Stabilisation, scanning and handle cancellation

In this note we describe a family of arguments that link the homotopy-type of a) the diffeomorphism group of the disc $D^n$, b) the space of co-dimension one embedded spheres in a sphere and c) the homotopy-type of the space of co-dimension two trivial knots in a sphere. We also describe some natural extensions to these arguments. We begin with Cerf's `upgraded' proof of Smale's theorem, that the diffeomorphism group of the 2-sphere has the homotopy-type of the isometry group. This entails a canceling-handle construction, related to the `scanning' maps of Budney-Gabai. We further give a Bott-style variation on Cerf's construction, and a related Embedding Calculus framework for these constructions. We use these arguments to prove that the monoid of Schoenflies spheres is a group with respect to the connect-sum operation. This last result is perhaps only interesting when in dimension four, as in other dimensions it follows from the resolution of the various generalized Schoenflies problems.


Introduction
In Cerf's landmark paper [10], somewhat overlooked is a novel proof of Smale's theorem, that the group of diffeomorphisms of the 2-sphere, Diff(S 2 ) has the homotopy-type of its linear subgroup O 3 .The core of Cerf's argument is the proof that the Smale-Hirsch map (pointwise derivative) Diff(D 2 ) → Ω 2 GL 2 (R) has a left homotopy-inverse.Cerf states his theorem in the language of homotopy groups, i.e. the homotopy groups of Diff(D 2 ) inject into the homotopy groups of Ω 2 GL 2 (R).Since the latter homotopy groups are trivial, and diffeomorphism groups of compact manifolds have the homotopy-type of countable CW-complexes [31], this allows Cerf to conclude Diff(D 2 ) is contractible via the Whitehead Theorem.In this paper we use the notation that if N is a manifold with boundary, Diff(N) denotes the group of diffeomorphisms of N that restrict to the identity on ∂N .We will use the same conventions for embedding spaces, i.e.Emb(N, M) denotes the space of smooth embeddings of N in M, and if N and M have boundary these maps will all restrict to one given map ∂N → ∂M.
Smale's proof that Diff(D 2 ) is contractible uses the Poincaré-Bendixson Theorem to guarantee the flow of the vector fields he uses terminate in finite time.As the Poincaré-Bendixson theorem is only available in dimension two, it limits the applicability of Smale's argument.We should note, there have been attempts to broaden the applicability of a Smale-type argument, by studying spaces of closed 1-forms.See for example the two papers of Laudenbach and Blank [25] [26] for a sampling.Since Cerf's argument does not depend on Poincaré-Bendixson, it allows for greater applicability.The headline consequences of Cerf's arguments are that the diffeomorphism group Diff(D n ) has the same homotopy-type as ΩEmb(D n−1 , D n ) and also that the embedding space Emb(D n−1 , D n ) is a homotopy-retract of ΩEmb(D n−2 , D n ).Putting these two results together, the homotopy groups of Diff(D n ) inject into the homotopy groups of Ω 2 Emb(D n−2 , D n ) for all n.While Cerf states these as his theorems, his techniques prove much more.It is the purpose of this paper to outline the consequences of his techniques.

Cerf's techniques
To give Cerf's results some context, we first mention how the spaces he studies are related to some more commonly-discussed objects.A linearization argument [3] shows that the diffeomorphism group Diff(S n ) has the homotopy-type of O n+1 × Diff(D n ), indeed the homotopy-equivalence comes from considering Diff(D n ) as the subgroup of Diff(S n ) that is the identity on a fixed hemisphere, and the homotopy-equivalence O n+1 × Diff(D n ) → Diff(S n ) is given by the group multiplication in Diff(S n ).There is an analogous homotopy-equivalence Emb(S j , S n ) ≃ SO n+1 × SO n−j Emb(D j , D n ) when n > j.If we let Emb(D n−1 , S 1 × D n−1 ) denote the space of smooth embeddings of D n−1 in S 1 × D n−1 which agree with the standard inclusion D n−1 → {1} × D n−1 on the boundary sphere, then there is a 'handle-filling' homotopy-equivalence Diff(S 1 × D n−1 ) ≃ Diff(D n ) × Emb(D n−1 , S 1 × D n−1 ).In Cerf's paper [10] the main results we highlight concern three maps: , this is the map given by attaching a 1-handle to D n so that the attaching sphere links the standard S n−2 in ∂D n ≡ S n−1 , i.e. we think of S 1 × D n−1 as D n with a 1-handle attached, thus the map comes from simply changing the co-domain of the embedding. ( The symbol ν indicates the embeddings are required to have an everywhere non-zero normal vector field, and the vector field are some standard (constant) on the boundary.
