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Home , Algebraic topology, Geometric topology, Homotopy sphere

Quantitative algebraic and Lipschitz

Steve Ferrya and Shmuel Weinbergerb,1

aDepartment of , Rutgers University, Piscataway, NJ 08854; and bDepartment of Mathematics, University of Chicago, Chicago, IL 60637

Edited by Alexander Nabutovsky, University of Toronto, Toronto, ON, Canada, and accepted by the Editorial Board January 14, 2013 (received for review May 16, 2012)

We consider when it is possible to bound the Lipschitz constant 1. M bounds iff ΦM is homotopic to a constant map. If M is the + + a priori in a homotopy between Lipschitz maps. If one wants uniform of W, one embeds W in Dm N 1, extending the em- + bounds, this is essentially a finiteness condition on homotopy. This bedding of M into Sm N. Extending Thom’s construction over contrasts strongly with the question of whether one can homotop this gives a nullhomotopy of ΦM. Conversely, one uses the the maps through Lipschitz maps. We also give an application to nullhomotopy and takes the transverse inverse of Gr(N, m + and discuss analogous isotopy questions. N) under a good smooth approximation to the homotopy to the constant map ∞ to produce the nullcobordism. amenable | uniformly finite 2. ΦM is homotopic to a constant map iff νM*([M]) = 0 ∈ Hm(Gr (N, m + N); Z2). This condition is often reformulated in – he classical paradigm of , exemplified by, terms of the vanishing of Stiefel Whitney numbers. Tat least, immersion theory, cobordism, smoothing and tri- It is now reasonable to define the complexity of M in terms of angulation, surgery, and theory is that of reduction to the volume of a Riemannian metric on M whose local structure (and perhaps some additional pure ). A is constrained [e.g., by having curvature and injectivity radius geometric problem gives rise to a map between spaces, and solving bounded appropriately, jKj ≤ 1, inj ≥ 1, or, alternatively, by fi the original problem is equivalent to nding a nullhomotopy or counting the number of simplices in a triangulation (again a of the map. Finally, this homotopical problem is solved whose local structure is bounded)]. In that case, supposing M typically by the completely nongeometric methods of algebraic bounds [i.e., the conclusion of Thom’s theorem holds (2)], can topology (e.g., localization theory, rational , we then bound the complexity of the that M bounds? spectral ). Note that the Lipschitz constant of ΦM isrelatedtothe Although this has had enormous successes in answering classical complexity that we have chosen, in that the curvature controls fi qualitative questions, it is extremely dif cult [as has been em- the local Lipschitz constant of ν, but that there is an additional phasized by Gromov (1)] to understand the answers quantitatively. global aspect that comes from the embedding. It is understood ’ One general type of question that tests one s understanding of the that because of expander graphs, for example, one might have to solution of a problem goes like this: Introduce a notion of com- have extremely thin tubular neighborhoods when one embeds plexity, and then ask about the complexity of the solution to the a manifold in a high-dimensional . This increases problem in terms of the complexity of the original problem. Other the Lipschitz constant accordingly when extending νM to ΦM.We possibilities can involve understanding typical behavior or the shall refer to the Lipschitz constant of a Thom map for M as the implications of making variations of the problem. Thom complexity of M. In this task, the complexity of the problem is often reflected, Of course, we can separate the problems and deal with the somewhat imperfectly, in the Lipschitz constant of the map. In- problem of understanding the complexity of embedded coboun- deed, one can often view the Lipschitz constant of the map as daries based on a complexity involving jKj and τ, which is, by a measure of the complexity of the geometric problem. definition, the feature size of computational topology, the smallest For concreteness, let us quickly review the classical case of size at which normal exponentials to M in a high-dimensional cobordism, following Thom (2). Let M be a compact smooth collide. Doing this, studying the Lipschitz constant of Φ n M manifold. The problem is: When is M the boundary of some essentially is the same as considering sup(jKj,1/τ) In this paper, n+1 other compact smooth W ? that is the approach we take; however, an alternative approach There are many possible choices of in this con- avoiding the issues associated with choosing an embedding can be struction, such as oriented manifolds, manifolds with some based on the method of Buoncristiano and Hacon (3). structure on their (stabilized) tangent bundles, Piecewise linear The second issue, then, is item (4). How does one get in- (PL) and topological versions, and so on. However, for now, we formation about the size of the nullhomotopy? Algebraic to- will confine our attention to this simplest version. pology does not directly help us because it reasons algebraically m Thom (2) embeds M in a high-dimensional Euclidean space involving many formally defined groups and their structures, m+N−1 m+N M ⊂ S ⊂ R and then classifies the normal bundle by constantly identifying objects with equivalence classes. Despite N amapνM: νM → E(ξ ↓ Gr(N, m + N)) from the normal bundle this, Gromov (1) has suggested the following. to the universal bundle of N-planes in m + N space. Including + + Rm N into Sm N via one- compactification, we can think of Optimistic Possibility νM as being a neighborhood of M in this sphere (via the tubular If X and Y are finite complexes and Y is simply connected, there + neighborhood theorem) and extend this map to Sm N if we in- is a constant K, such that if f, g: X → Y are homotopic Lipschitz clude E(ξN ↓ Gr(N, m + N)) into its one-point compactification E(ξN ↓ Gr(N, m + N))^, the of the universal bundle. Letuscallthismap Author contributions: S.F. and S.W. performed research and wrote the paper. À Á The authors declare no conflict of interest. m+N N ΦM : S → E ξ ↓GrðN; m + NÞ ̂: This article is a PNAS Direct Submission. A.N. is a guest editor invited by the Editorial Board. Thom (2) shows, among other things, that: 1To whom correspondence should be addressed. E-mail: [email protected].

