By Jose I. Burgos Gil
This ebook includes a entire facts 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 is determined by very comparable descriptions of the Chern-Weil morphisms and the van Est isomorphism. The ebook has varied components. the 1st one studies the fabric from algebraic topology and Lie team conception wanted for the comparability theorem. themes similar to 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 includes the comparability theorem and the categorical fabric wanted in its evidence, akin to particular descriptions of the Chern-Weil morphism and the van Est isomorphisms, a dialogue approximately small cosimplicial algebras, and a comparability of other definitions of Borel's regulator.
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Extra resources for The regulators of Beilinson and Borel
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. The other fact is that the vector bundle πS∗ F has a canonical rank one trivial subbundle L. The fibre of L over a point v is the line spanned by v. Let us write F0 = πS∗ F/L. 9. Let F be a rank n vector bundle over X. The integer valued Chern classes of F , bp (F ) ∈ H 2p (X, Z) are determined inductively by the following conditions: (1) bn (F ) = e(F ).
P, denote the projection over the ith factor of Ep G. 4. THE SUSPENSION IN THE WEIL ALGEBRA where t0 , . . , tp are baricentric coordinates of H p . Since 45 ti = 1, the form 1 ∇E· G ∈ Esimp (E· G, R) is a connection, and it is called the canonical connection of the universal principal bundle. Let ϕ : H → G be a morphism of Lie groups. Then we have that (E· ϕ)∗ ∇E· G = ϕ∗ ◦ ∇E· H . 6) From the canonical connection we obtain a morphism ∗ ∗ f∇E· G : IG → Esimp (B· G, R). 6) is that this morphism is functorial on the Lie group G.
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. 3) θ([h, k]) = θ(h) ◦ θ(k) − θ(h) ◦ θ(k), θ(h) = i(h) ◦ d + d ◦ i(h), d ◦ θ(h) = θ(h) ◦ d. In particular, when the base is a point and E = G, we have operators i(h) and θ(h) defined in E ∗ (G, R). We may restrict these operators to the subalgebra E ∗ (G, R)L , that is, to E ∗ (g, R).