Cerf's result is that the maps (1) and ( 3) are homotopy-equivalences, while ( 2) is a homotopyretract, i.e. has a left homotopy-inverse.The definition of the maps (1) and ( 3) are analogous, and will be described precisely in Section 2. The rough idea of these maps is to fiber the domain of the embedding by a 1-parameter family of co-dimension one discs, and restrict the map to these fibers, and appropriately changing the co-domain of the family of embeddings, via a filling.In the case of (3) the fibering construction would give a 1-parameter family of embeddings of D n−2 in S 1 × D n−1 but we carefully fill with a canceling 2-handle, to construct an element of ΩEmb(D n−2 , D n ).

Extrapolating from Cerf
In Section 2 we observe that Cerf's argument, unchanged, gives a homotopy-equivalence Cerf's results (1) and (3) above correspond to the j = n and j = n − 1 case of this homotopyequivalence.If we think of S n−j × D j as D n union an (n − j)-handle, then the domain of our map, Emb(D j , S n−j × D j ) is the space of all cocores, i.e. smooth embeddings of D j in S n−j × D j that agree with a standard linear inclusion D j → { * } × D j on the boundary.The codomain is the loop space of the space of smooth embeddings D j−1 → D n that carry a nowhere-zero normal vector field, moreover the embedding and the vector field are standard linear embeddings on the boundary.The base-point of the embedding space Emb ν (D j−1 , D n ) is the linear (i.e.boundary parallel) embedding.
It is here where authors noticed a connection to recent 'lightbulb theorems' in low-dimensional topology [13] [4] [21].The above equivalence can be recast slightly -using the same argument but applying it to a strictly larger class of spaces.Let N be a compact n-manifold with non-empty boundary, and let ♮ denote the boundary connect-sum operation.Think of the boundary connect sum N♮(S n−j × D j ) as N with a trivial (n − j)-handle attached, then the space of cocores of this attached handle, which we could denote Emb(D j , N♮(S n−j × D j )) has the same homotopy-type as the loop space ΩEmb ν (D j−1 , N), which is the loop space of the space of embedded D j−1 discs with normal vector field in the manifold N -the space of embeddings we give the base-point of a boundary-parallel embedding.This version is emphasized in [21].

Related expositions
Another way to look at this paper is that it is both an addendum to [3], and a paper that highlights methods from [4] and [10] that deserve to be singled-out.Both [4] and [10] are long papers with preprint many results, so it is easy to overlook this technique.We hope a shorter-format paper devoted to one tool does the ideas the justice they deserve.In [3], an attempt was made to describe the most basic relations between the homotopy-type of diffeomorphism groups and embedding spaces for the smallest manifolds, such as spheres and discs.These Cerf techniques were known to the author, but perhaps indicative of the techniques, the only consequences the author knew at the time were already known, by other methods.So they were removed from the paper before publication.
For example, the connection between the homotopy-type of the component of the unknot Emb u (S 1 , S 3 ), with the homotopy-type of Diff(S 3 ), which is immediate from Cerf's perspective, is historically derived using Hatcher's work on spaces of incompressible surfaces [18] (see the final pages).In Section 3 we describe the relation between Cerf's half-disc fibrations and the more commonly used restriction fibration Diff(S n ) → Emb(S j , S n ).

Schönflies
An interesting observation in [4] is that the 'stacking' operation, while appearing to be just a monoid structure on the space Emb(D n−1 , S 1 × D n−1 ), using Cerf's argument one can show the space is group-like, i.e. the induced monoid structure on π 0 Emb(D n−1 , S 1 × D n−1 ) is that of a group, for all n ≥ 2. One consequence of this is an argument the monoid of Schönflies spheres π 0 Emb(S n−1 , S n ) is a group using the relative connect-sum operation.There is a classical argument due to Kervaire-Milnor that this monoid has inverses.Our argument is characteristically different, in that we construct an onto homomorphism from a group, i.e. in a weak sense we give a presentation of the monoid of Schönflies spheres.This appears in Section 4.

