon
PSh ( Disc ∂ ) [BdBW13, Section 3.1]. That is, the functor (7) models the homotopy rightKan-extension along the inclusion Disc ∂ ⊂ Man ∂ [BdBW13, Section 4.2]. As a consequence,the functor (7) preserves homotopy limits in the projective model structures, which are com-puted objectwise.(b) J ∞ -covers and descent: If F is the restriction of a presheaf F ∈ PSh ( Man ∂ ) then T ∞ F canbe seen alternatively as the homotopy J ∞ -sheafification of F : for 1 ≤ k ≤ ∞ (we will onlyuse 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 ∂ M and every finite subset of cardinality ≤ k of int( M ) iscontained in some element of U . An enriched presheaf on Man ∂ is a homotopy J k -sheaf if itsatisfies descent for Weiss k -covers in sense of [BdBW13, Definition 2.2]. Note that homotopy J -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 PSh ( Man ∂ ) ∋ F T ∞ F ∈ PSh ( Man ∂ )together with the natural transformation id PSh ( Man ∂ ) ⇒ T ∞ is a model for the homotopy J ∞ -sheafification. In particular, if F is already a J ∞ -sheaf, then F → T ∞ F is a weak equivalence,so any map F → G in PSh ( Man ∂ ) with G a homotopy J k -sheaf for some 1 ≤ k ≤ ∞ factorscanonically over F → T ∞ F up to weak equivalence.It is often more 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 anyfinite intersection of elements in U . Regarding such a cover U as a poset ordered by inclusion,the map induced by restriction T ∞ F ( M ) −→ holim U ∈U T ∞ F ( U )is a weak homotopy equivalence [KK20, Lemma 6.7].1.3. Properties of embedding calculus.
We explain various features of embedding calculusthat illustrate that T ∞ Emb ∂ ( M, N ) has formally similar properties to Emb ∂ ( M, N ) even whenembedding calculus need not converge.(a)
Postcomposition with embeddings:
Given triads M , N , and K with dim( M ) = dim( N ), andboundary conditions e ∂ M : ∂ M ֒ → ∂ N and e ∂ N : ∂ N ֒ → ∂ K , there is a map T ∞ Emb ∂ ( M, N ) × Emb ∂ ( N, K ) −→ T ∞ Emb ∂ ( M, K )that are associative in the evident sense and compatible with the composition maps for em-beddings spaces, both up to higher coherent homotopy.In the model of Section 1.2.1, these maps are given by applying the mapEmb ∂ ( N, K ) −→ Map
PSh ( Disc ∂ M ) loc (cid:0) Emb ∂ ( − , N ) , Emb ∂ ( − , K ) (cid:1) (8)induced by postcomposition in the second factor, followed by composition in PSh ( Disc ∂ M ) loc .Note that the codomain of (8) does in general not agree with T ∞ Emb ∂ ( N, K ).(b)
Naturality and isotopy invariance:
In the situation of (a), if we assume dim( M ) = dim( N ),then there are composition maps T ∞ Emb ∂ ( M, N ) × T ∞ Emb ∂ ( N, K ) −→ T ∞ Emb ∂ ( M, K ) (9)that are associative in the evident sense and compatible with (8) and the composition forembeddings, up to higher coherent homotopy. Combining this with (a), we see that likespaces of embeddings, T ∞ Emb ∂ ( − , − ) is isotopy-invariant in source and target: if M ⊂ M ′ MANUEL KRANNICH AND ALEXANDER KUPERS is a sub-triad with ∂ M ⊂ ∂ M ′ such that there is an embedding of triads M ′ ֒ → M whichis inverse to the inclusion up to isotopy of triads, then the maps T ∞ Emb ∂ ( M ′ , N ) → T ∞ Emb ∂ ( M, N ) and T ∞ Emb ∂ ( L, M ) → T ∞ Emb ∂ ( L, M ′ )induced by restriction and inclusion are weak homotopy equivalences. Here L is any othertriad with a boundary condition e ∂ : ∂ L ֒ → ∂ M .In the model described in Section 1.2.1, the composition map (9) can implemented asfollows: the codimension 0 embedding e ∂ M : ∂ M ֒ → ∂ N induces enriched functor( e ∂ M ) ∗ : Disc • ∂ M −→ Disc • ∂ N and ( e ∂ M ) ∗ : PSh ( Disc • ∂ N ) → PSh ( Disc • ∂ M )Writing d := dim( M ) = dim( N ), the functor ( e ∂ M ) ∗ sends on objects ∂ M × [0 , ⊔ n × R d to ∂ N × [0 , ⊔ n × R d . On morphisms, ( e ∂ M ) ∗ keeps the parameter s fixed and sends anembedding e to the embedding given by id ∂ N × ( s · ( − )) on ∂ M × [0 ,
1) and by ( e ∂ M × [0 , ⊔ id n × R d ) ◦ e | n × R d on n × R d . The functor ( e ∂ M ) ∗ is given by precomposition with( e ∂ M ) ∗ . The restriction mapsEmb ∂ N ( ∂ N × [0 , ⊔ n × R d , N ) −→ Emb ∂ M ( ∂ M × [0 , ⊔ n × R d , N )are weak homotopy equivalences by the contractibility of spaces of collars, and similarly forEmb ∂ ( − , K ), so we have weak homotopy equivalences in PSh ( Disc • ∂ M )( e ∂ M ) ∗ Emb ∂ N ( − , N ) ≃ −→ Emb ∂ M ( − , N ) and( e ∂ M ) ∗ Emb ∂ N ( − , K ) ≃ −→ Emb ∂ M ( − , K ) . Using the model T ∞ Emb ∂ ( M, N ) ≃ Map
PSh ( Disc • ∂ M ) loc (cid:0) Emb ∂ ( − , M ) , Emb ∂ ( − , N ) (cid:1) , the composition (9) is given by applying ( e ∂ M ) ∗ to the second factor, composition in thecategory PSh ( Disc • ∂ M ) loc , and using the weak equivalences of presheaves above.(c) Convergence on disjoint unions of discs:
Embedding calculus converges if the domain M isdiffeomorphic (as a triad) to ∂ M × [0 , ⊔ S × R d for a finite set S , where ∂ (cid:0) ∂ M × [0 , ⊔ S × R d (cid:1) = ∂ M × { } . This follows from the corresponding fact for manifold calculus (see Section 1.2.1). By isotopyinvariance, it remains true with S × R d replaced by S × R d ⊔ T × D d for T a finite set.(d) Comparison to bundle maps:
The derivative map Emb ∂ ( M, N ) → Bun ∂ ( T M, T N ) fits intoa natural commutative diagram (up canonical homotopy) of the formEmb ∂ ( M, N ) Bun ∂ ( T M, T N ) Map ∂ ( M, N ) T ∞ Emb ∂ ( M, N ) . (10)which is compatible with composition maps from (9) up to higher coherent homotopy. Thisfollows from the discussion in Section 1.2.3 (b) by observing the target in the natural trans-formation Emb ∂ ( − , N ) → Bun ∂ ( − , T N ) is a homotopy J -sheaf.(e) Extension by the identity:
Suppose that we have another triad Q with an identification of ∂ Q with a codimension zero submanifold of ∂ M . Then we can form, up to smoothing corners,the triad M ∪ Q = M ∪ ∂ Q Q with ∂ ( M ∪ Q ) = ( ∂ M \ int( ∂ Q )) ∪ ∂ Q. If M and N are of the same dimension and we are further given a boundary condition e ∂ : ∂ M ֒ → ∂ N , we can form N ∪ Q in the same manner. Extending embeddings by the MBEDDING CALCULUS FOR SURFACES 7 identity gives a map Emb ∂ ( M, N ) → Emb ∂ ( M ∪ Q, N ∪ Q ) (strictly speaking this requiresthe addition of collars to the definitions to guarantee the glued map is smooth but we foregothe addition of this contractible space of data), which can be shown to fit into a commutativediagram up to canonical homotopyEmb ∂ ( M, N ) Emb ∂ ( M ∪ Q, N ∪ Q ) T ∞ Emb ∂ ( M, N ) T ∞ Emb ∂ ( M ∪ Q, N ∪ Q ) . (11)The existence of the dashed map in (11) is proved by noting that T ∞ Emb ∂ ( − ∪ Q, N ∪ Q ) isa homotopy J ∞ -sheaf on Disc ∂ M , see Section 1.2.3 (b).(f) Isotopy extension:
