Degrees of self-maps of products

TL;DR

This paper studies the set of degrees of self-maps on product manifolds, providing conditions under which these sets are exactly the products of the individual sets, and explores implications for chirality and flexibility.

Contribution

It offers new criteria for the self-mapping degrees of product manifolds and characterizes strongly chiral hyperbolic manifolds via their products.

Findings

Identifies conditions where $D(M imes N)$ equals the product of $D(M)$ and $D(N)$

Constructs manifolds with specific self-mapping degree properties like chirality and inflexibility

Characterizes odd-dimensional strongly chiral hyperbolic manifolds through their product degrees

Abstract

Every closed oriented manifold $M$ is associated with a set of integers $D(M)$, the set of self-mapping degrees of $M$. In this paper we investigate whether a product $M\times N$ admits a self-map of degree $d$, when neither $D(M)$ nor $D(N)$ contains $d$. We find sufficient conditions so that $D(M\times N)$ contains exactly the products of the elements of $D(M)$ with the elements of $D(N)$. As a consequence, we obtain manifolds $M\times N$ that do not admit self-maps of degree $-1$ (strongly chiral), that have finite sets of self-mapping degrees (inflexible) and that do not admit any self-map of degree $dp$ for a prime number $p$. Furthermore we obtain a characterization of odd-dimensional strongly chiral hyperbolic manifolds in terms of self-mapping degrees of their products.