High co-dimension scanning
A scanning technique was proposed for studying the homotopy-type of Diff(S 1 × D n ), by considering the chain of maps in the sequence [4], [5].Interestingly, an infinitely-generated subgroup of π n−4 Diff(S 1 × D n−1 ) survives to the end of the sequence for all n ≥ 4. At present little is known about Cerf's scanning maps Diff(D n ) → Ω j Emb(D n−j , D n ) when j ≥ 3, but these results suggest such maps have the potential to be homotopically non-trivial, and could be used to deduce results even about π 0 Diff(D n ) for n ≥ 4. Although, we now know the map Diff( is perhaps of greatest interest, as the target space can be studied with techniques such as the Embedding Calculus, while we have little in the way of general theory to study the homotopytype of Emb(D n−2 , D n ).
where the space on the right is the terminal j = n case.The map Diff(D n ) → Ω n GL n (R) is known as the Smale-Hirsch map, i.e. the pointwise derivative map.Whether or not this Smale-Hirsch map is homotopically non-trivial has been an open problem for some time.Interestingly, it has recently been shown to be homotopically non-trivial in the n = 11 case [11].
One other impetus for studying such scanning maps is that these embedding spaces are highly structured objects.For example, Diff(D n ) is homotopy-equivalent to the space EC(n, * ), called the 'cubically-supported embedding space'.If M is a compact manifold, EC(j, M) denotes the space of smooth embeddings f : R j × M → R j × M where the support supp( f ) is constrained to be a subset of The space EC(j, M) admits an action of the operad of (j + 1)-cubes, thus it is not far away from being an (j + 1)-fold loop-space.The way to think about this operad action is there is an action of the j-cubes operad on EC(j, M), due to the affine structure on the R j factors of R j × M. The space EC(j, M) is also a monoid under composition of functions, and these two operations can be promoted naturally to a (j + 1)-cubes action, described in [3].
The space EC(j, D n−j ) fibers over Emb(D j , D n ) with fiber Ω j SO n−j -indeed, the spaces EC(j, D n−j ) are homotopy-equivalent to Emb f r (D j , D n ).There are scanning maps which commute with the action of the (n + 1)-cubes operad.While the reference [3] allows one to see these cubes actions explicitly, there are also ways of describing the iterated loop space structure using smoothing theory.Thus the ability of scanning maps to detect homotopy in diffeomorphism groups and embedding spaces is closely connected to the question of to what extent the Smale-Hirsch map for Diff(D n ) is non-trivial.To add some additional context, iterated loop spaces are highly structured objects, and finding maps between them is somewhat analogous to finding a homomorphism between other highly-structured objects like rings or modules: if the map is not zero, it is often highly non-trivial.
In this paper we outline what is known about such scanning maps, and where some potentially interesting future computations sit.
The author would like to thank David Gabai, Robin Koytcheff, Victor Turchin, Hyam Rubinstein and Alexander Kupers for helpful comments.In particular, this paper is largely exposition of a subset of results from a joint paper with David Gabai [4].

Canceling handles
The space Emb(D j , N) denotes the space of embeddings of D j in N where the boundary of D j is mapped to ∂N in some fixed, prescribed manner.In the case of Emb(D j , D n ) the embedding is required to restrict to the standard inclusion x −→ (x, 0) on the boundary. preprint Cerf constructs an isomorphism [10] (Proposition 5, pg.128) for all i ≥ 0 and n ≥ 1 (see also Theorem 4 of [9]) Equivalently, this homotopy-equivalence can be stated as a description of the classifying space of Diff(D n ) The subscript u indicates the component of the unknot in Emb(D n−1 , D n ), i.e. the component of the linear embedding.The above results were stated at least as far back as [3], but it would not be surprising if this observation had been written down earlier.The map back ΩEmb( ) is defined by an elementary isotopy-extension construction.Following Cerf, let HD j denote the j-dimensional half-disc, i.e.
The boundary ∂HD j consists of the two subspaces: The subspace (1) ∂D j ∩ HD j , called the round face, and the subspace (2) satisfying x 1 = 0 called the flat face.Let Emb(HD n , D n ) be the space of embeddings of HD n into D n that restrict to the identity map on HD n ∩ ∂D n , i.e. acting as the identity on the round face.The map given by restriction to the flat face is a Serre fibration (see Figure 1) [8] Diff

PSfrag replacements
preprint Moreover, via an argument directly analogous to the homotopy classification of collar neighborhoods or tubular neighborhoods, one can show Emb(HD n , D n ) is contractible [9].The rough idea is that every such embedding is isotopic to its restriction to a small neighborhood of the round face, where you can approximate the embedding by the standard linear inclusion -indeed, the straight-line homotopy between the embedding and the standard linear inclusion is an isotopy, at least in a sufficiently-small neighborhood of the round face.
The proof that the above map is a Serre fibration is a version of the isotopy-extension theorem 'with parameters', i.e. the proof of isotopy extension given in Hirsch's text [19] suffices to also prove such maps are Serre fibrations.We should also mention that Palais also has shown [31] that a broad class of spaces of embeddings and diffeomorphism groups, including all the spaces discussed in this paper, have the homotopy-type of countable CW-complexes.The rough idea of the proof is that such embedding spaces are homeomorphic to open subsets of a Hilbert cube (consider for example representing smooth functions via something like a Fourier expansion), and open subsets of Hilbert cubes admit CW-structures, in a manner analogous to open subsets of R n .
