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Topological lower bounds on the sizes of simplicial complexes and simplicial sets

Sergey Avvakumov, Roman Karasev

Abstract

We prove that if an $n$-dimensional space $X$ satisfies certain topological conditions then any triangulation of $X$ as well as any its representation as a simplicial set with contractible faces has at least $2^n$ faces of dimension $n$. One example of such $X$ is the $n$-dimensional torus $(S^1)^n$.

Topological lower bounds on the sizes of simplicial complexes and simplicial sets

Abstract

We prove that if an -dimensional space satisfies certain topological conditions then any triangulation of as well as any its representation as a simplicial set with contractible faces has at least faces of dimension . One example of such is the -dimensional torus .

Paper Structure

This paper contains 2 sections, 2 theorems, 3 equations.

Table of Contents

  1. Introduction
  2. Proofs

Key Result

Theorem 1.1

Let $G$ be a finite group and let $V$ be an $n$-dimensional real $G$-representation. Let $X$ be an $n$-dimensional simplicial set with a simplicial action of $G$. Assume that Then $X$ has at least $2^n$$G$-orbits of faces of dimension $n$.

Theorems & Definitions (5)

  • Theorem 1.1
  • Corollary 1.2
  • Example 1.3
  • proof : Proof of Theorem \ref{['theorem:orbits']}
  • proof : Proof of Corollary \ref{['corollary:torus-like']}