By Nigel Higson

ISBN-10: 0198511760

ISBN-13: 9780198511762

Analytic K-homology attracts jointly principles from algebraic topology, practical research and geometry. it's a instrument - a method of conveying info between those 3 matters - and it's been used with specacular luck to find amazing theorems throughout a large span of arithmetic. the aim of this publication is to acquaint the reader with the fundamental rules of analytic K-homology and boost a few of its purposes. It encompasses a exact advent to the required useful research, by way of an exploration of the connections among K-homology and operator conception, coarse geometry, index concept, and meeting maps, together with an in depth remedy of the Atiyah-Singer Index Theorem. starting with the rudiments of C - algebra thought, the booklet will lead the reader to a few vital notions of latest study in geometric practical research. a lot of the fabric integrated right here hasn't ever formerly seemed in ebook shape.

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12). Let R be an n-dimensional regular loeal domain, let] and I be nonzero principal ideals in R, let S be a positive-dimensional element in m(R) such that (S, I) has a pseudonormal crossing at R, let (R', j', I') be a monoidal trans/orm 0/ (R,], I, S), and let SI E m(R) f1 m(R') such that dirn S' ~ n - 1, {S, S'} has anormal crossing at R, and (S', I) has a pseudonormal crossing at R. Then (S', I') has a pseudonormal crossing at R'. PROOF. Let d = ords ], e = ord~, and m = dirn S. For a moment suppose that m = 1; then R' = R; we can take XE R such that R f1 M(S) = xR; then ordRx = 1 and I' = xdI; since {S, S'} has anormal crossing at R, we get that (S', xdR') has a normal crossing at R'; sinceI' = xdI and (S', I) has a pseudonormal crossing at R, we conclude that (S', I') has a pseudonormal crossing at R'.

X~nR and R f"'I M(S) = y'r;R for 22 1. LocAL THEORY each 8 E E. Given E C m(R) and a nonzero principal ideal I in R, we say that (E, I) has a strict normal crossing at R if (E, I) has a normal crossing at Rand E contains at most two elements. Given 8 E m(R) and a nonzero principal ideal I in R, we say that (8, I) has anormal crossing at R if({8}, I) has anormal crossing at R. Given nonzero principal ideals J and I in R, we say that (j, I) has a quasinormal crossing at R if I has anormal crossing at Rand for every nonzero principal prime ideal P in R with Je P we have that PI has anormal crossing at R.

### Analytic K-Homology by Nigel Higson

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