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Functorial seminorms on singular homology and (in)flexible manifolds

Diarmuid Crowley and Clara Löh

Algebraic & Geometric Topology 15 (2015) 1453–1499

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  10 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.

mapping degrees, simply connected manifolds, functorial seminorms on homology
Mathematical Subject Classification 2010
Primary: 57N65, 55N10
Secondary: 55N35, 55P62
Received: 11 February 2014
Accepted: 5 November 2014
Published: 19 June 2015
Diarmuid Crowley
Institute of Mathematics
University of Aberdeen
Aberdeen AB24 3UE
Clara Löh
Fakultät für Mathematik
Universität Regensburg
93040 Regensburg