19246–19250 | PNAS | November 26, 2013 | vol. 110 | no. 48 www.pnas.org/cgi/doi/10.1073/pnas.1208041110 Downloaded by guest on September 26, 2021 < ; ...; > maps with Lipschitz constant L, they are homotopic through KL- sending the barycenter of each ei0 eik to + ... + RN+1 SPECIAL FEATURE Lipschitz maps. ei0 eik , which is a vertex of the unit cube in . The rest of this paper makes some initial comments regarding Extending linearly throws our complex onto a subcomplex of this problem. We shall first discuss a stronger problem, that of the unit cube via a whose bi-Lipschitz con- constructing a KL-Lipschitz homotopy. We give necessary and stant is controlled by d. It is now an easy matter to subdivide the sufficient conditions for there to be a KL-Lipschitz homotopy be- cube into smaller congruent pieces. Here, by the “bi-Lipschitz tween homotopic L-Lipschitz maps with constant K only depending constant,” we mean the sup of the Lipschitz constants of the fi on dim(X). The hypotheses of this situation are suf cient for un- embedding and its inverse. oriented cobordism and give a linear increase of Thom complexity We now proceed to the converse. Assuming that Y is a finite for the problem of unoriented smooth embedded cobordism be- complex with at least one nonfinite , we will fi fi cause the Thom space is a nite complex with nite homotopy show that there is a nullhomotopic Lipschitz map Rn → Y for groups in that case. some n that is not Lipschitz nullhomotopic. The argument shows We also give some contrasting results, where, essentially for that there is no constant C(n) that works for all subcubes of Rn. homological reasons, one cannot find Lipschitz , but ~ Suppose that π1(Y)isinfinite. We give Y the metric homotopies through Lipschitz maps are possible. obtained by pulling the metric in Y up locally. Then, by König’s Finally, we make some comments and about the ~ lemma (7), the 1-skeleton of Y contains an infinite path R that related problem of isotopy of in both the Lipschitz ~ has infinite diameter in Y. Let X = [0, ∞) with the usual sim- setting and the Ck setting. Both of these contrast with the C1 plicial structure. The map f: X → R is 1-Lipschitz and nullho- setting considered by Gromov (5). motopic. If there were a Lipschitz homotopy F from f to a j × Constructing Lipschitz Homotopies constant, the length of each path~ F {x} I would then be boun- Theorem 1. Let Y be a finite complex with finite homotopy groups in ded independent of x.LiftingtoY would give uniformly bounded ≤d. Then, there exists C(d) so that for all simplicial path paths from f(x) to the basepoint for all x, contradicting the fact fi metric spaces X with X ≤ d, such that the restriction of that R has in nite diameter. π fi ≤ − ≥ the metric to each simplex of X is standard, if f, g: X → Y are Assume that k(Y)is nite for k n 1, n 2, and assume that π fi fi homotopic L-Lipschitz maps with L ≥ 1, there is then a C(d)L- n(Y)isin nite. Let us rst consider the case in which Y is simply Lipschitz homotopy F from f to g. Conversely, if a Lipschitz connected. By Serre’s extension of the , the fi fi homotopy always exists, the homotopy groups of Y are then finite in rst in nite homotopy group of Y is isomorphic to the corre- MATHEMATICS dimensions ≤d. sponding group modulo torsion; thus, we have maps Proof: Sn → Y → K(Z, n), such that the generator of Hn(K(Z, n)) ≅ Z We begin with the positive direction; the argument is a slight pulls back to a generator of Hn(Sn). The image of Y is compact; adaptation of that given by Siegel and Williams (6), which can be thus, the image of Y lies in a finite skeleton of K(Z, n), which, for referred to for more detail and for a generalization. We may definiteness, we could take to be a finite symmetrical product of k n n assume that Y is a compact smooth manifold embedded in R+ for S s. The composition is homotopic to a map into S ⊂ K(Z, n), − some k, with ∂Y ⊂ Rk 1. The normal bundle projection from