The total space Emb(HD n , D n ) is contractible, as sketched above and proven by Cerf [10].This tells us that the connecting map A smoothing construction [19] allows us to perturb this map to be globally smooth, not affecting the the restriction of the map to the boundary of [0, 1] × D n−1 .It is with this smoothing that we apply the isotopy-extension construction.Specifically, this smoothing argument tells us the subspace of ΩEmb(D n−1 , D n ) such that the associated map [0, 1] × D n−1 → D n is smooth, this subspace has the same homotopy-type as ΩEmb(D n−1 , D n ).There is an alternative approach that is formally analogous to the result that the loop space of a manifold has the same homotopytype as its subspace of smooth loops.We view Emb(D n−1 , D n ) as a smooth Banach or Fréchet manifold (depending on the order of differentiability of the embeddings, C k with k finite or infinite respectively).From this perspective a smooth map [0 This has been made precise in several places in the literature, see [20] or [28].
We can justify why scanning Diff( To extrapolate, let Emb ν (D j−1 , N) denote the space of smooth embeddings of D j−1 in N such that the boundary is sent to the boundary in a prescribed manner, and the embedding comes equipped with a normal vector field (standard on the boundary), then we have a restriction (Serre) fibration The total space is the space of smooth embeddings of HD j in N such that the round face is sent to ∂N in a prescribed manner.The space νD j−1 indicates an open tubular neighborhood in N corresponding to the base-point element of Emb ν (D j−1 , N).We keep track of the normal vector field in the base space, as otherwise the fiber would be an embedding space where the discs are not neatly embedded.One can of course argue the above is not literally the fiber -it should be the subspace of Emb(HD j , N) that agrees with a fixed embedding on the flat boundary.That said, blowing up the flat boundary or a tubular neighborhood argument together with drilling the open tubular neighborhood completes the identification of the fibre.
PSfrag replacements If one drills a tubular neighborhood of a linearly embedded D j−1 → D n one has a manifold diffeomorphic to S n−j × D j , which gives the equivalence The space N \ νD j−1 is N with a (j − 1)-handle drilled-out, and the embedding of D j is a canceling handle for the (j − 1)-handle, thus the (j − 1)-handle is parallel to the boundary.As another model for N \ νD j−1 , we turn the handle upside-down and think of this manifold as N ∪ H n−j , i.e.N union a (n − j)-handle.Since the handle attachment is trivial, this manifold is diffeomorphic to (S n−j × D j )♮N .From this perspective, the embeddings of Emb(D j , (S n−j × D j )♮N) can be thought of as a space of embeddings of cocores for the (n − j)-handle attachment of the boundary connectsum (S n−j × D j )♮N , i.e. these cocores are allowed to reach into the N summand.
This last interpretation is perhaps the most convenient for stating the homotopy-equivalence ΩEmb ν (D j−1 , N) ≃ Emb(D j , (S n−j × D j )♮N) as the boundary condition on the latter embedding space sends ∂D j to {p} × ∂D j ⊂ S n−j × D j .By design, the embeddings in Emb ν (D j−1 , N) are isotopically trivial on the boundary S j−2 → ∂N .
Theorem 2.1 There is a homotopy-equivalence where Emb ν (D j−1 , N) is the space of smooth embeddings of D j−1 in N such that the pre-image of the boundary of N is the boundary of D j−1 .The embedding of ∂D j−1 is required to be a fixed embedding, and isotopically trivial, i.e. bounding an embedded D j−1 → ∂N .The base-point of Emb ν (D j−1 , N) can be chosen to be any embedding that is parallel to an embedding in ∂N (rel ∂).The ν indicates the embedding comes equipped with a normal vector field, standard on the boundary.The space Emb(D j , (S n−j × D j )♮N) is a space of cocores for the handle attachment (S n−j × D j )♮N = N ∪ H n−j , i.e. it is the space of smooth embeddings of D j in (S n−j × D j )♮N such that the boundary of D j is sent to { * } × ∂D j where * ∈ S n−j is some point disjoint from the mid-ball of the boundary connect-sum.