Suppose that the triads M and N are both d -dimensional, ∂ M = ∂M ,and e ∂ : ∂M ֒ → ∂N is a boundary condition. Fix a compact d -dimensional sub-manifoldtriad P ⊂ M (so in particular ∂ P = ∂M ∩ ∂P ) and consider the induced boundary condition e ∂ : ∂M ⊃ ∂ P ֒ → ∂N . Suppose that embedding calculus converges for triad embeddings oftriads of the form P ⊔ S × R d ֒ → N for finite sets S . Then, fixing a triad embedding e : P ֒ → N disjoint from ∂N \ e ∂ ( ∂ P ), there is a map of fibration sequencesEmb ∂ ( M \ int( P ) , N \ int( e ( P ))) Emb ∂ ( M, N ) Emb ∂ ( P, N ) T ∞ Emb ∂ ( M \ int( P ) , N \ int( e ( P ))) T ∞ Emb ∂ ( M, N ) T ∞ Emb ∂ ( P, N ) ≃ whose right square results from (9) and whose left square is an instance of the diagram (11).The homotopy fibres are taken over the embedding e and its image in T ∞ Emb ∂ ( P, N ), and ∂ ( M \ int( P )) := ∂ P ∪ ∂M \ int( ∂ P ) . with boundary condition is induced by e and e ∂ . For the upper row, this is a form of theusual parametrised isotopy extension theorem. For the lower row, this is a result of Knudsenand the second-named author [KK20, Theorem 6.1, Remarks 6.4 and 6.5]. Note that, everytriad embedding P ֒ → N is disjoint from ∂N \ e ∂ ( ∂ P ) up to isotopy of triad embeddings, soif we would like to draw conclusions about all homotopy fibres of the right horizontal maps,it suffices to restrict to embeddings of this form. Remark . Boavida de Brito and Weiss [BdBW13] restrict their attention to the case ∂ M = ∂M (see Section 9 loc. cit.), but this turns out to be no less general: given a manifold triad M , themanifold triad M \ ∂ M with ∂ ( M \ ∂ M ) = int( ∂ M ) = ∂ ( M \ ∂ M ) is isotopy equivalent to M ,so there is a weak homotopy equivalence T ∞ Emb ∂ ( M, N ) ≃ T ∞ Emb ∂ ( M \ ∂ M, N \ ∂ N )by item (b) above. Remark . As a consequence of property (d) above, to show that the map of Corollary C π Diff ∂ (Σ) → π T ∞ Emb ∂ (Σ , Σ) is injective, it suffices to prove that π Diff ∂ (Σ) −→ π hAut ∂ (Σ) (12)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 closedconnected surfaces are homotopy equivalent if and only if they are diffeomorphic, which is aconsequence of the fact that closed surfaces are classified by orientability and the Euler character-istic, and both of these are preserved by homotopy equivalences relative to the boundary. In theconnected case, the claimed injectivity is proved for instance in [Bol09, Theorem 4.6], with the MANUEL KRANNICH AND ALEXANDER KUPERS exception of Σ = S and Σ = R P . These two cases can settled using the fibre sequence resultingfrom restricting to an embedded 2-disc and the fact that the mapping class groups of a disc anda M¨obius strip are trivial (see [Sma59, Theorem B], [Eps66, Theorem 3.4]).In fact, the forgetful map (12) is often an isomorphism: for closed orientable surfaces of positivegenus this is an instance of the Dehn–Nielsen–Baer theorem [FM12, Theorem 8.1], but there isalso an argument for most surfaces with boundary [Bol09, Theorem 1.1 (1)]. Remark . It follows from item (c) above that the mapEmb(
M, N ) −→ T ∞ Emb(
M, N )is a weak homotopy equivalence whenever M is 0-dimensional. In particular, Theorem A and itscorollaries are valid in dimension 0.The proof of Theorem A relies on some additional properties of embedding calculus whichwe establish in the ensuing subsections. Though these properties are not very surprising, to ourknowledge they have not appeared in the literature before.1.4. Thickened embeddings.
Fix manifold triads M and N and consider M × R k as a triadvia ∂ ( M × R k ) = ∂ M × R k . Fixing a boundary condition e ∂ : ∂ ( M × R k ) ֒ → ∂ N , we obtain aboundary condition e ′ ∂ : ∂ M ֒ → ∂ N by restricting along the inclusion M = M × { } ⊂ M × R k .From (10), we obtain the solid arrows in the diagramEmb ∂ ( M × R k , N ) Emb ∂ ( M, N ) T ∞ Emb ∂ ( M × R k , N ) T ∞ Emb ∂ ( M, N )Bun ∂ ( T ( M × R k ) , T N ) Bun ∂ ( T M, T N ) . (13) Lemma 1.5.
There exists a dashed map in (13) such that the diagram commutes up to preferredhomotopy and so that the two subsquares are homotopy cartesian.Proof.
Taking derivatives induces a homotopy cartesian squareEmb ∂ ( − × R k , N ) Emb ∂ ( − , N )Bun ∂ ( T ( − × R k ) , T N ) Bun ∂ ( T ( − ) , T N ) , of enriched presheaves on Man ∂ , so the resulting square T ∞ Emb ∂ ( − × R k , N ) T ∞ Emb ∂ ( − , N ) T ∞ Bun ∂ ( T ( − × R k ) , T N ) T ∞ Bun ∂ ( T ( − ) , T N ) , obtained by applying T ∞ is homotopy cartesian as well, by Section 1.2.3 (a). Observing thatBun ∂ ( T ( − × R m ) , T N ) is a homotopy J -sheaf, we conclude that the map of presheavesBun ∂ ( T ( − × R m ) , T N ) ≃ −→ T ∞ Bun ∂ ( T ( − × R m ) , T N )is a weak equivalence for all m ≥
0, by Section 1.2.3 (b).
MBEDDING CALCULUS FOR SURFACES 9
Applying this for m = 0 and m = k , we obtain a diagram as in (13) with homotopy cartesianbottom square, but with T ∞ Emb ∂ ( M × R k , N ) replaced by T ∞ Emb ∂ ( − , N )( M × R k ). As theouter square is homotopy cartesian, to finish the proof, it suffices to provide an equivalence T ∞ Emb ∂ ( − × R k , N )( M ) ≃ T ∞ Emb ∂ ( − , N )( M × R k )which is compatible with the maps from the embedding spaces and the maps to the spaces ofbundle maps. To do so, we consider the complete Weiss ∞ -cover U of M given by open subsets U ⊂ M that are equal to a collar on ∂ M and a finite disjoint union of open discs. Then U ′ = { U × R k | U ∈ U} is a complete Weiss k -cover of M × R k and we have a diagram T ∞ Emb ∂ ( − × R k , N )( M ) T ∞ Emb ∂ ( − , N )( M × R k )holim U ∈U T ∞ Emb ∂ ( − × R k , N )( U ) holim U ∈U T ∞ Emb ∂ ( − , N )( U × R k )holim U ∈U Emb ∂ ( U × R k , N ) holim U ∈U Emb ∂ ( U × R k , N ) ≃ ≃≃ ≃ using descent for J ∞ -covers and convergence on discs (see Section 1.2.3 (b) and Section 1.3 (c)).This provides a dashed equivalence as desired. (cid:3) Lifting along covering maps.