and this composition Sn → Sn has degree one. a tubular neighborhood N of Y to Y retracts N to Y by a Lipschitz Consider the map Rn → Tn → Sn, where the last map is the map, which we may assume to have a Lipschitz constant less than degree one map obtained by squeezing the complement of the 2. Choose an e > 0 so that the straight line connecting points y′ top cell to a point. The n-form representing the generator of and y″ in Y is contained in N whenever d(y′, y″) < e. n Z ℓ H (K( , n)) pulls back to a form cohomologous to the volume The space of μ-Lipschitz maps ∂Δ → Y is compact for each ℓ form on Sn, and thus to a closed form on Tn cohomologous and μ > 0; thus, for each ℓ ≤ d + 1 and μ > 0, we can choose to a positive multiple of the volume form. This pulls back to fi ϕ ∂Δℓ → μ a nite collection { i,ℓ,μ: Y}of -Lipschitz maps that is aclosedformonRn boundedly cohomologous to a multiple of e μ ∂Δℓ → -dense in the space of all -Lipschitz maps Y. For each thevolumeform. Δℓ such map that extends to , choose Lipschitz extensions, Suppose that the map Rn → Y is Lipschitz nullhomotopic. The ϕ : Δℓ → ϕ i;ℓ;μ;j Y in each homotopy class of extensions of i,ℓ,,μ. Lipschitz nullhomotopy can be approximated by a smooth Lip- Because the homotopy groups of Y are finite, there are only fi- ϕ ∂Δℓ → schitz nullhomotopy (e.g., ref. 4). By the proof of the Poincaré nitely many such extensions for each i,ℓ,μ.Iff: Y is Rn α μ ϕ lemma, this shows that the volume form on is d for some -Lipschitz, f is homotopic to some i0;ℓ;μ by a linear homotopy in α ’ k bounded form . This is easily seen to be impossible by Stokes R+. Retracting this into Y along the normal bundle gives a 2μ- ℓ theorem, because the integral of the volume form over an m × m × Lipschitz homotopy from f to ϕ ;ℓ;μ.Iff extends over Δ , some n i0 ...× m cube grows like m in m, whereas the integral of α over the ϕ ;ℓ;μ; can then be pieced together with the Lipschitz homotopy n−1 i0 j ℓ boundary grows like m . This contradiction shows that the from f to ϕ ;ℓ;μ to give a Lipschitz extension of f j∂Δ . i0 Rn → n → n → The result of this is that for every ℓ, μ, there is a ν so that if f: composition T S Y is not Lipschitz nullhomotopic. ℓ ℓ ℓ fi Δ → Y is a map with f j∂Δ μ-Lipschitz, f j∂Δ has a ν-Lipschitz In case Y has~ a nontrivial nite , the uni- ℓ extension to Δ , which is homotopic to f. It is now an induction on versal cover Y is compact and simply connected. As above, we Rn → the skeleta of X to show that there is a ν, such that if f and g are assume that our map Y is~ Lipschitz nullhomotopic. The homotopic 1-Lipschitz maps from Xd to Y, f and g are ν-Lipschitz Lipschitz~ nullhomotopy lifts to Y. Applying the argument above homotopic. The rest of the argument in the forward direction is in Y produces a contradiction. fi a rescaling and subdivision argument. If f and g are homotopic L- Block and Weinberger (8) discuss uniformly nite cohomology Lipschitz maps, L ≥ 1, as in the statement of the theorem, we can theory for manifolds of bounded . This theory uses rescale the metric and subdivide to obtain maps with Lipschitz cochains that are uniformly bounded on simplices in a tri- constant 1. After extending, we rescale back to the original metric, angulation of finite complexity or, in the de Rham version, uses obtaining the desired result. k-forms that are uniformly bounded on k-tuples of unit vectors. For the rescaling argument, it is helpful to work with cubes The argument above then shows there is an obstruction in this rather than standard simplices because subdivisions are easier to theory that must vanish in order for a nullhomotopic map to be handle. We consider X as a subcomplex of the standard simplex, Lipschitz nullhomotopic. embedded as a subcomplex of the standard N-simplex, thought Under favorable circumstances, it is also possible to use this + of as the convex hull of unit vectors in RN 1.Thereisamap theory to construct Lipschitz nullhomotopies.