Alternatively, one could express the theorem in the 'reductionist' form . by writing M = (S n−j × D j )♮N , then N = M ∪ H n−j+1 , i.e. we derive N from M by adding a canceling handle.Thus for the homotopy-equivalence to hold we need M to admit a canceling handle, i.e. for an element f ∈ Emb(D j , M) the restriction to the boundary is an embedding f |∂D j : S j−1 → ∂M and there must admit an embedding S n−j → ∂M with a trivial normal bundle that transversely intersects f |∂D j in a single point.In the recent 'light-bulb theorem' literature the embedded S n−j is simply called a transverse sphere [13].This version of Theorem 2.1 appears in [21].
A homotopy-equivalence can be expressed as a map in either direction.The map ΩEmb ν (D j−1 , N) → Emb(D j , N \ νD j−1 ) is induced by isotopy extension i.e. one lifts the element of ΩEmb ν (D j−1 , N) to a path in Emb(HD j , N), starting at the base-point of Emb(HD j , N). Drilling the flat face from the endpoint of this path gives the element of Emb(D j , N \ νD j−1 ).
The map back Emb(D j , N \ νD j−1 ) → ΩEmb ν (D j−1 , N) involves thinking of D j as fibered by parallel copies of D j−1 and taking those restrictions, and composing with the inclusion N \ νD j−1 → N .The paper [4] gives a detailed account in the Emb(HD j , D n ) case, and [21] gives a detailed account using the 'reductionist' perspective for Emb(HD j , N).

Proposition 2.2
The co-dimension 2 scanning map induces a split injection on all homotopy and homology groups, for n ≥ 2. The map admits a left homotopyinverse.
When n > 3 the forgetful map Emb ν (D n−2 , D n ) → Emb(D n−2 , D n ) is a homotopy-equivalence, since the fiber has the homotopy-type of Ω n−2 S 1 .When n = 2 or n = 3, the double-looping of the map Given that the unit normal fibers are copies of S 1 , we can discard the normal vector fields, i.e. the forgetful map The left homotopy-inverse of the map in the middle comes from thinking of the universal cover of S 1 × D n−1 as a copy of R × D n−1 which could also be thought of as as D n remove two points on its boundary, i.e. we have a map back Emb( Since the two maps on the ends are homotopy-equivalences, this gives us the result. Cerf's proof of Smale's theorem (Corollary 2.3) is also highlighted in [24] §6.2.4.When the codimension of the embeddings are three or larger, sharp connectivity estimates for the scanning map exist.See for example [7] pgs.23-25, and the initial pages of Goodwillie's Ph.D thesis [14].The paper [16] also includes a detailed analysis of scanning maps for spaces of concordence embeddings, when the co-dimension is at least three.
The deloopings of the spaces Diff(D n ) and Emb(D j , D n ) are studied in [32] and [33].It would be interesting to see if there are analogous retraction results for the deloopings of the scanning maps Diff(D n ) → Ω n−j Emb f r (D j , D n ).It is perhaps unlikely, but it is a basic question that deserves investigation.

Theorem 2.4
The scanning map Emb( is the inclusion portion of a homotopy-retraction, i.e. it induces split injections on all homotopy-groups for all n ≥ 2. When n > 2 the ν can be dropped from the target space, i.e. the theorem remains true for embeddings without a normal vector field.
Proof By Theorem 2.1, scanning gives us a homotopy equivalence Emb( by attaching a trivial 1-handle to D n , i.e. thinking of S 1 × D n−1 as D n union a 1-handle.This inclusion is the inclusion portion of a homotopy-retract, i.e. it has a left homotopy-inverse.The left homotopy inverse comes from lifting an embedding D n−2 → S 1 × D n−1 to the universal cover, which we identify with a copy of D n with two points removed from the boundary.
The proof of the above theorem is largely a duplicate of the proof of Theorem 2.2.Similarly, this argument allows us to identify the map Emb( with the scanning map.
Notice when n = 2, the above scanning map is a homotopy-equivalence by Gramain [17].When n = 3 it follows by Hatcher's work [18] that the scanning map is a homotopy-equivalence, indeed, both spaces are contractible.
When n ≥ 4 far less is known about such scanning maps.In [4] and [5] the mapping-class group π 0 Diff(S 1 × D 3 ) was shown to be not finitely generated via the map [4].The study of our scanning map ), i.e. the loop space functor applied to the scanning map preprint One might attempt to apply the reductionist version of Theorem 2.1 to construct a homotopyequivalence Emb(D 2 , ), but since the boundary circle for the embeddings of Emb(D 2 , S 1 × D 3 ) is homologically trivial, it does not have the required 2sphere intersecting the embedding transversely in a single point.Alternatively, the embeddings in Emb(D 2 , S 1 × D 3 ) are not the cocore of a 2-handle attachment, so we can not appeal to the primary version of Theorem 2.1, either.That said, we do know that the map Emb(D 2 , ) is homotopically non-trivial ( [4], [5]) as the induced map on π 1 maps onto an infinitely-generated subgroup, so further study of these scanning maps is warranted.
