Functorial seminorms on singular homology and (in)flexible manifolds

Diarmuid Crowley, Clara Löh

Research output: Contribution to journalArticle

4 Citations (Scopus)
4 Downloads (Pure)

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.
Original languageEnglish
Pages (from-to)1453-1499
Number of pages48
JournalAlgebraic & Geometric Topology
Volume15
Issue number3
DOIs
Publication statusPublished - 19 Jun 2015

Keywords

  • mapping degrees
  • simply connected manifolds
  • functorial seminorms on homology

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