Ferry and Weinberger PNAS | November 26, 2013 | vol. 110 | no. 48 | 19247 Downloaded by guest on September 26, 2021 x

C1 C2 y M

Fig. 1. A one-ended manifold.

Theorem 2. Let M be a closed orientable manifold with non- the maps used in the construction were chosen from a finite amenable fundamental group. Then, the composition collection, this nullhomotopy can be taken to be globally Lipschitz. The converse follows from the de Rham argument used in the ~ M → M → Sn proof of Theorem 1 applied to a Følner . We note that the theory of homotopies ht that are Lipschitz for is Lipschitz nullhomotopic. Here, M → Sn is the degree one map every t is quite different from the theory of Lipschitz homotopies. obtained by crushing out the n − 1-skeleton of M and sending the For instance, contracting the domain in itself shows that every n Rn → n interior of one n-simplex homeomorphically onto S − {*},asin Lipschitz map S is nullhomotopic through Lipschitz maps, the proof of the first theorem. Everything else goes to *. Conversely, whereas the construction in Theorem 1 shows that such a map if the fundamental group is amenable, this composite is then not need not be Lipschitz nullhomotopic. Lipschitz nullhomotopic. Theorem 3. If M is a closed connected manifold with infinite fun- Proof: ~ Triangulate M by pulling up a triangulation of M. The composi- damental group, the map described in Theorem 2 is nullhomotopic tion in the statement of the theorem gives an element of the nth through Lipschitz maps. ~ Proof: uniformly finite cohomology of M with coefficients in π (Sn) = Z. n We may assume that M has dimension ≥2, because the is There is a duality theorem stated by Block and Weinberger~ (8) to the effect that the nth uniformly finite cohomology of M is the only 1D example and the theorem is clearly true in this case. We may also assume that M is 1- or 2-ended, because Stallings’ isomorphic to its 0th uniformly finite homology. One of the main structure theorem for ends of groups implies that, otherwise, theorems of Block and Weinberger (9) says that the 0th uniformly π (M) is nonamenable (10). We will begin by assuming that M is finite homology of the universal cover of a manifold is trivial if 1 1-ended, which means that for any compact C ⊂ M, there is and only if the fundamental group is nonamenable. a compact C ⊂ D ⊂ M, such that any two points in M − D are Because the proof of the duality theorem quoted above is connected by an arc in M − C. By induction, we can write M as previously undocumented, we outline a proof in the case we ~ a nested union of compact sets C ⊂ C + , such that any two used. Orient the cells of M,andthereforeofM,insuchaway i i 1 points in M − Ci+1 are connected by an arc in M − Ci (Fig. 1). that the sum of the positively oriented top-dimensional cells is Shrink the support of the map M → Sn to lie in a small ball in fi alocally~ nite cycle. Consider the dual cell decomposition on the interior of a top-dimensional simplex. Here, by support, we M. Each vertex in the dual cell complex is the barycenter of mean the inverse image of Sn − {*}. Run a proper ray out to π n ~ a top-dimensional cell; thus, we can assign an element of n(S ) infinity in the 1-skeleton of M, as in the proof of Theorem 1, and to each vertex in the dual complex. By the theorem of Block connect the ray by a geodesic segment to the center of a lift of and Weinberger (9) quoted above, this chain is the boundary of the support ball. Smooth the ray by rounding angles in the~ 1- a uniformly bounded 1-chain in the dual skeleton. Assigning the skeleton, and thicken the ray to a map of ½0; ∞Þ × Dn−1 → M. fi − coef cient of each 1-simplex to the (n 1)-cell it pierces, gives Because there~ are only finitely many different angles in the 1- a uniformly bounded cochain whose coboundary is the original skeleton of M, we can assume that this thickened ray has con- cocycle, up to sign, exactly as in the classical PL proof of stant thickness larger than~ the diameter of the lifted support ball. Poincaré duality. Homotop the map M → Sn to a map that is constant on this It follows, then, that there is a uniformly bounded (n − 1)- tube by pushing the support out to infinity during the interval t ∈ cochain whose coboundary is equal to the obstruction. One uses [0,1/2]. This homotopy is Lipschitz for every t because it is this cochain exactly as in ordinary to build a smooth and agrees with the original Lipschitz map outside homotopy from the given composition to a constant map. Because a compact set for every t. Repeat this construction for every

C1 C2

M

Fig. 2. The support of the degree one map is pushed to infinity.

