Algebraic Groups: Mathematisches Institut, by Yuri Tschinkel (Ed.)

By Yuri Tschinkel (Ed.)

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In general, let E be a contractible topological space with a free action of G, then E /G has the homotopy type of BG above. The space BG (modulo homotopy equivalence) can be characterized by the structure of its homotopy groups. Namely, it is a connected space, which means that π0 (BG) = 1, with G = π1 (BG) and πi (BG) = 0, for i 2. The universal covering EG of BG is connected and has trivial homotopy groups, hence is contractible. Thus the action of G on EG introduced above is the action of the fundamental group of the quotient space BG on its universal covering.

Thus, if a ∈ H 2 ((V L /G), Z/p) has a non-trivial value on a cycle C as above, then a does not vanish in H s2 ((V L /G), Z/p). In particular, the only negligible elements of H 2 ((V L /G), Z/p) are the codimension two cycles represented by divisors and hence belong to βH 1 (G, Z/p). 2 The group Hnr (G) = B 0 (G) can be defined inside H 2 (G, Q/Z) as a subgroup consisting of elements restricting to 0 on any abelian subgroup A ⊂ G with two generai (G, Z/p) restricts trivtors. (see [Bog89], [Bog87],[Sal84]).

However, the cohomology groups have several advantages: a) cup product b) existence of the universal object. (In homology we would only get a map into Hi (BG, F )). Recall that all cohomology classes are the images of certain universal classes: for any positive integers n, l > 1 there is a space K (n; Z/l ) and a cohomology class b ∈ H n (K (n; Z/l ), Z/l ), such that any cohomology class a ∈ H n (X , Z/l ) is induced from b ∈ H n (K (n; Z/l ), Z/l ) by a homotopically unique map f a : X → K (n; Z/l ) with f a∗ (b) = a.

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