By Jose I. Burgos Gil
This publication encompasses a whole evidence of the truth that Borel's regulator map is two times Beilinson's regulator map. the tactic of the evidence follows the argument sketched in Beilinson's unique paper and depends on very related descriptions of the Chern-Weil morphisms and the van Est isomorphism. The ebook has varied components. the 1st one stories the fabric from algebraic topology and Lie workforce conception wanted for the comparability theorem. subject matters comparable to simplicial items, Hopf algebras, attribute sessions, the Weil algebra, Bott's Periodicity theorem, Lie algebra cohomology, non-stop team cohomology and the van Est Theorem are mentioned. the second one half includes the comparability theorem and the explicit fabric wanted in its evidence, similar to specific descriptions of the Chern-Weil morphism and the van Est isomorphisms, a dialogue approximately small cosimplicial algebras, and a comparability of alternative definitions of Borel's regulator.
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Additional info for The Regulators of Beilinson and Borel (CRM Monograph Series)(draft)
Example text
17. Let G be a Lie group and let g be its Lie algebra. The Weil algebra of G is the bigraded algebra W (G) defined by W p,q (G) = W p,q (G, R) = S p (g) ⊗ E q−p (g). Initially we will consider the Weil algebra only as a graded algebra with the total degree. The map f∇ : E ∗ (g) → E ∗ (E, R) can be extended to a map defined on W (G). To this end, for x ∈ S 1 (g) = g∨ , we write f∇ (x) = φ(x) and we extend f∇ multiplicatively. By definition f∇ is a morphism of algebras. 3). We first define the operator i(h).
With the hypotheses of the above proposition, there is a natural isomorphism ∗ Hcont (G, R) → H ∗ (g, u, R). 3. Computation of Continuous Cohomology The van Est isomorphism can be used to compute the continuous cohomology of Lie groups. Let G be an algebraic reductive group defined over R. Let us assume that G(R) is a connected Lie group. Then G(C) is a complex connected reductive group. Let gR be the Lie algebra of G(R). Then gC = g ⊗ C is the Lie algebra of G(C). Let K be a maximal compact subgroup of G(R), let k be the Lie algebra of K and let gR = k ⊕ p be the Cartan decomposition of gR with respect to k.
Let Erp,q be the Leray spectral sequence of the bundle S. Since it is a S 2n−1 -bundle, then Erp,q = 0 for q = 0, 2n − 1. Thus the only nonzero differential is d2n . As in the previous section, the standard orientation of F as complex vector bundle defines a class σ2n+1 ∈ H 2n−1 (S 2n−1 , Z) = E20,2n−1 . The Euler class of F is the class e(F ) = d2n σ2n−1 ∈ E 2n,0 = H 2n (X, Z). To give an inductive definition of Chern classes we need two more facts. First, observe that, for j < 2n − 1, the morphism πS∗ : H j (X, Z) → H j (S, Z) is an isomorphism.