By Jose I. Burgos Gil
This publication incorporates a entire evidence of the truth that Borel's regulator map is two times Beilinson's regulator map. the method of the facts follows the argument sketched in Beilinson's unique paper and depends on very comparable descriptions of the Chern-Weil morphisms and the van Est isomorphism. The booklet has assorted components. the 1st one experiences the fabric from algebraic topology and Lie staff idea wanted for the comparability theorem. issues resembling simplicial items, Hopf algebras, attribute periods, the Weil algebra, Bott's Periodicity theorem, Lie algebra cohomology, non-stop staff cohomology and the van Est Theorem are mentioned. the second one half comprises the comparability theorem and the categorical fabric wanted in its evidence, reminiscent of particular 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
The continuous cohomology groups of GLn (C) are ∼ H ∗ (Un , R) = (α1 , α3 , . . , α2n−1 ). 12. Let n ≤ n and m < 2n be positive integers. Then the morphism m m (GLn (C), R) → Hcont (GLn (C), R) ϕ∗n ,n : Hcont is an isomorphism. CHAPTER 7 Small Cosimplicial Algebras In this section we will recall briefly the notion of cosimplicial algebras and we will review the theory of small cosimplicial algebras and small differential graded algebras. This theory was introduced by Beilinson in order to compare his regulator with Borel’s regulator.
19. The Chern-Weil morphism is the morphism ∗ ωE : IG → H ∗ (B, R). The next result is the heart of the de Rham realization of characteristic classes (cf. 20. The Chern–Weil morphism is independent of the connection. As a consequence of this theorem, the image ωE (IG ) is a subalgebra of H ∗ (B, R) which is characteristic of the principal G-bundle E. 21. Since the Weil algebra and the subspace of invariant elements only depend on the Lie algebra g we will sometimes write W (g) and I(g) for W (G) and IG .
For each x ∈ E, the map g → xg induces a morphism νx : g → Tx E. Let Xh be the fundamental vector field generated by h. This vector field is determined by the condition (Xh )x = νx (h). We will denote by i(h) the substitution operator by the vector field Xh and by θ(h) the Lie derivative with respect to the vector field Xh . Explicitly, if Φ ∈ E p (E, V ), i(h)Φ(X2 , . . , Xp ) = Φ(Xh , X2 , . . , Xp ), p θ(h)Φ(X1 , . . , Xp )= Xh Φ(X1 , . . , Xp ) − Φ(X1 , . . , [Xh , Xi ], . . , Xp ). i=1 The operators i(h) and θ(h) are derivations (in the graded sense) of degree −1 and 0.
The regulators of Beilinson and Borel by Jose I. Burgos Gil