The second property is concerned with the problem of liftingembeddings of triads
M ֒ → N along covering maps π : e N → N . To state the result, we consider e N as a triad by setting ∂ e N := π − ( ∂ N ) and ∂ e N := π − ( ∂ N )and fix a boundary condition e ∂ : ∂ M ֒ → ∂ N as well as a lift ˜ e ∂ : ∂ M ֒ → ∂ e N . We moreoverpick a homotopy class [ α ] ∈ π Map ∂ ( M, N ) such that there exists a lift [ ˜ α ] ∈ π Map ∂ ( M, e N ),and moreover that ∂ M → M is 0-connected, so that this lift is unique. We writeEmb ∂ ( M, N ) α ⊂ Emb ∂ ( M, N ) and T ∞ Emb ∂ ( M, N ) α ⊂ T ∞ Emb ∂ ( M, N )for the collections of path components that map to [ α ] ∈ π Map ∂ ( M, N ) via the maps in (10).We similarly define Emb ∂ ( M, e N ) ˜ α ⊂ Emb ∂ ( M, e N ) and T ∞ Emb ∂ ( M, e N ) ˜ α ⊂ T ∞ Emb ∂ ( M, e N ). Lemma 1.6.
In this situation, there exists a dashed map making the diagram
Emb ∂ ( M, N ) α Emb ∂ ( M, e N ) ˜ α T ∞ Emb ∂ ( M, N ) α T ∞ Emb ∂ ( M, e N ) ˜ α commute up to homotopy. Here the top map is given by sending an embedding β ∈ Emb ∂ ( M, N ) α to its unique lift e β ∈ Emb ∂ ( M, e N ) ˜ α extending ˜ e ∂ .Proof. Let Emb π∂ ( − , e N ) ⊂ Emb ∂ ( − , e N ) be the presheaf on Disc ∂ M of those embeddings thatremain an embedding after composition with π . This fits in a pullback diagramEmb π∂ ( − , e N ) Emb ∂ ( − , N )Map ∂ ( − , e N ) Map ∂ ( − , N ) π ◦− π ◦− of presheaves on Disc ∂ M with vertical maps given by inclusion. This is homotopy cartesian inthe projective model structure on PSh ( Disc ∂ M ), since ( π ◦ − ) : Map ∂ ( − , e N ) → Map ∂ ( − , N )is a objectwise fibration by the lifting property of covering maps. Using that T ∞ ( − ) preserveshomotopy limits by Section 1.2.3 (a), and evaluating at M , we arrive at a commutative cubeEmb π∂ ( M, e N ) Emb ∂ ( M, N ) T ∞ Emb π∂ ( M, e N ) T ∞ Emb ∂ ( M, N )Map ∂ ( M, e N ) Map ∂ ( M, N ) T ∞ Map ∂ ( M, e N ) T ∞ Map ∂ ( M, N ) ≃ ≃ with front and back faces homotopy cartesian, and bottom diagonal maps weak homotopy equiv-alences as Map ∂ ( − , e N ) and Map ∂ ( − , N ) are homotopy J -sheaves (see Section 1.2.3 (b)). Bythe uniqueness of lifts (this uses that ∂ M → M is 0-connected), the bottom horizontal mapsbecome weak homotopy equivalences when we restrict domain and target to the path componentsof [ ˜ α ] and [ α ] respectively. Doing so and using the homotopy pullback property, the top of thecube provides a commutative square with horizontal weak homotopy equivalencesEmb π∂ ( M, e N ) ˜ α Emb ∂ ( M, N ) α T ∞ Emb π∂ ( M, e N ) ˜ α T ∞ Emb ∂ ( M, N ) α . ≃≃ The top map is in fact a homeomorphism, by the uniqueness of lifts. Using the inclusion ofpresheaves Emb π∂ ( − , e N ) ⊂ Emb ∂ ( − , e N ), we obtain a commutative diagramEmb ∂ ( M, N ) α Emb π∂ ( M, e N ) ˜ α Emb ∂ ( M, e N ) ˜ α T ∞ Emb ∂ ( M, N ) α T ∞ Emb π∂ ( M, e N ) ˜ α T ∞ Emb ∂ ( M, e N ) ˜ α . ∼ = ≃ whose top composition is given by sending an embedding to its unique lift extending ˜ e ∂ , so weobtain a map T ∞ Emb ∂ ( M, N ) α → Emb ∂ ( M, e N ) ˜ α as desired. (cid:3) Remark . If α has no lift, then there is no component of Emb ∂ ( M, e N ) mapping to [ α ] undercomposition with π . In this case, the above argument shows that there is also no component of T ∞ Emb ∂ ( M, e N ) mapping to [ α ] under the map of Section 1.3 (d) and composition with π .1.6. Adding a collar to the source.
The next property concerns the behaviour of embeddingcalculus when adding a disjoint collar to the domain.We fix triads M and N and a boundary condition e ∂ : ∂ M ֒ → ∂ N . Given a compact(dim( M ) − K , we replace M by the triad M ⊔ K × [0 ,
1) with ∂ ( M ⊔ K × [0 , ∂ M ⊔ K × { } and fix an extension e ′ ∂ : ∂ ( M ⊔ K × [0 , ֒ → ∂N of e ∂ as boundary condition.By contractibility of the space of collars, the restriction mapEmb ∂ M ⊔ K ×{ } ( M ⊔ K × [0 , , N ) −→ Emb ∂ ( M, N ) MBEDDING CALCULUS FOR SURFACES 11 is a weak homotopy equivalence. This remains true upon applying embedding calculus:
Lemma 1.8.
In this situation, in the diagram induced by restriction
Emb ∂ M ⊔ K ×{ } ( M ⊔ K × [0 , , N ) Emb ∂ M ( M, N ) T ∞ Emb ∂ M ⊔ K ×{ } ( M ⊔ K × [0 , , N ) T ∞ Emb ∂ M ( M, N ) , ≃≃ both horizontal maps are weak homotopy equivalences.Proof. Let U be the open cover of M ⊔ K × [0 ,
1) given by subsets of the form U = V ⊔ K × [0 , V ⊂ M is the union of a open subset diffeomorphic to a collar on ∂ M and a finite disjointunion of open balls. This is a complete Weiss ∞ -cover of M ⊔ K × [0 , U ′ = { U ∩ M | U ∈ U} is a complete Weiss ∞ -cover of M . Restriction thus induces a commutative diagramEmb ∂ M ⊔ K ×{ } ( M ⊔ K × [0 , , N ) holim U ∈U Emb ∂ M ⊔ K ×{ } ( U, N ) T ∞ Emb ∂ M ⊔ K ×{ } ( M ⊔ K × [0 , , N ) holim U ∈U T ∞ Emb ∂ M ⊔ K ×{ } ( U, N ) ≃≃ whose horizontal maps are weak homotopy equivalences by Section 1.2.3 (b) and whose rightvertical map is a weak homotopy equivalence by Section 1.3 (c). Similarly we have a squareEmb ∂ M ( M, N ) holim U ∈U Emb ∂ M ( U ∩ M, N ) T ∞ Emb ∂ ( M, N ) holim U ∈U T ∞ Emb ∂ M ( U ∩ M, N ) , ≃≃ which receives a map from the former square by restriction, so it suffices to show that the mapsEmb ∂ M ⊔ K ×{ } ( U, N ) −→ Emb ∂ M ( U ∩ M, N )are weak homotopy equivalence This follows from the contractibility of spaces of collars. (cid:3)
Taking disjoint unions.
