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The Chern character in the simplicial de Rham complex. (English) Zbl 1351.58003
Let \(G\) be a Lie group. It is well known that the characteristic classes of principal \(G\)-bundles are in one-one correspondence with elements in \(H^*(BG)\), where \(BG\) is the classifying space of \(G\). Since it is well known that \(BG\) is not a manifold, \(H^*(BG)\) cannot be described using de Rham theory. Instead, we can consider the simplicial manifold \(NG(\ast)\) and consider a double complex \(\Omega^{p, q}(NG)=\Omega^q(NG(p))\) introduced by Dupont on it whose cohomology is isomorphic to \(H^*(BG)\). The article under review gives an explicit cocycle representative of the universal Chern character for \(G=\mathrm{GL}(n; \mathbb{C})\) in \(\Omega^{*, *}(NG)\).
Reviewer: Man Ho (Hong Kong)
58A12 de Rham theory in global analysis
57R20 Characteristic classes and numbers in differential topology
19L10 Riemann-Roch theorems, Chern characters
14F40 de Rham cohomology and algebraic geometry
Full Text: arXiv Euclid
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