Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups
type: Item
instance of: scholarly article
title: Self-homotopy equivalences which induce the identity on homology, cohomology or homotopy groups
zbMATH Open document ID: 0929.55008
DOI: 10.1016/S0166-8641(97)00162-4
author: Martin Arkowitz
author: Ken-ichi Maruyama
published in: Topology and its Applications
publication date: 2000-01-25T00:00:00Z
review text: In [the reviewer, ``Some research problems on homotopy-self-equivalences'', Lect. Notes Math. 1425, 204--207 (1990; Zbl 0704.55005), p. 206] the basic problem of finding the relation between the homotopy-self-equivalences which fix homology and those which fix homotopy is presented. The present paper explores questions for special cases of Moore and co-Moore spaces, and Poincaré duality complexes. In the first two cases, the results are complete, as well as including those self-equivalences which fix cohomology. When \(X\) is a Poincaré duality complex, the self-equivalences which fix homology are those which fix cohomology. The authors close with some very interesting examples. If \(X=\mathbb{C} P^n\vee S^{2n}\), the self-equivalences which fix homotopy through dimension \(2n\) is \(\mathbb{Z}\), while those which fix homotopy through dimension \(2n+1\) is the trivial group.
Mathematics Subject Classification ID: 55P10
Mathematics Subject Classification ID: 57P10
zbMATH DE Number: 1307957
zbMATH Keywords: Moore spaces
zbMATH Keywords: homotopy-self-equivalences
zbMATH Keywords: Poincaré duality complexes
reviewed by: Donald W. Kahn
MaRDI profile type: Publication
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full work available at URL: s0166-8641(97)00162-4
OpenAlex ID: W2061217731
description: scientific article; zbMATH DE number 1307957