19248 | www.pnas.org/cgi/doi/10.1073/pnas.1208041110 Ferry and Weinberger Downloaded by guest on September 26, 2021 lift of the support ball, being~ careful to choose each~ ray so that be any larger than the number of e-balls it takes to cover the SPECIAL FEATURE if the support ball lies in M − Ci+1, the ray lies in M − Ci. Pa- space of maps with Lipschitz constant at most CL. In our situation, rameterizing these pushes to infinity to occur on intervals [1/2, 3/ if X is compact and d-dimensional, this observation would allow 4], [3/4, 7/8], etc. The resulting homotopies are constant on a Lipschitz homotopy that is roughly of size eLd . Our examples of larger and larger compact sets and converge to the constant map homotopic Lipschitz maps that are not at all Lipschitz homotopic when t → 1. Note that the Lipschitz constants of these Lipschitz thus, of course, require the noncompactness of the domain. maps are globally bounded (Fig. 2). The argument for the two-ended case is similar, except that the Some Remarks on Isotopy Classes complements of the Cis will have two unbounded components. If X and Y are manifolds, we can then consider analogous One must be careful that for i > 1, a ball that lies in an unbounded problems for embeddings rather than just maps. In his seminal fl ’ component of M − Ci+1 should be pushed to infinity in that paper, Gromov (5) used Hae iger s reduction of metastable component after dropping back no further than into Ci − Ci−1. embedding theory to homotopy theory (i.e., the theory of Remark 1: embeddings M → N when the dimensions satisfy 2n > 3m + 2) to Calder and Siegel (11) have shown that if Y is a finite complex show that a bound on the bi-Lipschitz constant of an embedding with finite fundamental group, for each n, there is then a b,such cuts the possible number of isotopy classes down to a finite that if X is n-dimensional and f, g: X → R are homotopic maps, (polynomial) number when the target is simply connected. there is a homotopy ht from f to g, such that the path {ht(x) j 0 ≤ In general, he pointed out that because of the existence of t ≤ 1} is b-Lipschitz for every x ∈ X. In the case of our map R2 → Haefliger knots, that is, infinite families of smooth embeddings − S2, such a homotopy can be obtained by lifting to S3 via the Hopf of S4k 1 ⊂ S6k, there are infinite smooth families with a bound on map and contracting the image along geodesics emanating from the bi-Lipschitz constant. a point not in the image of R2. One way to prove the general It is possible to continue this line of thought into much lower case uses a construction from Ferry (1). If Y is a finite simplicial codimension with the following theorems, by changing the cat- complex with finite fundamental group, given n > 0, Ferry’s egories (Fig. 3). The picture represents a sequence of Haefliger- theorem 2′ (1) produces a PL map q from a contractible finite knotted with bi-Lipschitz convergence to an embedding polyhedron to Y that has the approximate for n- that is bi-Lipschitz but not C1. If the Haefliger knots are replaced dimensional spaces. Taking a regular neighborhood in some by ordinary codimension 2 knots, this becomes an example high-dimensional Euclidean space gives a space PL homeo- showing that Theorem 4 is false in codimension 2. morphic to a standard ball. Composing q with this PL homeo- MATHEMATICS morphism and the regular neighborhood collapse gives a PL Theorem 