Coboundary expanders

We describe a natural topological generalization of edge expansion for graphs to regular CW complexes and prove that this property holds with high probability for certain random complexes.Comment: Version 2: significant rewrite. 18 pages, title changed, and main theorem extended to more general random complexe

On Expansion and Topological Overlap

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 $X\rightarrow \mathbf{R}^d$ there exists a point $p\in \mathbf{R}^d$ that is contained in the images of a positive fraction $\mu>0$ of the $d$-cells of $X$. More generally, the conclusion holds if $\mathbf{R}^d$ is replaced by any $d$-dimensional piecewise-linear (PL) manifold $M$, with a constant $\mu$ that depends only on $d$ and on the expansion properties of $X$, but not on $M$.Comment: Minor revision, updated reference

LIPIcs

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 → ℝd there exists a point p ∈ ℝd whose preimage intersects a positive fraction μ > 0 of the d-cells of X. More generally, the conclusion holds if ℝd is replaced by any d-dimensional piecewise-linear (PL) manifold M, with a constant μ that depends only on d and on the expansion properties of X, but not on M

On expansion and topological overlap

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 (Formula presented.) there exists a point (Formula presented.) that is contained in the images of a positive fraction (Formula presented.) of the d-cells of X. More generally, the conclusion holds if (Formula presented.) is replaced by any d-dimensional piecewise-linear manifold M, with a constant (Formula presented.) that depends only on d and on the expansion properties of X, but not on M

The (Co)isoperimetric Problem in (Random) Polyhedra

We consider some aspects of the global geometry of cellular complexes. Motivated by techniques in graph theory, we develop combinatorial versions of isoperimetric and Poincare inequalities, and use them to derive various geometric and topological estimates. This has a progression of three major topics: 1. We define isoperimetric inequalities for normed chain complexes. In the graph case, these quantities boil down to various notions of graph expansion. We also develop some randomized algorithms which provide (in expectation) solutions to these isoperimetric problems. 2. We use these isoperimetric inequalities to derive topological and geometric estimates for certain models of random simplicial complexes. These models are generalizations of the well-known models of random graphs. 3. Using these random complexes as examples, we show that there are simplicial complexes which cannot be embedded into Euclidean space while faithfully preserving the areas of minimal surfaces.Ph

2-Complexes with Large 2-Girth

The 2-girth of a 2-dimensional simplicial complex X is the minimum size of a non-zero 2-cycle in H[subscript 2](X,Z/2) . We consider the maximum possible girth of a complex with n vertices and m 2-faces. If m=n[superscript 2+α] for α1/2, the 2-girth is at most Cα . 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. Keywords: Random simplicial complexes, Homology, Counting surfacesSimons Foundation (Investigator Grant

Quantitative null-cobordism

For a given null-cobordant Riemannian n n -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n n . In the appendix the bound is improved to one that is O ( L 1 + ε ) O(L^{1+\varepsilon }) for every ε > 0 \varepsilon >0 . This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y 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 L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y

QUANTITATIVE NULL-COBORDISM

For a given null-cobordant Riemannian n n -manifold, how does the minimal geometric complexity of a null-cobordism depend on the geometric complexity of the manifold? Gromov has conjectured that this dependence should be linear. We show that it is at most a polynomial whose degree depends on n n . In the appendix the bound is improved to one that is O ( L 1 + ε ) O(L^{1+\varepsilon }) for every ε > 0 \varepsilon >0 . This construction relies on another of independent interest. Take X X and Y Y to be sufficiently nice compact metric spaces, such as Riemannian manifolds or simplicial complexes. Suppose Y 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 L -Lipschitz maps f , g : X → Y f,g:X \to Y are homotopic via a C L CL -Lipschitz homotopy. We present a counterexample to show that this is not true for larger classes of spaces Y Y