The next property of embedding calculus concerns taking disjointunions of the domain and target. Its full strength is not needed to prove the main results of thispaper—only Corollary 1.10 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 ′ ). We formnew triads M ⊔ M ′ with ∂ i ( M ⊔ M ′ ) = ∂ i M ⊔ ∂ i M ′ for i = 0 ,
1, and N ⊔ N ′ with ∂ i ( N ⊔ N ′ ) = ∂ i N ⊔ ∂ i N ′ for i = 0 ,
1. Given boundary conditions e ∂ : ∂ M ֒ → ∂ N and e ′ ∂ : ∂ M ′ ֒ → ∂ N ′ ,we get a boundary condition e ∂ ⊔ e ′ ∂ : ∂ ( M ⊔ M ′ ) ֒ → ∂ ( N ⊔ N ′ ). Disjoint union of embeddingsinduces a map Emb ∂ ( M, N ) × Emb ∂ ( M ′ , N ′ ) −→ Emb ∂ ( M ⊔ M ′ , N ⊔ N ′ )which is a weak homotopy equivalence (in fact, a homeomorphism) if both inclusions ∂ M ֒ → M and ∂ M ′ ֒ → M ′ are 0-connected. This remains true upon applying embedding calculus: Lemma 1.9.
In this situation, there is a dashed weak homotopy equivalence making
Emb ∂ ( M, N ) × Emb ∂ ( M ′ , N ′ ) Emb ∂ ( M ⊔ M ′ , N ⊔ N ′ ) T ∞ Emb ∂ ( M, N ) × T ∞ Emb ∂ ( M ′ , N ′ ) T ∞ Emb ∂ ( M ⊔ M ′ , N ⊔ N ′ ) ≃≃ commute up to homotopy. Proof.
As in the proof of Lemma 1.8, we will use that T ∞ has descent for complete Weiss ∞ -coversby Section 1.2.3 (b).We take U M to be the open cover of M given by open subsets U ⊂ M that are diffeomorphicto a collar on ∂ M and a finite disjoint union of open balls, and similarly for U M ′ . Let us 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 ′ . These are all complete Weiss ∞ -covers.We consider U M ⊔ M ′ as a poset ordered by inclusion and let Emb ⊔ ∂ ( − , N ⊔ N ′ ) be the presheafon U M ⊔ M ′ that sends U ⊔ U ′ with U ∈ U M and U ′ ∈ U M ′ to the subspace Emb ⊔ ∂ ( U ⊔ U ′ , N ⊔ N ′ ) ⊂ Emb ∂ ( U ⊔ U ′ , N ⊔ N ′ ) which map U into N and U ′ into N ′ . Defining Map ⊔ ∂ ( − , N ⊔ N ′ ) similarly,we have a homotopy pullback diagram of presheaves on U M ⊔ M ′ Emb ⊔ ∂ ( − , N ⊔ N ′ ) Emb ∂ ( − , N ⊔ N ′ )Map ⊔ ∂ ( − , N ⊔ N ′ ) Map ∂ ( − , N ⊔ N ′ ) , . (14)which remains a homotopy pullback when taking homotopy limits over U M ⊔ M ′ .To identify the term holim U ⊔ U ′ ∈U M ⊔ M ′ Emb ⊔ ∂ ( U ⊔ U ′ , N ⊔ N ′ )we note that there are isomorphisms U M ⊔ M ′ ∼ = U M × U M ′ of categories, and Emb ⊔ ∂ ( − , N ⊔ N ′ ) ∼ =Emb ⊔ ∂ ( − , N ) × Emb ∂ ( − , N ′ ) of presheaves, so the Fubini theorem for homotopy limits impliesthat this homotopy limit is given byholim U ∈U M Emb ∂ ( U, N ) × holim U ′ ∈U M Emb ∂ ( U ′ , N ′ ) . Combining descent for T ∞ with the fact that embedding calculus converges on U ∈ U M and U ′ ∈ U M ′ by Section 1.3 (c), we conclude thatholim U ⊔ U ′ ∈U M ⊔ M ′ Emb ⊔ ∂ ( U ⊔ U ′ , N ⊔ N ′ ) ≃ T ∞ Emb ∂ ( M, N ) × T ∞ Emb ∂ ( M ′ , N ′ ) . The same analysis holds for Map ⊔ ∂ ( − , M ⊔ M ′ ) and since this is a homotopy J -sheaf (see Sec-tion 1.2.3 (b)), we conclude thatholim U ⊔ U ′ ∈U M ⊔ M ′ Map ⊔ ∂ ( U ⊔ U ′ , N ⊔ N ′ ) ≃ Map ∂ ( M, N ) × Map ∂ ( M ′ , N ′ ) . By the same argument (using descent for T ∞ , convergence on U ⊔ U ′ ∈ U M ⊔ M ′ , and thatMap ∂ ( − , N ⊔ N ′ ) is a homotopy J -sheaf), we have weak homotopy equivalencesholim U ⊔ U ′ ∈U M ⊔ M ′ Emb ∂ ( U ⊔ U ′ , N ⊔ N ′ ) ≃ T ∞ Emb ∂ ( M ⊔ M ′ , N ⊔ N ′ ) andholim U ⊔ U ′ ∈U M ⊔ M ′ Map ∂ ( U ⊔ U ′ , N ⊔ N ′ ) ≃ Map ∂ ( M ⊔ M ′ , N ⊔ N ′ ) , so altogether we obtain a homotopy pullback diagram of the form T ∞ Emb ∂ ( M, N ) × T ∞ Emb ∂ ( M ′ , N ′ ) T ∞ Emb ∂ ( M ⊔ M ′ , N ⊔ N ′ )Map ∂ ( M, N ) × Map ∂ ( M ′ , N ′ ) Map ∂ ( M ⊔ M ′ , N ⊔ N ′ ) . The condition that ∂ M ֒ → M and ∂ M ′ ֒ → M ′ are both 0-connected implies that the bottommap is a weak homotopy equivalence, so the top map is a weak homotopy equivalence as well.The proof is finished by tracing through the weak homotopy equivalences used to see that thismakes the square in the statement homotopy commute. (cid:3) Taking M ′ = ∅ , which is the only case used in this paper, Lemma 1.9 says: MBEDDING CALCULUS FOR SURFACES 13
Corollary 1.10.
In this situation, in the diagram induced by the inclusion N ⊂ N ⊔ N ′ Emb ∂ ( M, N ) Emb ∂ ( M, N ⊔ N ′ ) T ∞ Emb ∂ ( M, N ) T ∞ Emb ∂ ( M, N ⊔ N ′ ) , ≃≃ both horizontal maps are weak homotopy equivalences.Remark . Corollary 1.10 admits an alternative proof along the lines of Lemma 1.6: oneobserves there is a homotopy pullback diagram of presheaves on
Disc ∂ M given byEmb ∂ ( − , N ) Emb ∂ ( − , N ⊔ N ′ )Map ∂ ( − , N ) Map ∂ ( − , N ⊔ N ′ ) . Taking T ∞ and evaluating at M yields a homotopy pullback diagram of spaces and if ∂ M → M is 0-connected, the map Map ∂ ( M, N ) → Map ∂ ( M, N ⊔ N ′ ) is a weak homotopy equivalencehence so is the map T ∞ Emb ∂ ( M, N ) → T ∞ Emb ∂ ( M, N ⊔ N ′ ).2. Spaces of arcs in surfaces and embedding calculus
As further preparation to the proof of Theorem A, we prove the following convergence resultof independent interest on spaces of (thickened) arcs in surfaces. It includes Theorem D.
Theorem 2.1.
