Algebraic & Geometric Topology

Functorial seminorms on singular homology and (in)flexible manifolds

Diarmuid Crowley and Clara Löh

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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.

Article information

Algebr. Geom. Topol., Volume 15, Number 3 (2015), 1453-1499.

Received: 11 February 2014
Accepted: 5 November 2014
First available in Project Euclid: 16 November 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 57N65: Algebraic topology of manifolds 55N10: Singular theory
Secondary: 55N35: Other homology theories 55P62: Rational homotopy theory

mapping degrees simply connected manifolds functorial seminorms on homology


Crowley, Diarmuid; Löh, Clara. Functorial seminorms on singular homology and (in)flexible manifolds. Algebr. Geom. Topol. 15 (2015), no. 3, 1453--1499. doi:10.2140/agt.2015.15.1453.

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