Explicit construction of characteristic classes. (English) Zbl 0809.57016

Goncharov, A. B.

Gelfand, Sergej (ed.) et al., I. M. Gelfand Seminar. Part 1: Papers of the Gelfand seminar in functional analysis held at Moscow University, Russia, September 1993. Providence, RI: American Mathematical Society. Adv. Sov. Math. 16(1), 169-210 (1993).

Let \(E\) denote a vector bundle over the algebraic manifold \(X\). Using maps from \(X\) to Grassmannians the author defines a bicomplex of sheaves on \(X\). The hypercohomology of the total complex associated with this bicomplex is denoted by \(H^* (X, BC^* (n))\). The paper contains an explicit construction of characteristic classes \(c_ n (E) \in H^{2n} (X,BC^* (n))\). In the special case \(n = \dim E\) one obtains a construction of A. Beilinson, R. MacPherson and V. Schechtman [Duke Math. J. 54, 679-710 (1987; Zbl 0632.14010)]. Furthermore a construction of Chern classes \(c_ n (E) \in H^ n (X,K^ M_ n)\) is given, where \(k^ M_ n\) is the sheaf of Milnor’s \(K\)-groups.
For the entire collection see [Zbl 0777.00035].

Reviewer: R.Kultze (Frankfurt / Main)

MSC:

57R20	Characteristic classes and numbers in differential topology
55R40	Homology of classifying spaces and characteristic classes in algebraic topology
55R50	Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
19D45	Higher symbols, Milnor \(K\)-theory

Keywords:

vector bundle over algebraic manifold; sheaf of Milnor’s \(K\)-groups; bicomplex of sheaves; hypercohomology; characteristic classes; Chern classes

Citations:

Zbl 0632.14010