Geometric \(K\)-homology with coefficient. I: \(\mathbb Z / k\mathbb Z\)-cycles and Bockstein sequence
type: Item
instance of: scholarly article
zbMATH Open document ID: 1256.19005
author: Robin J. Deeley
publication date: 2012-08-27T00:00:00Z
full work available at URL: https://arxiv.org/abs/1101.0697
Mathematics Subject Classification ID: 19K33
Mathematics Subject Classification ID: 55N22
Mathematics Subject Classification ID: 55N20
Mathematics Subject Classification ID: 19K56
Mathematics Subject Classification ID: 55N15
zbMATH DE Number: 6072572
zbMATH Keywords: \(K\)-homology
zbMATH Keywords: geometric cycles
zbMATH Keywords: \(\mathbb Z / k \mathbb Z\)-manifolds
zbMATH Keywords: index theory
MaRDI profile type: Publication
OpenAlex ID: W2132889062
arXiv ID: 1101.0697
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DOI: 10.1017/IS011010022JKT170
published in: Journal of \(K\)-Theory
title: Geometric \(K\)-homology with coefficient. I: \(\mathbb Z / k\mathbb Z\)-cycles and Bockstein sequence
review text: The author uses \(\mathrm{spin}^c\mathbb Z / k \mathbb Z\)-manifolds, introduced in \textit{J. W. Morgan} and \textit{D. P. Sullivan} [Ann. Math. (2) 99, 463--544 (1974; Zbl 0295.57008)], to give a geometric definition of \(K\)-homology with \(\mathbb Z / k \mathbb Z\) coefficients that is modeled on the \textit{P. Baum} and \textit{R. G. Douglas} [Proc. Symp. Pure Math. 38, Part 1, Kingston/Ont. 1980, 117--173 (1982; Zbl 0532.55004)] definition of \(K\)-homology. The definition allows a geometric proof of the exactness of the Bockstein sequence. The author's version of \(K\)-homology with \(\mathbb Z / k \mathbb Z\) coefficients has the expected relationships with bordism theory and with the topological index of \textit{D. S. Freed} and \textit{R. B. Melrose} [Invent. Math. 107, No. 2, 283--299 (1992; Zbl 0760.58039)].
reviewed by: Peter Haskell
description: scientific article; zbMATH DE number 6072572