Functorial seminorms on singular homology and (in)flexible manifolds

Abstract

A functorial semi-norm on singular homology is a collec- tion of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in ho- mology. 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 prop- erties. 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.