Functorial semi-norms on singular homology and (in)flexible manifolds

TL;DR

This paper develops new functorial semi-norms on singular homology using manifold degree constraints, answering Gromov's question by constructing semi-norms that are finite and positive on certain simply connected spaces.

Contribution

It introduces novel functorial semi-norms based on manifold degree properties and extends the existence of inflexible manifolds to a broader class.

Findings

Constructed functorial semi-norms with finite positive values on specific homology classes.

Extended the class of known inflexible manifolds using new methods.

Provided an answer to Gromov's question regarding semi-norms on simply connected spaces.

Abstract

A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree -1, 0, or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.