4. The number of topological isotopy classes of embeddings map from a standard ball PL ball, B,toY, which has the ap- of M → N represented by locally flat embeddings with a given bi- proximate lifting property for n-dimensional spaces. If dim X ≤ Lipschitz constant is finite whenever m − n ≠ 2. n and f: X → R is a nullhomotopic map, there is a map f : X → B, This is proved in codimension >2inMaher’sdoctoraldisser- such that p ∘ f is e-close to f, with e as small as we like. Coning tation (14) by using the Browder–Casson–Haefliger–Sullivan–Wall off in B gives a nullhomotopy in B with lengths of paths bounded analysis of topological embeddings in terms of Poincaré embed- by the diameter of B. Composing with p gives a nullhomotopy of dings (15), together with the observation that any such embedding f, where the lengths of the tracks of the homotopy are bounded will be topologically locally flat using the 1-LC Flattening Theo- by the diameter of B times the~ Lipschitz constant of p. rem (ref. 16, corollary 5.7.3, p. 261), and the fact that the image n Thus, in our construction M → M → S , we can achieve Lip- has Hausdorff codimension at least 3. More precisely, let {fi}be – schitz nullhomotopies in the t-direction~ whenever n ≥ 2and a sequence of K bi-Lipschitz embeddings. Using the Arzela- Lipschitz nullhomotopies in the M-direction whenever the Ascoli theorem, we can extract a convergent subsequence that fundamental group of M is infinite, but we can achieve both remains K–bi-Lipschitz. Because the codimension is at least 3, simultaneously if and only if the fundamental group of M this image with have Hausdorff codimension 3, and therefore will is nonamenable. be 1-LC, as observed by Siebenmann and Sullivan (17). All the fis Remark 2: sufficiently C0 close to this limit will be topologically isotopic to As mentioned in the introduction, for geometric applications, it is it because they induce the same Poincaré embedding. In codi- much more useful to have Lipschitz homotopies than homotopies mension 1, a similar argument works, except that the limit cannot through Lipschitz maps. However, as pointed out by Weinberger beassumedtobelocallyflat. However, by theorem 7.3.1 of (ref. 13, pp. 102–104), it is possible to turn the latter into the Daverman and Venema (16; see also 19), any two locally flat former at the cost of increasing the length of time the homotopy embeddings C0 close enough to the limit must be isotopic; thus, takes. More precisely, if one is a situation where any two e-close the conclusion follows in this case as well. maps are homotopic, the length of the homotopy does not have to Theorem 5. The same is true in the smooth (in all codi- mensions) if one bounds the C2 norms of the embeddings. A linear homotopy between sufficiently C1-close C1 embeddings gives an isotopy of embeddings that extends to an . Thus, a sequence of C2 embeddings in which C1 converges con- tains only finitely many topological isotopy classes of embeddings. Both of these theorems then give rise to interesting quanti- tative questions, both in terms of bounding the number of embeddings and in terms of understanding how large the Lip- schitz constants must grow during the course of an isotopy. At the moment, unlike the situation for maps, we do not even see any effective bounds on the number of isotopy classes. Nevertheless, based on Gromov’s ideas (1, 5), the following seems plausible.

