By Nigel Higson
Analytic K-homology attracts jointly principles from algebraic topology, practical research and geometry. it's a device - a way of conveying info between those 3 matters - and it's been used with specacular luck to find awesome theorems throughout a large span of arithmetic. the aim of this ebook is to acquaint the reader with the basic principles of analytic K-homology and boost a few of its functions. It contains a distinct creation to the required sensible research, via an exploration of the connections among K-homology and operator idea, coarse geometry, index idea, and meeting maps, together with an in depth remedy of the Atiyah-Singer Index Theorem. starting with the rudiments of C - algebra conception, the publication will lead the reader to a couple primary notions of up to date learn in geometric useful research. a lot of the fabric incorporated right here hasn't ever formerly seemed in e-book shape.
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Extra resources for Analytic K-Homology
Moreover this representation is compatible with the differential. Therefore we have induced representations in the subspace of cycles, denoted Z ∗ (g), the subspace of boundaries, denoted B ∗ (g) and the cohomology, H ∗ (g). The representation θ is semisimple, therefore there is a direct sum decomposition E ∗ (g) = E ∗ (g)θ=0 ⊕ θ E ∗ (g) . The relationship between invariant forms and Lie algebra cohomology is given by the following result. 2. 9. Let g be a reductive Lie algebra. Then Z ∗ (g) = E ∗ (g)θ=0 ⊕ B ∗ (g), B ∗ (g) = θ Z ∗ (g) = Z ∗ (g) ∩ θ E ∗ (g) .
Therefore, for j ≤ 2m, the morphisms (ϕm )∗ : Hj (GLm (C), Z) → Hj (GL(C), Z), (ϕm )∗ : H j (GL(C), Z) → H j (GLm (C), Z) are isomorphisms. This result is called the stability of the homology and cohomology of the general linear group. All the classical series of Lie groups enjoy a similar property. 1) H ∗ (GL(C), Z) = (α1 , α3 , . . 2) H∗ (GL(C), Z) = (β1 , β3 , . . ), where the elements α2p−1 , β2p−1 , p = 1, 2, . . are primitive of degree 2p − 1. There is also a similar stability result for the homology and cohomology of the classifying space B· GL(C).
Then G(C) is a complex connected reductive group. Let gR be the Lie algebra of G(R). Then gC = g ⊗ C is the Lie algebra of G(C). Let K be a maximal compact subgroup of G(R), let k be the Lie algebra of K and let gR = k ⊕ p be the Cartan decomposition of gR with respect to k. Then gu = k ⊕ ip ⊆ gC is a compact form of gR . Thus, the corresponding Lie group Gu is a maximal compact subgroup of G(C) that contains K. Let us write X = K\G(R) and Xu = K\Gu . Strictly speaking, Xu is the compact twin of the symmetric space X and is determined by the pair (G(R), K).
Analytic K-Homology by Nigel Higson