The work of Fresse-Turchin-Willwacher [12] describes the delooping of the homotopy-fiber of the map from embeddings to immersions Emb(D j , D n ) → Imm(D j , D n ) ≃ Ω j V n,j , giving a fairly concrete description of its rational homotopy-type in the language of graph complexes when n − j > 2. In principle this should give us some useful information on the co-dimension one scanning map Emb(D j , D n ) → ΩEmb(D j−1 , D n ) in rational homotopy, although our lack of understanding of the induced map Emb(D j , D n ) → Imm(D j , D n ) in rational homotopy (when j > 1) may be a limiting factor at present.A related topic is the 'Freudenthal Suspension map' Emb(D 1 , D n ) → ΩEmb(D 1 , D n+1 ) [3] which is defined via two canonical unknotting operations.This map is known to be zero on rational homotopy (unpublished at this time), yet the map itself could potentially be homotopically non-trivial.

Bott handles and miscellany
The homotopy-equivalence Diff(D n ) ≃ ΩEmb(D n−1 , D n ) can be extrapolated to a homotopyequivalence Diff(I × N) ≃ ΩEmb({ 1 2 } × N, I × N), and scanning maps Whereas the scanning of Section 2 could be viewed as an argument where the intermediate space is that of the space of canceling handles, i.e. vanilla Morse theory, the scanning above has intermediate space the space Bott-style canceling handles, i.e. the kinds of handles that occur with Bott-style Morse functions (functions on manifolds where the critical point sets are manifolds and the Hessian is non-degenerate on these critical submanifolds [1]).For Bott-style Morse functions 'handle' attachments are disc bundles over manifolds, whereas in standard Morse theory one attaches disc bundles over points, i.e. plain discs.Specifically, an adjunction where one attaches a disc-bundle over M, M ⋉ D k to another manifold N along an embedding M ⋉ ∂D k → ∂N is what is called a Bott-style handle attachment [1], as these sorts of attachments occur for Bott-type Morse functions, i.e. functions W → R whose critical points are manifolds and the Hessian is nondegenerate on the normal bundle fibers.Bott-style Morse functions typically occur when functions have symmetry, for example the trace of a matrix is a Bott-style Morse function on the orthogonal group O n .The critical points of this function are the square roots of the identity matrix I , thus copies of Grassman manifolds.As a concrete example, the trace functional expresses SO 3 as the tautological line bundle over RP 2 union a 3-handle.
The analogue to Theorem 2.1 in the Bott case has the form of a homotopy-equivalence Given that our scanning maps are highly structured, they would appear to be a potentially useful device for exploring the homotopy-types of diffeomorphism groups like Diff(D n ), Diff(S 1 × D n ) and generally product manifolds Diff(N × D k ), in particular for studying spaces of pseudoisotopies.From this perspective there is perhaps a similarly overlooked element of Embedding Calculus [29] [36] that is relevant.
For example, given a manifold M, let O k (M) be the category of open subsets of M diffeomorphic to a disjoint union of at most k open balls, arrows given by inclusion maps.Given U ∈ O k (M) let F(U) be Emb(U × D j , M × D j ), i.e. smooth embeddings of U × D j in M × D j that restrict to the standard inclusion on U × ∂D j .The k-th stage of the Taylor tower could be taken to be From this perspective, the scanning map is the evaluation map to the first stage of the Taylor tower.Higher stages of the Taylor tower are built from spaces of generalized string links (in the sense that the Goodwillie-Weiss-Klein embedding calculus is built from configuration spaces), and similarly the layers will be a relative section space.This Taylor tower maps to the GWK-Taylor tower so it should converge when the co-dimension of the embeddings are sufficiently large.Minimally from the above it will have embeddings as a homotopy retract.The rate of convergence of this Taylor tower we suspect will often be greater -for example by Cerf's theorem Diff(D n ) ≃ ΩEmb(D n−1 , D n ) the first stage when M = I is homotopy-equivalent to Diff(I × D j−1 ) ≃ Diff(D j ).The potential for this framework is that it may provide more manageable inductive steps for practical computations of homotopy and homology groups of embedding spaces, as one is no longer comparing an embedding space directly with configuration spaces.Spaces of string links have been the subject of some recent investigations by Koytcheff [22], Turchin and Tsopméné [35] [22], including a description of some of their lowdimensional homotopy groups [22] as well as an operad action [23], so we may not be far removed from being able to analyze these string link Taylor towers.
