Characteristic classes of augmented bundles
type: Item
instance of: scholarly article
title: Characteristic classes of augmented bundles
zbMATH Open document ID: 0734.55008
DOI: 10.1007/BF01444645
author: Albrecht Dold
published in: Mathematische Annalen
publication date: 1992-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164938
review text: The purpose of the paper is to extend the theory of Chern classes of complex sphere bundles \(S\to B\) to other bundles with a fibre-wise action of the circle group C. The classical Chern-Grothendieck definition uses the associated projective bundle \(\pi\) : P\(=S/C\to B\), the canonical class \(u\in H^ 2P\), and the fact that \(1,u,...,u^ n\) is an \(H^*B\)- basis of \(H^*P\) (Leray-Hirsch property). The scheme of the paper is to avoid the Leray-Hirsch condition by using transfer methods for (\(\pi\),u). Two approaches are presented. Both assume fibre bundles \(\pi\) : \(P\to B\) and a cohomology class \(u\in H^ 2P\), the augmentation. In the 1st approach the fibres \(\pi^{-1}(b)\) are assumed to be compact ENRs. The fixed point transfer \(\tau\) : \(H^*P\to H^*B\) is used to define characteristic classes \(e_ j\in H^{2j}B\), \(j\geq 0\). These can be thought of as generalizations of the power sum classes in Chern theory (with integral coefficients). They can also be thought of as generalized traces of the iterates of the \(H^*B\)-endomorphism \(u^{\cup}\) of \(H^*P\). The corresponding properties of \(\{e_ i\}\) are established, resulting in a satisfactory theory. Several special cases are discussed. In the 2nd approach the fibres are assumed to be 2n-circuits, and \(u^ n| \pi^{-1}(b)\neq 0\). Integration over the fibres \(\int_{\pi}: H^*P\to H^*B\) is used to define characteristic classes \(a_ i=\int_{\pi}u^{n+i}\in H^{2i}(B;{\mathbb{R}})\) resp. \(c_ j\in H^{2j}(B;R)\), i,j\(\geq 0\), by \((\sum_{i}a_ i)(\sum_{j}c_ j)=a_ 0\), where \(R={\mathbb{R}}[a_ 0^{-1}]\subset {\mathbb{Q}}\). The classes \(c_ j\) can be though of as generalized Chern classes. Some of the corresponding properties of \(\{c_ j\}\) are established, others are left open. One also thinks of \(c(t)=\sum_{j\geq 0}(-1)^ jc_ jt^{n+1-j}\) as a generalized characteristic polynomial of the \(H^*B\)-endomorphism \(u^{\cup}\) of \(H^*P\). In general, however, c(t) is not polynomial \((j>n+1\nrightarrow …
Mathematics Subject Classification ID: 55R40
zbMATH DE Number: 4214943
zbMATH Keywords: Chern classes
zbMATH Keywords: complex sphere bundles
zbMATH Keywords: characteristic classes
zbMATH Keywords: characteristic polynomial
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4214943