We describe a higher-dimensional generalization of edge expansion for graphs which applies to arbitrary cell complexes. This generalization relies on a type of co-isoperimetric inequality. We utilize these inequalities to analyze the topological and geometric behavior of some families of random simplicial complexes.

For a given null-cobordant Riemannian n-manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? In [Gro99], Gromov conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n.This construction relies on another of independent interest. Take X and Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y is simply connected and rationally homotopy equivalent to a product of Eilenberg-MacLane spaces: for example, any simply connected Lie group. Then two homotopic L-Lipschitz maps f, g : X → Y are homotopic via a CL-Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y .A beautiful example of this paradoxical state of affairs is the result of Nabutovsky that despite the result of Smale (proved inter alia in the proof of the high-dimensional Poincaré conjecture) that every smooth codimension one sphere in the unit n-disk (n > 4) can be isotoped to the boundary, the minimum complexity of the embeddings required in the course of such an isotopy (measured by how soon normal exponentials to the embedding intersect) cannot be bounded by any recursive function of the original complexity of the embedding. Effectively, an easy isotopy would give such a sphere a certificate of its own simple connectivity, which is known to be impossible.In other situations, such as those governed by an h-principle, a hard logical aspect of this sort does not arise. In this paper we introduce some tools of quantitative algebraic topology which we hope can be applied to showing that various geometric problems have solutions of low complexity.As a first, and, we hope, typical example, we study the problem, emphasized by Gromov, of trying to understand the work of Thom 1 on cobordism. Given a closed smooth (perhaps oriented) manifold, the cobordism question is whether it bounds a compact (oriented) manifold. The answer to this is quite checkable: it is determined by whether the cycle represented by the manifold in the relevant (i.e. Z or Z/2Z) homology of a Grassmannian (where the manifold is mapped in via the Gauss map classifying the manifold's stable normal bundle) is trivial.This raises two questions: the first is how the geometry of a manifold is reflected in the algebraic topological problem, and second is, how difficult is it to find the nullhomotopy predicted by the algebraic topology. As a test of this combined problem, Gromov suggested the following question: Given a manifold, assume away small scale problems by giving it a Riemannian metric whose injectivity radius is at least 1, and whose sectional curvature is everywhere between −1 and 1. These properties can be achieved through a rescaling. A manifold possessing these properties will be said to have bounded local geometry. The geometric complexity of such a manifold can be measured by its volume.If M is a smooth compact manifold, without a specified me...

We give a detailed and easily accessible proof of Gromov's Topological Overlap Theorem. Let X be a finite simplicial complex or, more generally, a finite polyhedral cell complex of dimension d. Informally, the theorem states that if X has sufficiently strong higher-dimensional expansion properties (which generalize edge expansion of graphs and are defined in terms of cellular cochains of X ) then X has the following topological overlap property: for every continuous map X → R d there exists a point p ∈ R d that is contained in the images of a positive fraction μ > 0 of the d-cells of X . More generally, the conclusion holds if R d is replaced by any d-dimensional piecewise-linear manifold M, with a constant μ that depends only on d and on the expansion properties of X , but not on M.

The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in H 2 (X, Z/2). We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m = n 2+α for α < 1/2, then we show that the 2-girth is at most 4n 2−2α and we prove the existence of complexes with 2-girth at least c α,ǫ n 2−2α−ǫ . On the other hand, if α > 1/2, the 2-girth is at most C α . So there is a phase transition as α passes 1/2.Our results depend on a new upper bound for the number of combinatorial types of triangulated surfaces with v vertices and f faces.

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