The combinatorics of hyperbolized manifolds

TL;DR

This paper examines the combinatorial aspects of hyperbolized manifolds, verifying the Euler Characteristic Sign Conjecture for various hyperbolization methods and exploring their combinatorial properties related to generating functions.

Contribution

It provides a detailed analysis of the combinatorics of hyperbolization functors and confirms the Euler Characteristic Sign Conjecture for multiple hyperbolized manifolds.

Findings

Verified the Euler Characteristic Sign Conjecture for several hyperbolization methods.

Analyzed combinatorial properties related to generating functions.

Explored connections between hyperbolizations and well-studied combinatorial structures.

Abstract

A topological version of a longstanding conjecture of H. Hopf, originally proposed by W. Thurston, states that the sign of the Euler characteristic of a closed aspherical manifold of dimension $d=2m$ depends only on the parity of $m$. Gromov defined several hyperbolization functors which produce an aspherical manifold from a given simplicial or cubical manifold. We investigate the combinatorics of several of these hyperbolizations and verify the Euler Characteristic Sign Conjecture for each of them. In addition, we explore further combinatorial properties of these hyperbolizations as they relate to several well-studied generating functions.