Functorial seminorms on singular homology and (in)flexible manifolds

Crowley, Diarmuid; Löh, Clara

doi:10.2140/agt.2015.15.1453

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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

DOI: 10.2140/agt.2015.15.1453

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

References

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

Volume 15, issue 3 (2015)