Quantitative algebraic topology and Lipschitz homotopy

Abstract

We consider when it is possible to bound the Lipschitz constant a priori in a homotopy between Lipschitz maps. If one wants uniform bounds, this is essentially a finiteness condition on homotopy. This contrasts strongly with the question of whether one can homotop the maps through Lipschitz maps. We also give an application to cobordism and discuss analogous isotopy questions.amenable group | uniformly finite cohomology T he classical paradigm of geometric topology, exemplified by, at least, immersion theory, cobordism, smoothing and triangulation, surgery, and embedding theory is that of reduction to algebraic topology (and perhaps some additional pure algebra). A geometric problem gives rise to a map between spaces, and solving the original problem is equivalent to finding a nullhomotopy or a lift of the map. Finally, this homotopical problem is solved typically by the completely nongeometric methods of algebraic topology (e.g., localization theory, rational homotopy theory, spectral sequences).Although this has had enormous successes in answering classical qualitative questions, it is extremely difficult [as has been emphasized by Gromov (1)] to understand the answers quantitatively. One general type of question that tests one's understanding of the solution of a problem goes like this: Introduce a notion of complexity, and then ask about the complexity of the solution to the problem in terms of the complexity of the original problem. Other possibilities can involve understanding typical behavior or the implications of making variations of the problem.In this task, the complexity of the problem is often reflected, somewhat imperfectly, in the Lipschitz constant of the map. Indeed, one can often view the Lipschitz constant of the map as a measure of the complexity of the geometric problem.For concreteness, let us quickly review the classical case of cobordism, following Thom (2). Let M be a compact smooth manifold. The problem is: When is M n the boundary of some other compact smooth W n+1 ? There are many possible choices of manifolds in this construction, such as oriented manifolds, manifolds with some structure on their (stabilized) tangent bundles, Piecewise linear (PL) and topological versions, and so on. However, for now, we will confine our attention to this simplest version.Thom (2) embeds M m in a high-dimensional Euclidean space M ⊂ S m+N−1 ⊂ R m+N and then classifies the normal bundle by a map νM: νM → E(ξ N ↓ Gr(N, m + N)) from the normal bundle to the universal bundle of N-planes in m + N space. Including R m+N into S m+N via one-point compactification, we can think of νM as being a neighborhood of M in this sphere (via the tubular neighborhood theorem) and extend this map to S m+N if we include E(ξ N ↓ Gr(N, m + N)) into its one-point compactification E(ξ N ↓ Gr(N, m + N))^, the Thom space of the universal bundle. Let us call this mapThom (2) , extending the embedding of M into S m+N . Extending Thom's construction over this disk gives a nullhomotopy of Φ M . Conversely, one uses the nullhomotop...

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“…Given a suitable measure of complexity for the maps in question, a typical question is: What is the relation between the complexity of a given nullhomotopic map f : X → Y and the minimum complexity of a nullhomotopy witnessing this? In quantitative homotopy theory, complexity is often quantified by assuming that the spaces are metric spaces and by considering Lipschitz constants (which are closely related to the sizes of the simplicial representatives of maps and homotopies [12]). …”

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