By Alexei Skorobogatov

The topic of this publication is mathematics algebraic geometry, a space among quantity conception and algebraic geometry. it's approximately utilising geometric the right way to the research of polynomial equations in rational numbers (Diophantine equations). This ebook represents the 1st entire and coherent exposition in one quantity, of either the speculation and purposes of torsors to rational issues. a few very fresh fabric is integrated. it truly is tested that torsors offer a unified method of numerous branches of the speculation which have been hitherto constructing in parallel.

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This special case of the generalized Baum-Connes Conjecture is usually known as the Connes-Kasparov Conjecture, and has been proved for connected linear reductive Lie groups by Wassermann [Was]. , [Pl3], [BHP], [BCH]). For arbitrary closed subgroups G of amenable connected Lie groups, of SO(n, 1), or of SU (n, 1), the generalized Baum-Connes Conjecture follows from still stronger results of Kasparov et al. ([Kas4], [Kas5], [Kas7], [JuK]). ) Around 1980, Gromov and Lawson ([GL1], [GL2]) began to notice an interesting parallel between the Novikov Conjecture and a problem in Riemannian geometry, that of determining what smooth manifolds admit Riemannian metrics of positive scalar curvature.

Without great loss of generality, suppose the dimension n of M is even, n = 2k. We compare a certain analytic invariant of M , which one can call the analytic higher signature, with an a priori homotopy invariant, the Mishchenko symmetric signature. The former is the generalized index of a certain (generalized) elliptic operator; it plays the role of the index of a family of twisted signature operators in Lusztig’s proof. Recall that the index of a family of operators parameterized by a compact space Y is a certain formal difference of vector bundles over Y , in other words an element of the Grothendieck group of vector bundles, K 0 (Y ).

Applying the conjecture again, f is homotopic to a homeomorphism rel ∂W . Since Whitehead torsion is a topological invariant and τ (M × [0, 1], M × {0}) = 0, τ = 0. Proposition. The Borel Conjecture holds for a group Γ if and only if the algebraic surgery assembly map AΓ : H∗ (BΓ; L• (Z)) → L∗ (Z[Γ]) is an isomorphism and Wh(Γ) = 0, with L∗ = Lh∗ = Ls∗ . Thus if the Borel Conjecture holds for Γ then S∗ (BΓ) = 0, and if (M, ∂M ) is an n-dimensional manifold with boundary such that π1 (M ) = Γ and M is aspherical then S T OP (M rel ∂) = Sn+1 (M ) = Sn+1 (BΓ) = 0 .

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