Characteristic classes of augmented bundles
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 \((j>n+1\nrightarrow c_ j=0)\). Examples are given, with fibres \(\pi^{- 1}(b)\approx {\mathbb{C}}P_ m\times {\mathbb{C}}P_ n\), such that \(c_ j\neq 0\) for infinitely many j.
scientific article; zbMATH DE number 4214943