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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)
##### MSC:
 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
##### Keywords:
Chern character; simplicial de Rham complex
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##### References:
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