Geometric \(K\)-homology with coefficient. I: \(\mathbb Z / k\mathbb Z\)-cycles and Bockstein sequence (Q2905337)
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scientific article; zbMATH DE number 6072572
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Geometric \(K\)-homology with coefficient. I: \(\mathbb Z / k\mathbb Z\)-cycles and Bockstein sequence |
scientific article; zbMATH DE number 6072572 |
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27 August 2012
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\(K\)-homology
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geometric cycles
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\(\mathbb Z / k \mathbb Z\)-manifolds
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index theory
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Geometric \(K\)-homology with coefficient. I: \(\mathbb Z / k\mathbb Z\)-cycles and Bockstein sequence (English)
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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)].
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