String links appear in two essential ways in both [4] and [5].Specifically, barbell diffeomorphism families are the induced diffeomorphisms coming from the low-dimensional homotopy groups of spaces of 2-component string-links.Moreover, the map we use to detect our diffeomorphisms of S 1 × D n−1 has the form Diff(S 1 × D n−1 ) → Ω n−2 Emb(D 1 , S 1 × D n−1 ).If we take the lifts of an element of Emb(D 1 , S 1 × D n−1 ) to the universal cover, we get an equivariant, infinite-component string link in R × D n−1 .Thus string links would appear to be a relatively efficient machine for investigating embedding spaces and diffeomorphism groups.It would be very interesting to see the relative rate of convergence of the above Taylor towers, compared to the standard Embedding Calculus [15].
A small comment on the relationship between the restriction maps Diff(S n ) → Emb(S j , S n ) and the Cerf half-disc fibrations.When j < n these fibrations are null-homotopic via a 'shrinking support' argument [3].This is closely related to the half-disc fibration.Specifically, if we replace the above diffeomorphism group and embedding space with their 'long' version, and require the embeddings to have trivialized normal bundles we have the fibration Diff(D n ) → Emb f r (D j , D n ).This fibration has fiber homotopy-equivalent to Diff(S n−j−1 × D j+1 ).There is a cancelling-handle homotopy-equivalence Lastly, let Emb f r (HD j+1 , D n ) be the half-disc embedding space where the half-discs are equipped preprint with trivialized normal bundles.Then we have a fibre sequence Like in the unframed case, the space Emb f r (HD j+1 , D n ) is contractible.
This gives us a little commutative diagram of homotopy fiber sequences (three top vertical maps fibrations, three rightmost horizontal maps are also fibrations) i.e. we are asserting that the fibration Diff( ) but where we have inserted a trivial Diff(D n ) factor in the total space and fiber.

The Schönflies monoid
We end with the observation, implicit in [4], that the monoid π 0 Emb(S n−1 , S n ) using the connectsum operation, that this is a group for all n ≥ 2, as it is unclear if a proof of this statement exists in the literature.For n = 4 this group is known to be isomorphic to π 0 Diff(D n−1 ).In dimension n = 4 the Schönflies problem is equivalent to stating this group is trivial.
The connect-sum operation on π 0 Emb(S n−1 , S n ) has a description as a relative surgery (i.e.performing surgery on both the ambient manifold and submanifold at the same time).One embeds pairs (D n , D n−1 ) in the pairs (S n , f (S n−1 )) and (S n , g(S n−1 )) respectively.Given that our embeddings are parametrized this requires a linearization operation relative to the functions f and g about the embeddings D n−1 → f (S n−1 ) and D n−1 → g(S n−1 ) respectively, as well as an identification of S n #S n with S n .
To minimize the overhead of formalism we will assume the homotopy-equivalence [3] Emb which follows from a linearization argument.
The space Emb(D n−1 , D n ) can be thought of as the smooth embeddings R n−1 → R n that agrees with the standard inclusion {0} × R n−1 ⊂ R n outside of D n−1 , and maps D n−1 into D n .We endow Emb(D n−1 , D n ) with a binary operation (indeed many such) by stacking embeddings.To stack two elements of Emb(D n−1 , D n ) one needs rescaling and translation to make the operation precise [3].
If one combines all such operations on has an action of the operad of (n − 1)-discs on Emb(D n−1 , D n ).
The connect-sum operation is the induced monoidal structure on π 0 Emb(D n−1 , D n ).The neutral element is the linear embedding.This connect-sum operation generalizes directly to all embedding spaces π 0 Emb(S j , S n ).When n = j + 2 it is the classical connect-sum operation on co-dimension two knots, and when j = n it is the composition operation on π 0 Diff(S n ).
The proof is a small variation on the proofs of Proposition 2.2 and Theorem 2.4.The inclusion from Proposition 2.2 is compatible with stacking, i.e. it induces a map of monoids on path-components.The space Emb(D n−1 , S 1 × D n−1 ) has the homotopy-type of ΩEmb ν (D n−2 , D n ) by Theorem 2.1.The space ΩEmb ν (D n−2 , D n ) has two stacking operations, i.e. one can 'stack' using the loop-space parameter, or stack using the analogous stacking operation on the space Emb ν (D n−2 , D n ).These two operations are homotopic.In introductory algebraic topology courses, one uses this type of argument to show the fundamental group of a topological group must be abelian.It is often called an Eckmann-Hilton argument.Another way to say this is that the space ΩEmb ν (D n−2 , D n ) has an action of the operad of 2-cubes, where the action restricts to either concatenation construction, depending on the position of the cubes.When n = 1, the set π 0 Emb(S 0 , S 1 ) is also known to be a group, as it has only a single-element.The group π 1 Emb ν (D n−2 , D n ) is known to be non-trivial when n = 4 [4] although all presentlyknown elements map to zero in π 0 Emb(D n−1 , D n ).