Let Σ be a compact surface.(i) For any boundary condition e ∂ : { , } ֒ → ∂ Σ , the map Emb ∂ ( I, Σ) −→ T ∞ Emb ∂ ( I, Σ) is a weak homotopy equivalence.(ii) For any boundary condition e ∂ : { , } × [0 , ֒ → ∂ Σ and a finite (possibly empty) set S , Emb ∂ ( I × [0 , ⊔ S × R , Σ) −→ T ∞ Emb ∂ ( I × [0 , ⊔ S × R , Σ) is a weak homotopy equivalence. Convention 2.2.
Throughout this section, we fix a compact surface Σ, write I = [0 , I × [0 ,
1] as a manifold triad with ∂ ( I × [0 , { , }× [0 , I × [0 ,
1] into a surface Σ will always be assumed toextend a boundary condition e ∂ : { , } × [0 , ֒ → Σ which will either be specified or is clear fromthe context. We consider I as a submanifold of I × [0 ,
1] via the inclusion { / } × [0 , ⊂ I × [0 , e ∂ as above in particular induces a boundary condition e ∂ : { , } ֒ → Σfor embedding of the form
I ֒ → Σ by restriction.As a preparation to the proof of Theorem 2.1, we establish three lemmas that interpolatebetween the two parts of the theorem.
Lemma 2.3.
For a boundary condition e ∂ : { , } × [0 , ֒ → ∂ Σ , the map Emb ∂ ( I × [0 , , Σ) −→ T ∞ Emb ∂ ( I × [0 , , Σ) is a weak homotopy equivalence if the map Emb ∂ ( I, Σ) → T ∞ Emb ∂ ( I, Σ) is a weak homotopyequivalence for the boundary condition e ∂ obtained by restricting e ∂ . Proof.
Combining Lemma 1.5 with isotopy invariance (see Section 1.3 (b)) provides a commutative(up to canonical homotopy) diagramEmb ∂ ( I × [0 , , Σ) Emb ∂ ( I × { / } , Σ) T ∞ Emb ∂ ( I × [0 , , Σ) T ∞ Emb ∂ ( I × { / } , Σ)Bun ∂ ( T ( I × [0 , , Σ) Bun ∂ ( T ( I × { / } ) , Σ) (15)whose two subsquares are homotopy cartesian. The claim follows from the top square. (cid:3)
The converse requires us to consider all extensions of the boundary condition e ∂ : { , } ֒ → ∂ Σto a boundary condition e ∂ : { , } × [0 , ֒ → ∂ Σ. Lemma 2.4.
For a boundary condition e ∂ : { , } ֒ → ∂ Σ , the map Emb ∂ ( I, Σ) −→ T ∞ Emb ∂ ( I, Σ) is a weak homotopy equivalence if the map Emb ∂ ( I × [0 , , Σ) → T ∞ Emb ∂ ( I × [0 , , Σ) is a weakhomotopy equivalence for all boundary conditions e ∂ : { , } × [0 , ֒ → ∂ Σ extending e ∂ .Proof. Consider once more the commutative diagram (15), for an extension of the boundarycondition e ∂ that we have yet to specify. For [ β ] ∈ π Bun ∂ ( T ( I ×{ / } ) , T Σ), we write Emb ∂ ( I ×{ / } , Σ) β and T ∞ Emb ∂ ( I × { / } , Σ) β the unions of components mapping to [ β ]. It suffices toprove that the map Emb ∂ ( I × { / } ) β −→ T ∞ Emb ∂ ( I × { / } ) β is a weak homotopy equivalence for all choices of [ β ].The homotopy fibre of the bottom map Bun ∂ ( T ( I × [0 , , T Σ) → Bun ∂ ( T ( I × { / } ) , T Σ)over β is given by a space of sections over I with fibre {± } relative to { , } . This space iseither empty or contractible, and we can always pick an extension of the boundary condition e ∂ for which it is contractible. Doing so, let [ ˜ β ] ∈ π Bun ∂ ( T ( I × [0 , , T Σ) be the unique lift of [ β ]so that we have a homotopy cartesian squareEmb ∂ ( I × [0 , , Σ) ˜ β Emb ∂ ( I × { / } , Σ) β T ∞ Emb ∂ ( I × [0 , , Σ) ˜ β T ∞ Emb ∂ ( I × { / } , Σ) β . with bottom horizontal and left vertical maps weak homotopy equivalences. This implies that theright vertical map is also a weak homotopy equivalence, so the claim follows. (cid:3) Lemma 2.5.
For a finite set S and a boundary condition e ∂ : { , } × [0 , ֒ → Σ , the map Emb ∂ ( I × [0 , ⊔ S × R , Σ) −→ T ∞ Emb ∂ ( I × [0 , ⊔ S × R , Σ) is a weak homotopy equivalence if the maps Emb ∂ ( I, Σ ′ ) → T ∞ Emb ∂ ( I, Σ ′ ) are weak homotopyequivalences for all compact surfaces Σ ′ and all boundary conditions e ∂ : { , } ֒ → ∂ Σ ′ . MBEDDING CALCULUS FOR SURFACES 15
Proof.
Lemma 2.3 gives the result when S = ∅ . If S = ∅ , we note that by isotopy extension andthe convergence on discs (see Section 1.3 (c) and (f)), there is a map of fibre sequncesEmb ∂ ( I × [0 , , Σ \ S × int( D )) Emb ∂ ( I × [0 , ⊔ S × R , Σ) Emb( S × D , Σ) T ∞ Emb ∂ ( I × [0 , , Σ \ S × int( D )) T ∞ Emb ∂ ( I × [0 , ⊔ S × R , Σ) T ∞ Emb( S × D , Σ) ≃ induced by restriction along the standard inclusion D ⊂ R , with homotopy fibres taking overany embedding e : S × D ֒ → int(Σ). Strictly speaking, the domain of the spaces in the fibre is I × [0 , ⊔ S × R \ int( D ) as opposed to I × [0 ,
1] but we can remove the collars by Lemma 1.8.The case S = ∅ shows that the left vertical map is a weak homotopy equivalence for any choiceof e , so we conclude that the middle vertical map is a weak homotopy equivalence as well. (cid:3) In view of Lemma 2.5, the second part of Theorem 2.1 is a consequence of its first part. Weprove the latter in the next two lemmas, first for arcs that connect two boundary componentsand then for those connecting a boundary component to itself. These arguments are inspired by[Gra73] and the exposition thereof in [Hat14].
Lemma 2.6.
For a boundary condition e ∂ : { , } ֒ → ∂ Σ that hits two distinct boundary compo-nents, the map Emb ∂ ( I, Σ) −→ T ∞ Emb ∂ ( I, Σ) is a weak homotopy equivalence.Proof. By Lemma 2.4 we may prove the statement for I × [0 ,
1] instead of I , for all extensions ofthe boundary condition e ∂ to a boundary condition e ∂ : { , } × [0 , ֒ → ∂ Σ.We glue a disc D to the boundary component hit by { } , and consider L = ( I × [0 , ∪ D ).Smoothing corners and applying isotopy extension isotopy extension and the convergence on discs(see Section 1.3 (c) and (f)), we have a map of fibre sequenceEmb ∂ ( I × [0 , , Σ) Emb I ×{ } ( L, Σ ∪ D ) Emb( D, Σ ∪ D ) T ∞ Emb ∂ ( I × [0 , , Σ) T ∞ Emb I ×{ } ( L, Σ ∪ D ) T ∞ Emb( D, Σ ∪ D ) ≃ ≃ with homotopy fibres taken over the standard inclusion. Observing that L is isotopy equivalent to I × [0 ,
1) relative to I × { } , the middle vertical map is a weak homotopy equivalence by isotopyinvariance and the convergence on discs (see Section 1.3 (b) and (c)). This implies that the leftvertical map is a weak homotopy equivalence as well, so the proof is finished. (cid:3) The case of arcs connecting the same boundary component is harder and its proof is the heartof this section. It relies on the lifting Lemma 1.6, which we specialise to the case arcs for thebenefit of the reader. The statement involves a covering map e N → N , a boundary condition e ∂ : { , } ֒ → ∂N , a path α ∈ Map ∂ ( I, N ), and a lift ˜ α : I → e N of α whose endpoints induce aboundary condition e ∂ : { , } ֒ → ∂ e N . Recall thatEmb ∂ ( I, N ) α ⊂ Emb ∂ ( I, N ) and T ∞ Emb ∂ ( I, N ) α ⊂ T ∞ Emb ∂ ( I, N )are the collections of path components that map to [ α ] ∈ π Map ∂ ( I, N ) via the maps in (10). αβ ΣΓ Figure 1.
