## Download e-book for iPad: Analytic K-Homology by Nigel Higson

By Nigel Higson

Analytic K-homology attracts jointly principles from algebraic topology, sensible research and geometry. it's a instrument - a method of conveying details between those 3 topics - and it's been used with specacular luck to find striking theorems throughout a large span of arithmetic. the aim of this publication is to acquaint the reader with the fundamental principles of analytic K-homology and advance a few of its purposes. It features a exact creation to the mandatory practical research, by means of an exploration of the connections among K-homology and operator conception, coarse geometry, index idea, and meeting maps, together with a close therapy of the Atiyah-Singer Index Theorem. starting with the rudiments of C - algebra conception, the booklet will lead the reader to a couple imperative notions of up to date examine in geometric practical research. a lot of the cloth integrated the following hasn't ever formerly seemed in booklet shape.

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**Extra info for Analytic K-Homology**

**Example text**

Thus the presheaf of mixed Hodge structures (9) can be sheafified to a Zariski Qt-mixed sheaf. 4]). 1. Denote Jl~(Qt(t)) the Qt-mixed sheaf obtained hereabove. For X. a simplicial C-scheme denote Je~ • the simplicial Qt-mixed sheaf given by Je~p on the component X P. If X has algebraic dimension n then all its Zariski open affines U do have dimension ::5 n, thus Je~ = 0 for r > n. 2. The Zariski cohomology groups H * (X, Je~) carry oo-mixed Hodge structures. Possibly non-zero Hodge numbers of H*(X, Jl~) are in the finite set [0, r] x [0, r].

4. Edge maps Recall that the classical cycle class maps can be obtained via edge homomorphisms in the coniveau spectral sequence. This is a consequences of Bloch's formula [8). Working simplicially we then construct certain cycle class maps for singular varieties via edge maps in the local-to-global spectral sequence. We first show that the results of [8] hold in the category of oo-mixed Hodge structures. 1. 8] the cohomology groups H~(X) (= H*(XmodXZ, Z) in Deligne's notation) carry a mixed Hodge structure fitting into long exact sequences .

Gillet, M. Hanamura, U. Jannsen, M. D. Lewis, J. Murre, A. Rosenschon, M. Saito, C. Soule and V. Voevodsky for discussions on some matters treated herein. Finally, I would like to thank Paolo Francia for his helpful insight and invaluable guidance in the vast field of algebraic geometry by dedicating this paper to his memory. Note that this research was carried out with smooth efficiency thanks to several foundations. I would like to thank the Tata Institute of Fundamental Research and the Institut Henri Poincare for their support and hospitality.