## Analytic K-Homology by Nigel Higson

By Nigel Higson

Analytic K-homology attracts jointly rules from algebraic topology, useful research and geometry. it's a software - a way of conveying details between those 3 topics - and it's been used with specacular good fortune to find impressive theorems throughout a large span of arithmetic. the aim of this ebook is to acquaint the reader with the fundamental rules of analytic K-homology and enhance a few of its functions. It features a unique creation to the required useful research, by means of an exploration of the connections among K-homology and operator thought, coarse geometry, index thought, and meeting maps, together with an in depth therapy of the Atiyah-Singer Index Theorem. starting with the rudiments of C - algebra idea, the e-book will lead the reader to a few critical notions of latest examine in geometric useful research. a lot of the cloth incorporated right here hasn't ever formerly seemed in ebook shape.

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We will show in Chapter III that this topology can be defined intrinsically (rather t h a n in terms of the representation as an inverse limit); in fact, it turns out to be precisely the compact-open topology we considered in the preceding section. The reason this is the correct topology to consider (rather than, say the p-adic topology) is that it is the one for which the classical duality between modular forms and Hecke operators can be extended to the p-adic situation, as we will see in the next chapter.

Then, /or any integer k and any r 6 B such that ord(r) < 1/(p + 1), we have U ( M ( B , k , N ; r ) ® K) C M ( B , k , N ; r p) ® K. We interpret this as saying that, up to tensoring with K, the operator U improves overconvergence. It is not immediately clear, however, that giving M ( B , k , N ; r ) ® K and M(B, k , N ; r p) ® K their "natural" p-adic topologies (in which M(B, k , N ; r ) and M(B, k, N; r p) are the closed unit balls) makes the linear map U : M(B, k, N; r) ® K M(B, k, N, r p) @ K a bounded map.

This requires a delicate analysis of the structure of the formal group of the curve. See [Ka73, Thin. 1], where b o t h this and the following result are a t t r i b u t e d to Lubin. , one must check whether the quotient curve is more or less supersingu]ar t h a n the one we started with. 2. The Frobenius O p e r a t o r 37 only possible to give the quotient curve an rP-structure, so that the valuation of a lifting of the Hasse invariant is multiplied by p in the passage from the curve to its quotient by the fundamental subgroup.