The surface Γ. The original surface Σ is the region within the dotted circle.
Lemma 2.7.
In this situation, there exists a dashed map making the diagram
Emb ∂ ( I, N ) α Emb ∂ ( I, e N ) ˜ α T ∞ Emb ∂ ( I, N ) α T ∞ Emb ∂ ( I, e N ) ˜ α commute up to homotopy. Here the top map is given by sending an arc γ ∈ Emb ∂ ( I, N ) α to theunique lift e γ ∈ Emb ∂ ( I, e N ) ˜ α starting at ˜ α (0) ∈ e N .Proof. This is Lemma 1.6 for the triad M = I with ∂ I = { , } = ∂I . (cid:3) Lemma 2.8.
For a boundary condition e ∂ : { , } ֒ → ∂ Σ that hits a single boundary component, Emb ∂ ( I, Σ) −→ T ∞ Emb ∂ ( I, Σ) is a weak homotopy equivalence.Proof. By Corollary 1.10, we may assume that Σ is connected. We glue a 1-handle I × [0 , { , } , such that I × { } and I × { } are separatedon that boundary component by { , } and are embedded with opposite orientation, resultingin a new surface Γ with an additional boundary component; see Figure 1. The composition { , } ֒ → Σ ⊂ Γ now hits two distinct boundary components, so the right vertical map in thehomotopy-commutative diagram induced by the inclusion Σ ⊂ Γ (see Section 1.3 (a))Emb ∂ ( I, Σ) Emb ∂ ( I, Γ) T ∞ Emb ∂ ( I, Σ) T ∞ Emb ∂ ( I, Γ) ≃ (16)is a weak homotopy equivalence by Lemma 2.6. MBEDDING CALCULUS FOR SURFACES 17
We next investigate the set of path-components. To do so, we will use that the dashed map inthe commutative diagram π Emb ∂ ( I, Σ) (cid:0) π Map ∂ ( I, Σ) (cid:1) × π Map ∂ ( I, Γ) (cid:0) π Emb ∂ ( I, Γ) (cid:1) π Emb ∂ ( I, Γ) π Map ∂ ( I, Σ) π Map ∂ ( I, Γ) , is surjective: if an embedding I ֒ → Γ is homotopic to a map I → Σ, then it is isotopic to anembedding
I ֒ → Σ within the homotopy class of I → Σ. To see this, use the bigon criterion[FM12, Sections 1.2.4, 1.2.7] to isotope
I ֒ → Γ so that its geometric intersection number withthe cocore β of the 1-handle is equal to the algebraic intersection number, which is 0 since it ishomotopic to a map I → Σ. With this in mind, a diagram chase in the factorisation π Emb ∂ ( I, Σ) π Emb ∂ ( I, Γ) π T ∞ Emb ∂ ( I, Σ) π T ∞ Emb ∂ ( I, Γ) π Map ∂ ( I, Σ) π Map ∂ ( I, Γ) ∼ =2 shows that the maps 1 and 2 have the same image.Let us now fix a class [ α ] ∈ π Map ∂ ( I, Σ) in this image. As the map 1 is injective because twoembedded arcs are isotopic relative to the endpoints if and only if they are homotopic relative tothe endpoints [Feu66], there is a unique path component Emb ∂ ( I, Σ) α of Emb ∂ ( I, Σ) mapping to[ α ]. Denoting by T ∞ Emb ∂ ( I, Σ) α ⊂ T ∞ Emb ∂ ( I, Σ) the union of all path components that mapto [ α ], it suffices to show that the mapEmb ∂ ( I, Σ) α −→ T ∞ Emb ∂ ( I, Σ) α is a weak homotopy equivalence for all choices of [ α ]. Since the left side is contractible by [Gra73,Th´eor`eme 5], it suffices to prove that the right side is contractible as well.To do so, we will construct a homotopy-commutative diagramEmb ∂ ( I, Σ) α Emb ∂ ( I, Γ) α Emb ∂ ( I, Σ) α T ∞ Emb ∂ ( I, Σ) α T ∞ Emb ∂ ( I, Γ) α T ∞ Emb ∂ ( I, Σ) α ( e ◦− ) ◦ lift ≃ ( e ◦− ) ◦ lift (17)whose horizontal compositions are homotopic to the identity. This will finish the proof, since itexhibits T ∞ Emb ∂ ( I, Σ) α as a retract of the contractible space T ∞ Emb ∂ ( I, Γ) α ≃ Emb ∂ ( I, Γ) α .The left square in (17) is obtained by restricting the path-components of the homotopy com-mutative square (16). The right square arises as the composition of two squaresEmb ∂ ( I, Γ) α Emb ∂ ( I, e Γ) ˜ α Emb ∂ ( I, Σ) α T ∞ Emb ∂ ( I, Γ) α T ∞ Emb ∂ ( I, e Γ) ˜ α T ∞ Emb ∂ ( I, Σ) α . lift ≃ e ◦− lift e ◦− which we explain now. ˜ α Π Figure 2.
The surface Π.
The surface e Γ is an appropriate covering space of Γ: the construction of Γ gives a decomposition π (Γ) ∼ = π (Σ) ∗ Z and e Γ is the cover corresponding to the subgroup π (Σ). Explicitly, the cover e Γ can be constructed by cutting Γ along β to obtain a surface Π (see Figure 2) and gluing twocopies of the universal cover e Π of this surface to the two intervals in the boundary resulting from β . Note that Π contains a preferred lift ˜ α of α and hence so does e Γ. We denote the endpoints of α and e α in the various surfaces generically by { , } . The cover e Γ has the property that the mapΣ → Γ lifts uniquely to e Γ so that { , } is fixed and (using that the interior of e Π is diffeomorphicto R ) that there is an embedding e : e Γ ֒ → Σ fixing { , } such that the composition Σ → e Γ → Σis isotopic to the identity relative to { , } .The right square is induced by post-composition with e , so homotopy commutes in view ofSection 1.3 (a). The homotopy commutative left square is obtained by invoking the lifting lemmaLemma 2.7 for the covering map e Γ → Γ. The top composition in (17) is homotopic to the identityby construction, but it remains to justify this for the bottom composition.Justifying this requires the details of the proof of Lemma 1.6. Inspecting this proof, we see itsuffices to note that e Γ contains a copy of Σ such that the restriction π of Σ to this copy is injective,so that we get a dashed map of presheaves on Disc ∂I making the triangle in the following diagramcommute up natural homotopyEmb ∂ ( − , Σ) Emb ∂ ( − , Γ)Emb π∂ ( − , e Γ) Emb ∂ ( − , e Γ) Emb ∂ ( − , Σ) π ◦− e ◦ ( − ) It then suffices to observe that the composition Emb ∂ ( − , Σ) → Emb ∂ ( − , Σ) along the bottom isnaturally homotopic to the identity, as Σ → e Γ → Σ is isotopic to the identity. (cid:3)
Proof of Theorem 2.1.
