# On the bounded index property for products of aspherical polyhedra

## TL;DR

This paper investigates the Bounded Index Property for homotopy equivalences in products of aspherical polyhedra, establishing conditions under which such products, including hyperbolic manifolds, possess this property.

## Contribution

It provides sufficient conditions for products of compact polyhedra to have BIPHE and confirms this property for products of negatively curved Riemannian manifolds.

## Key findings

- Products of negatively curved manifolds have BIPHE
- Sufficient conditions for BIPHE in polyhedral products
- Affirmative answer to a special case of Jiang's question

## Abstract

A compact polyhedron $X$ is said to have the Bounded Index Property for Homotopy Equivalence (BIPHE) if there is a finite bound $\mathcal{B}$ such that for any homotopy equivalence $f:X\rightarrow X$ and any fixed point class $\mathbf{F}$ of $f$, the index $|\mathrm{ind}(f,\mathbf{F})|\leq \mathcal{B}$. In this note, we consider the product of compact polyhedra, and give some sufficient conditions for it to have BIPHE. Moreover, we show that the products of closed Riemannian manifolds with negative sectional curvature, in particular hyperbolic manifolds, have BIPHE, which gives an affirmative answer to a special case of a question asked by Boju Jiang.