The homomorphism π 1 Emb ν (D n−2 , D n ) → π 0 Emb(D n−1 , D n ) has this description.Take a linearlyembedded copy of HD n−1 in D n , i.e. the half-disc in D n−1 × {0} ⊂ D n .Given a loop of embeddings of D n−2 (with normal vector field) in D n , lift that path of embeddings to a path in Emb(HD n−1 , D n ) that begins at the linear embedding.At the end of this path, we have a smooth embedding HD n−1 → D n which agrees with our standard inclusion on the boundary, including its normal derivative.Via a small isotopy, we can ensure this embedding HD n−1 → D n agrees with the standard inclusion in a neighborhood of the boundary.Drill the flat face of the embedded HD n−1 from D n , this results in a copy of S 1 × D n−1 together with a smoothly-embedded D n−1 → S 1 × D n−1 which agrees with the standard inclusion {1} × D n−1 ⊂ S 1 × D n−1 on the boundary.Lift this embedding to the universal cover of S 1 × D n−1 and identify the universal cover with a subspace of D n ( D n with two boundary points removed).This embedding D n−1 → D n is the value of our map π 1 Emb ν (D n−2 , D n ) → π 0 Emb(D n−1 , D n ).
There is a Kervaire-Milnor style argument that the monoid π 0 Emb(S n−1 , S n ) has inverses.Given an embedding f : S n−1 → S n drill a small open ball from S n−1 and consider a tubular neighborhood of this manifold.It is diffeomorphic to D n−1 × I , and so the boundary of this manifold is diffeomorphic to the connect-sum of f (S n−1 ) with its mirror-reverse.Since the embedding preprint bounds a copy of D n−1 × I ≃ D n (after rounding corners), we have that f (S n−1 )# f (−S n−1 ) is standard, thus f and its mirror-reverse are inverses of each other.The relative advantage of Theorem 4.1 is that it provides a group π 1 Emb ν (D n−2 , D n ) that maps onto the Schönflies monoid π 0 Emb(S n−1 , S n ), i.e. it gives us a prescription for how one can construct all Schönflies spheres.
The resolution of the Schönflies problem in dimension different from four gives another argument that the monoid of Schönflies spheres π 0 Emb(S n−1 , S n ) is a group, when n = 4, as this tells us the reparametrizations of the linear embedding gives an onto homomorphism π 0 Diff(S n−1 ) → π 0 Emb(S n−1 , S n ).The triviality of π 0 Emb(S n−1 , S n ) when n = 4 is equivalent to the Schönflies Problem, as Diff(D 3 ) is contractible [18].
a simple description thinking of Diff(D n ) as the diffeomorphisms of R n with support contained in D n .One then considers D n as a subset of I × D n−1 .Restriction to the fibers {t} × D n−1 gives the 1-parameter family of embeddings of D n−1 into D n .After suitably translating and scaling the embedding family to have fixed boundary conditions, this is an element of ΩEmb(D n−1 , D n ).
a homotopy-equivalence via a rounding-the-corners argument.The definition of the connecting map ΩEmb(D n−1 , D n ) → Diff(HD n ) comes from observing that an element of ΩEmb(D n−1 , D n ) via currying can be thought of as a map [0, 1] × D n−1 → D n which is continuous globally, but smooth on the {t} × D n−1 fibers.

1 . 2 . 3 ( 3 .
a homotopy-equivalence, as the fiber has the homotopy-type of Ω 2 Ω n−2 S Corollary Smale) Diff(D 2 ) is contractible, i.e.Diff(S 2 ) has the homotopy-type of its linear subgroup O Proof (of Proposition 2.2) The proof follows from forming a composite of functions involving the homotopy-equivalence Diff(D n ) → ΩEmb(D n−1 , D n ) (i.e.Theorem 2.1, N = D n , j = n) with the induced map on loop spaces from Theorem 2.1, where N = D n and j = n − 1,
It would be more precise to to say we have general theory when n < 4 but when n ≥ 4, separating the path-components of Emb(D n−2 , D n ) is a difficult mathematical problem.Similarly, little is known about π 1 Emb(D 2 , D 4 ) at present.If one allows the embeddings to have trivialized normal bundles (normal framings) one has scanning maps of the form Diff