On Jiang's Bounded Index Property for products of nilmanifolds

TL;DR

This paper extends the concept of Bounded Index Property to iterates of selfmaps and demonstrates it for certain products of nilmanifolds, providing characterizations of fixed point invariants.

Contribution

It introduces BIP$_k$ for iterates of selfmaps and proves it for specific nilmanifold products, advancing understanding of fixed point properties in these spaces.

Findings

BIP$_k$ holds for certain products of nilmanifolds.

Characterizations of Lefschetz, Nielsen, and minimal fixed points numbers.

Extension of BIP to iterates of selfmaps.

Abstract

In 2023, Zhang and Zhao presented the first examples of aspherical manifolds lacking the Bounded Index Property (BIP) for fixed points. This answered a question posed by Jiang in 1998 in the negative. In this paper, we first extend the notion of BIP to that of iterates of selfmaps (BIP$_k$), and then demonstrate BIP$_k$ for certain products $M\times N$ of nilmanifolds. Finally, we give characterizations for the Lefschetz number, the Nielsen number, and the minimal number of fixed points of self-homotopy equivalences of $M\times N$.