The first part is Lemma 2.6 and Lemma 2.8. The second part follows fromthe first by Lemma 2.5. (cid:3) The proof of Theorem A
In this section, we prove Theorem A by a sequence of reductions that gradually simplify thedomain Σ, eventually to Σ = D , a case that we prove separately. Convention 3.1.
Throughout this section, we write I = [0 ,
1] and consider I × [0 ,
1] as a manifoldtriad with ∂ ( I × [0 , { , } × [0 , I × [0 ,
1] into a surface Σ willalways be assumed to extend a boundary condition e ∂ : { , } ֒ → Σ which will either be specifiedor is clear from the context.
MBEDDING CALCULUS FOR SURFACES 19
We begin with the following auxiliary lemma, which will be used in several induction steps.
Lemma 3.2.
Let Σ and Σ ′ be compact surfaces and e ∂ : ∂ Σ ֒ → ∂ Σ ′ be a boundary condition.(i) Fix D ⊂ int(Σ) . If for all embeddings e : D ֒ → int(Σ ′ ) , the map Emb ∂ (cid:0) Σ \ int( D ) , Σ ′ \ int( e ( D )) (cid:1) −→ T ∞ Emb ∂ (cid:0) Σ \ int( D ) , Σ ′ \ e (int( D )) (cid:1) is a weak homotopy equivalence, then so is the map Emb ∂ (Σ , Σ ′ ) −→ T ∞ Emb ∂ (Σ , Σ ′ ) . (ii) Fix an embedding I × [0 , ⊂ Σ with ∂ Σ ∩ ( I × [0 , { , } × [0 , . If for all embeddingsof triads e : I × [0 , → Σ ′ with im( e ) ∩ Σ ′ = { , } × [0 , , the map Emb ∂ (Σ \ ( I × (0 , , Σ ′ \ e ( I × (0 , −→ T ∞ Emb ∂ (Σ \ ( I × (0 , , Σ ′ \ e ( I × (0 , is a weak homotopy equivalence, then so is the map Emb ∂ (Σ , Σ ′ ) −→ T ∞ Emb ∂ (Σ , Σ ′ ) . Proof.
These are two instances of the fact that for a commutative square
E BE ′ B ′≃ whose right arrow is a weak homotopy equivalence, the map E → E ′ is a weak homotopy equiva-lence if and only if the map hofib( E → B ) → hofib( E ′ → B ′ ) is a weak homotopy equivalence forall choices of basepoint. For (i), we consider the squareEmb ∂ (Σ , Σ ′ ) Emb( D , Σ ′ ) T ∞ Emb ∂ (Σ , Σ ′ ) T ∞ Emb ∂ ( D , Σ ′ )whose right map is a weak homotopy equivalence by the convergence on discs (see Section 1.3(c)). This property also allows us to apply isotopy extension (see Section 1.3 (f)), which identifiesthe map on homotopy fibres over an embedding e : D ֒ → int(Σ ′ ) with one of the maps in theassumption, and the claim follows.The case (ii) proceeds analogously by considering the squareEmb ∂ (Σ , Σ ′ ) Emb ∂ ( I × [0 , , Σ ′ ) T ∞ Emb ∂ (Σ , Σ ′ ) T ∞ Emb ∂ ( I × [0 , , Σ ′ ) . and replacing the use of the convergence on discs by the second part of Theorem 2.1. (cid:3) Proof of Theorem A.
We now gradually simplify Σ to prove Theorem A.
Reduction to all components having boundary.
We first show that it suffices to assume that allcomponents of Σ have non-empty boundary. We do this by downwards induction on the numberof closed components. If Σ has a closed component, we pick an embedded disc D ⊂ Σ in a closedcomponent, so that Σ \ int( D ) has one less closed component and the assumptions of Lemma 3.2(i) are satisfied. Reduction to connected surfaces.
We next assume that Σ has no closed components and reduce tothe case that Σ is connected by downwards induction on the number of components. If Σ is not connected, we attach 1-handle I × [0 ,
1] along I × { , } to two different boundary components ofΣ to obtain a new surface Σ ∪ ( I × [0 , ′ along the composition of I × { , } ⊂ ∂ Σ with the boundarycondition e ∂ : ∂ Σ → ∂ Σ ′ to obtain a surface Σ ′ ∪ I × [0 ,
1] and consider the commutative squareEmb ∂ (Σ ∪ I × [0 , , Σ ′ ∪ I × [0 , ∂ ( I × [0 , , Σ ′ ∪ I × [0 , T ∞ Emb ∂ (Σ ∪ I × [0 , , Σ ′ ∪ I × [0 , T ∞ Emb ∂ ( I × [0 , , Σ ′ ∪ I × [0 , , ≃ ≃ where the right hand embedding spaces are formed with respect to the boundary condition { , }× [0 , ⊂ Σ ′ ∪ I × [0 , ∂ (Σ , Σ ′ ) −→ T ∞ Emb ∂ (Σ , Σ ′ )which is hence a weak homotopy equivalence as well. Reducing the genus.
By the classification of compact connected surfaces and the previous reduc-tions, we may assume that Σ is diffeomorphic to one of the following surfaces Σ g,n R P ♯ Σ g,n R P ♯ R P ♯ Σ g,n (18)for some g ≥ n ≥
1, where Σ g,n stands for a compact connected orientable surface of genus g with n boundary components. To reduce the case ( g + 1 , n ) to ( g, n + 1), we pick an embeddedstrip I × [0 , ⊂ Σ so that(i) we have ∂ Σ ∩ I × [0 ,
1] = { , } × [0 , { , } × [0 ,
1] lies in a single boundary component,(iii) with respect to some orientation of ∂ Σ, both maps { } × [0 , ֒ → ∂ Σ and { } × [0 , ֒ → ∂ Σare orientation-preserving, and(iv) Σ \ I × (0 ,
1) is connected.Then Σ \ ( I × (0 , g reduced by 1 and n increasedby 1, so an application of Lemma 3.2 (ii) finishes the reduction. Reducing the number of boundary components.
We are left with the case where Σ is diffeomorphicto a surface in the list (18) with ( g, n ) = (0 , n ) with n ≥
1. Now we reduce the the case (0 , n + 1)to (0 , n ) for n ≥
1. To do so, we pick an embedding I × [0 , ⊂ Σ such that I × [0 , ∩ ∂ Σ = { , } × [0 ,
1] hits two distinct components of Σ. Then Σ \ ( I × (0 , Reducing to a disc.
Thus it suffices to prove the case ( g, n ) = (0 , D , M , or M ♮M , where M = R P \ int( D ) denotes the M¨obius strip.To reduce the cases M and M ♮M to the case D , we use M ∼ = ([0 , × [0 , / ( x, ∼ (1 − x, . MBEDDING CALCULUS FOR SURFACES 21 and consider the embedding α : I × [0 , ֒ → M given by ( x, y ) ( x, (1 / y +1 / M ♮M away from α ( I × [0 , ∩ ∂M , we have M ♮M \ int( α ( I × [0 , ∼ = M and M \ int( α ( I × [0 , ∼ = D , so an application of Lemma 3.2 (ii) finishes the reduction. Proving the result for a disc.
