#### Volume 15, issue 3 (2015)

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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
##### Abstract

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 $-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.

##### Keywords
mapping degrees, simply connected manifolds, functorial seminorms on homology
##### Mathematical Subject Classification 2010
Primary: 57N65, 55N10
Secondary: 55N35, 55P62
##### Publication
Received: 11 February 2014
Accepted: 5 November 2014
Published: 19 June 2015
##### Authors
 Diarmuid Crowley Institute of Mathematics University of Aberdeen Aberdeen AB24 3UE UK http://www.dcrowley.net Clara Löh Fakultät für Mathematik Universität Regensburg 93040 Regensburg Germany http://www.mathematik.uni-r.de/loeh