Fig. 3. The Lipschitz norm is uniformly bounded, yet the C2 norm neces- Conjecture 6. If N is simply connected, and dim M < dim N-2, the sarily grows. number of L–bi-Lipschitz isotopy classes of embeddings of M in N

Ferry and Weinberger PNAS | November 26, 2013 | vol. 110 | no. 48 | 19249 Downloaded by guest on September 26, 2021 then grows like a polynomial in L. Furthermore, there is a C so that Gromov’s paper (5). This gives us some hope that this special any two such embeddings that are isotopic are isotopic by an iso- case of Conjecture 6 might be accessible. topic through CL–bi-Lipschitz embeddings. We now turn to the case of hypersurfaces: dim M = dim N − 1. A very interesting case suggested by the techniques of this Remarks: paper is the following. 1. For codimension 2 embeddings of the sphere, Nabutovsky and 2 Proposition 7. Let Σn be a rational . Then, the Weinberger (18) show that one cannot tell whether a C number of isotopy classes of topological bi-Lipschitz embeddings of is isotopic to the unknot. Therefore, there is no computable + Σn in Sn k is finite for k > 2. function that can bound the size of an isotopy. fi 2. By taking tubular neighborhoods, this gives an analogous re- The niteness follows from considerations about Poincaré 1 × n−2 n > embeddings. Standard facts about spherical fibrations give fi- sult for S S in S , for n 4. niteness of the normal data, and Alexander duality enormously 3. Perhaps more interesting from the point of view of this paper restricts the homology of the complement. The number of is that the same holds true in the smooth setting even if the hypersurface has nontrivial finite fundamental group accord- homotopy types is readily bounded via analysis of k-invariants – fi to be, at most, a tower of exponentials, where the critical pa- ing to Thom (ref. 2, pp. 83 85), such that niteness of a homo- rameter is the size of the torsion homology. Obstruction theory topy group is not enough to give a quantitative estimate on then allows only finitely many possibilities for each of these the size of an isotopy. It still seems possible that a version of Conjecture 6 can saved for embeddings in codimension 1 that (when taking into account that the total space of this Poincaré “ ” embedding is a ). are incompressible (i.e., that induce injections on their We note that the homotopy theory of this situation can be fundamental group). studied one prime at a time using a suitable pullback diagram. For large enough primes, the issues involved resemble ratio- ACKNOWLEDGMENTS. S.W. was partially supported by National Science nal homotopy theory—the core homotopical underpinning of Foundation Award 1105657.

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19250 | www.pnas.org/cgi/doi/10.1073/pnas.1208041110 Ferry and Weinberger Downloaded by guest on September 26, 2021