Quantitative algebraic topology and Lipschitz homotopy Steve Ferrya and Shmuel Weinbergerb,1 aDepartment of Mathematics, 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 boundary 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 disk 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 + cobordism and discuss analogous isotopy questions. N) under a good smooth approximation to the homotopy to the constant map ∞ to produce the nullcobordism. amenable group | uniformly finite cohomology 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 geometric topology, 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 embedding theory is that of reduction to the volume of a Riemannian metric on M whose local structure algebraic topology (and perhaps some additional pure algebra). 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 lift 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 homotopy theory, we then bound the complexity of the manifold that M bounds? spectral sequences). 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 Euclidean space. 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 sphere 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 manifolds 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-point 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 Thom space 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 simplex 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 homeomorphism 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 homotopy group, 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 homotopies, but ~ Suppose that π1(Y)isinfinite. We give Y the path 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 conjectures about the ~ lemma (7), the 1-skeleton of Y contains an infinite path R that related problem of isotopy of embeddings 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 dimensions ≤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 dimension 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.