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By Yuri Tschinkel (Ed.)
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Additional resources for Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005
Consider the induced map h A : A1 × Y˜ → A1 ×Y . n 1 ˜ Let U n = A1 × Y \ D ⊂ A1 × Y . Consider the premage h −1 A (U ) ⊂ A × Y . Then n 1 ˜ \ D˜ where D˜ consists of a finite number of sections over Y˜ . Inh −1 (U ) = A × Y A deed, the product Y˜ × D is equal to the union of the products Y˜ × D i and the latter consists of d eg (πin−1 ) number of copies of Y˜ . Note that π1 (U n ) contains π1 (U˜ n ) as subgroup of finite index, but the latter is a direct product π1 (Y˜ ) × F , where F is a free group.
Let us show the existence of a surjective map H s∗ (G, F )W → H s∗ (G, F )V . , ϕ(g x) = g (ϕ(x)) for all x ∈ V L and all g ∈ G) with the property that under the induced map ϕG : V L /G → W L /G the image −1 ϕG (V L /G) is not contained in D . Then, by taking D = ϕG (D ), we get a diagram: VL V L O /G ? (V L /G) \ D ϕ / WL / W L /G O ? / (W L /G) \ D Passing to cohomology we get: H ∗ (G, F ) → H ∗ (W L /G \ D , F ) = H ∗ (G, F )W → H ∗ (V L /G \ D, F ). Therefore, there is a surjection HS∗ (G, F )W → HS∗ (G, F )V .
It is clear that V L contains T which is isomorphic to the torus and T is G-invariant (see, for example, [Bog95a]). Thus T /G ⊂ V L /G is a K (π1 , 1)-space with π1 ((T /G)) being an extension of G by a finitely generated abelian group π1 (T ). The group π1 ((T /G)) has no torsion elements and π1 ((T /G)) = πˆ 1 ((T /G)). There is a natural surjective map τ : π1 (T /G) → G, where π1 (T /G) is an abelian extension of G, and hence the image τ∗ (H ∗ (G, F )) surjects onto HS∗ (G, F ). 2 below. F. Bogomolov: Stable cohomology 33 Unfortunately, contrary to the conjecture formulated in [Bog92], [Bog95a] these rings are different in general apart from dimensions 2 and 3 (see [Pey99],[Pey93] where the cohomology of dimension 3 were treated).
Algebraic Groups: Mathematisches Institut, Georg-August-Universitat Gottingen. Summer School, 27.6.-13.7.2005 by Yuri Tschinkel (Ed.)