A functorial seminorm on singular homology is a collection of seminorms on the
singular homology groups of spaces such that continuous maps between spaces induce
norm-decreasing maps in homology. Functorial seminorms 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 seminorms with interesting properties.
In particular, we answer a question of Gromov by providing a functorial
seminorm 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 ,
or .
The existence of such manifolds was first established by Arkowitz and Lupton; we
extend their methods to produce a wide variety of such manifolds.
Keywords
mapping degrees, simply connected manifolds, functorial
seminorms on homology