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Quillen’s relative Chern character. (English) Zbl 1207.19004

Miwa, Tetsuji (ed.) et al., Algebraic analysis and around in honor of Professor Masaki Kashiwara’s 60th birthday. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-51-8/hbk). Advanced Studies in Pure Mathematics 54, 255-291 (2009).
For \(Y\) a subset of \(X\), the relative Chern character defines a map \(K^{0}(X,Y) \rightarrow H^{*}(X,Y)\). The statement that the relative Chern character is multiplicative is the assertion that, for subsets \(Y\) and \(Z\) of \(X\), the product in \(K\)-theory, the relative cup product in cohomology, and the relative Chern characters for the pairs \((X,Y)\), \((X,Z)\), and \((X,Y\bigcup Z)\) form a commutative diagram. In the early days of \(K\)-theory, multiplicativity of the relative Chern character was established for pairs of finite CW-complexes. D. Quillen [Topology 24, 89–95 (1985; Zbl 0569.58030)] used superconnections to construct, from a smooth morphism between complex vector bundles over a smooth manifold, a relative Chern character taking values in relative de Rham cohomology. The relevant pair is formed by the manifold and the complement of the vector-bundle morphism’s support. There are no compactness assumptions. The paper under review proves that Quillen’s relative Chern character is multiplicative. The proof is independent of the more general results of B. Iversen [Ann. Sci. Éc. Norm. Supér. (4) 9, 155–169 (1976; Zbl 0328.14006)]. The explicit nature of the constructions in the paper under review permits extension to equivariant Chern characters with generalized coefficients, as described by P.-E. Paradan and M. Vergne [“Equivariant Chern character with generalized coefficients”, arXiv:0801.2822 (2008)].
The authors of the paper under review describe the images of the relative Chern character in other theories. When the vector-bundle morphism has compact support, there is an image in de Rham cohomology with compact supports. When the underlying manifold is the total space of a vector bundle and when the vector-bundle morphism’s support has compact intersection with each fiber of the underlying vector bundle, the relative Chern character has the Gaussian appearance introduced by V. Mathai and D. Quillen [Topology 25, 85–110 (1986; Zbl 0592.55015)] and defines a class in rapidly decreasing cohomology.
Finally, for real, oriented, Euclidean vector bundles, the authors give explicit formulas for Thom differential forms in relative cohomology, cohomology with compact support in the fibers, and rapidly decreasing cohomology. In the even-rank, spin case, they prove the Riemann-Roch result in which, in all of these theories, an \(\widehat{A}\)-form establishes, at the level of differential forms, the relationship between Chern forms and Thom forms.
For the entire collection see [Zbl 1160.32002].

MSC:

19L10 Riemann-Roch theorems, Chern characters
57R20 Characteristic classes and numbers in differential topology
53C05 Connections (general theory)
58A10 Differential forms in global analysis
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
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