By the previous sequence of reductions, we may restrict to the caseΣ = D . Moreover, as ∂ D = ∂D ֒ → D is 0-connected, we may assume by Corollary 1.10 thatΣ ′ is connected. If Σ ′ is connected and not diffeomorphic to D , then Emb ∂ ( D , Σ ′ ) = ∅ , so wehave to prove that in this case also T ∞ Emb ∂ ( D , Σ ′ ) is empty.If it were non-empty, then the target of the canonical map T ∞ Emb ∂ ( D , Σ ′ ) −→ Map ∂ ( D , Σ ′ )from Section 1.3 (d) is nonempty as well, so Σ ′ is a connected surface with a boundary componentwhose inclusion is null-homotopic. We claim this is impossible unless Σ ′ ∼ = D . Firstly, if Σ ′ =Σ ♮ · · · ♮ Σ n , then π (Σ ′ ) splits as a free product π (Σ ) ∗ · · · ∗ π (Σ n ) and we may choose thisdecomposition so that the homotopy class of the boundary inclusion represents the free productof the homotopy classes of boundary inclusions of those components at which we perform theboundary connected sums. By the classification of connected compact surfaces, it then sufficesto observe that all boundary inclusions are non-trivial in the fundamental group of the surfacesΣ , , Σ , , and M . For Σ , , each boundary inclusion represents a generator of π (Σ , ) ∼ = Z , forΣ , the boundary inclusion represents xyx − y − ∈ π (Σ , ) ∼ = h x, y i , and for M it representstwice a generator in π ( M ) ∼ = Z . This finishes the argument that T ∞ Emb ∂ ( D , Σ ′ ) is empty forall connected surfaces Σ ′ not diffeomorphic to D .Finally, it remains to show that the mapEmb ∂ ( D , D ) −→ T ∞ Emb ∂ ( D , D )a weak homotopy equivalence. For this we follow the proof of what is sometimes called the Cerf Lemma [Cer63, Proposition 5] (which leads to a proof of Smale’s theorem [Cer63, Th´eor´eme4]). To this end, we consider the triad H = D ∩ (( − / , ∞ ) × R ) with ∂ H = H ∩ ∂D and ∂ H = H ∩ ( { } × R ), see Figure 3. Inside of this, we have the strip J = H ∩ ([ − / , / × R )with ∂ J = J ∩ ∂D .We will use that Emb ∂ ( J, D ) is path-connected, a consequence of the classification of surfaces.Writing H = H \ (( − / , / × R ) ∩ H and D = D \ (( − / , / × R ) ∩ D , we combineisotopy extension (see Section 1.3 (f)) with the second part of Theorem 2.1 to obtain a map offibre sequences Emb ∂ ( H , D ) Emb ∂ ( H, D ) Emb ∂ ( J, D ) T ∞ Emb ∂ ( H , D ) T ∞ Emb ∂ ( H, D ) T ∞ Emb ∂ ( J, D ) ≃ ≃ with connected weakly equivalent bases and homotopy fibres taken over the standard inclusion J ֒ → D . As H is a closed collar on ∂ H , the middle terms are contractible by the contractibilityof the space of collars and Lemma 1.8, so the left vertical map is a weak homotopy equivalenceas well. By Lemma 1.8 we may discard the collar H ∩ (( −∞ , × R ), and obtain that for H ′ = H ∩ ([1 / , ∞ ) × R ) the mapEmb ∂ ( H ′ , D ) −→ T ∞ Emb ∂ ( H ′ , D )is a weak homotopy equivalence. Invoking Corollary 1.10 to neglect D \ H from the target andidentifying H ′ with a disc upon smoothing corners, we conclude thatEmb ∂ ( D , D ) −→ T ∞ Emb ∂ ( D , D ) (19)is a weak homotopy equivalence. D J H Figure 3.
The triads
J, H ⊂ D . Here H the union of J and H , and H ′ ⊂ H is thecomponent to the right of J . Automorphisms of the E -operad. The above argument does not determine the homo-topy type of T ∞ Emb ∂ ( D , D ); by Smale’s theorem [Sma59] already mentioned above, the spaceDiff ∂ ( D ) is contractible and hence so is T ∞ Emb ∂ ( D , D ) by our result. Combining Theorems1.2, 1.4, and 6.4 of [BdBW18], the space T ∞ Emb ∂ ( D d , D d ) can be identified with the ( d + 1)stloop space Ω d +1 Aut h ( E d ) / O( d ) of the homotopy fibre of the canonical map B O( d ) −→ B Aut h ( E d )obtained by delooping the canonical action of O( d ) on the little discs operad by derived operadautomorphisms, so we get the following corollary. Corollary 3.3. Ω Aut h ( E ) / O(2) ≃ ∗
Remark . Horel [Hor17, Theorem 8.5] proved Aut h ( E ) / O(2) ≃ ∗ with different methods.4.
Embedding calculus in dimension
Theorem 4.1.
For compact -dimensional manifolds M and M ′ and a boundary condition e ∂ : ∂M ֒ → ∂M ′ , the map Emb ∂ ( M, M ′ ) −→ T ∞ Emb ∂ ( M, M ′ ) is a weak homotopy equivalence. The proof is by reduction to M = D , which is dealt with similarly to D in the proof ofTheorem A. We use an analogue of Lemma 3.2 (i) which is proved by the same argument. Lemma 4.2.
Let M and M ′ be compact -manifolds and e ∂ : ∂M ֒ → ∂M ′ a boundary condition.Fix D ⊂ int( M ) . If for all embeddings e : D ֒ → int( M ′ ) , the map Emb ∂ (cid:0) M \ int( D ) , M ′ \ int( e ( D )) (cid:1) → T ∞ Emb ∂ (cid:0) M \ int( D ) , M ′ \ e (int( D )) (cid:1) is a weak homotopy equivalence, then so is the map Emb ∂ ( M, M ′ ) −→ T ∞ Emb ∂ ( M, M ′ ) . MBEDDING CALCULUS FOR SURFACES 23
Proof of Theorem 4.1.
Using Lemma 4.2, we prove Theorem 4.1 by simplifying M . Reduction to all components having boundary.
We first reduce the claim to the case that allcomponents of M have non-empty boundary by downwards induction over the number of closedcomponents. To do so, we proceed analogously to the corresponding step for surfaces by picking D ⊂ M in the interior of a closed component so that M \ int( D ) has one less closed componentand applying Lemma 4.2. Reduction to an interval.
Assuming that M has no closed components, we reduce to the casewhere M is also connected by downwards induction over the number of components. If M isnot connected, we can attaching a 1-handle [0 ,
1] along { , } to the boundary of two differentcomponent of M , so that M ∪ [0 ,
1] has one less component. Using the image of these pointsunder the boundary condition e ∂ , we also attach [0 ,
1] to M ′ and consider the squareEmb ∂ ( M ∪ [0 , , M ′ ∪ [0 , , , M ′ ∪ [0 , T ∞ Emb ∂ ( M ∪ [0 , , M ′ ∪ [0 , T ∞ Emb([0 , , M ′ ∪ [0 , , ≃ ≃ which allows us to finish the reduction in the same way as in the corresponding step for surfaces. Proving the result for an interval. If M = D , we may use Corollary 1.10 as in the case of surfacesto assume that M ′ is connected, and hence M ′ = D as well. One then takes H = ( − / , J = [ − / , / D = D \ int( J ) and develops a map of fibre sequencesEmb ∂ ( H , D ) Emb ∂ ( H, D ) Emb ∂ ( J, D ) T ∞ Emb ∂ ( H , D ) T ∞ Emb ∂ ( H, D ) T ∞ Emb ∂ ( J, D ) . ≃ ≃ similar to that in the last step of the proof for surfaces. We use Lemma 1.8 and Corollary 1.10to identify the map on fibres withEmb ∂ ( D , D ) −→ T ∞ Emb ∂ ( D , D ) , which is thus a weak homotopy equivalence, so the claim follows. Remark . Theorem 4.1 has similar consequences in dimension 1 as Theorem A has in dimension2. For example, the A ∞ -space T ∞ Emb ∂ ( M, M ) is group-like for compact 1-manifolds M (see thefirst remark of the introduction) and Diff ∂ ( D ) ≃ ∗ has Ω Aut h ( E ) / O(1) ≃ ∗ as a consequence(see Section 3.2).
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Email address : [email protected]@utoronto.ca