By William S. Massey

ISBN-10: 038797430X

ISBN-13: 9780387974309

"This ebook is meant to function a textbook for a direction in algebraic topology before everything graduate point. the most subject matters lined are the category of compact 2-manifolds, the elemental team, protecting areas, singular homology conception, and singular cohomology thought. those themes are constructed systematically, averting all pointless definitions, terminology, and technical equipment. anyplace attainable, the geometric motivation at the back of a few of the ideas is emphasised. The textual content involves fabric from the 1st 5 chapters of the author's prior e-book, ALGEBRAIC TOPOLOGY: AN creation (GTM 56), including just about all of the now out-of-print SINGULAR HOMOLOGY conception (GTM 70). the fabric from the sooner books has been conscientiously revised, corrected, and taken as much as date."

Searchable DJVU with a little askew pages.

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Part (c) follows since stable conjugacy coincides with conjugacy in a compact unitary group. Suppose that E /Fis local. Let / E M be a singular, non-central element. , = H. 2, there exists a Cartan subgroup T of type (1) contained in H such that O(T, G) = Z /2. Suppose that 1' E T has the same eigenvalues as/· Then 1' is stably conjugate to I in G, but since OF(T, G) = S 3 , we can choose 1' so that it is not conjugate to / in G. In particular, / 1 is regular as an element of H. Let bH be the nontrivial element in 'DH(T/F) and set 1" = (-y') 0n.

This gives rise to an F-isomorphism 'lj;: TH----) T, the inverse of the map naturally associated to X*(TH) ----) X*(T). Assume that TH and T are defined over F. Then the r- actions on X *( T) and X*(T) differ by a twisting with values in flc(T). In fact, if Bis defined 42 Chapter 4 over F, then the r-actions coincide. On the other hand, if T' is another Cartan subgroup of G defined over F, then T' = gTg- 1 for some g E G(F). The 1-cocycle {g- 1 T(g)} takes values in na(T) and ad(g) intertwines the r-action on X*(T') with the twist of the r-action on X*(T) by {g- 1 T(g )}.

This program has been carried in various cases, (for example, in the real case [S 1 ]) but is not yet known in general. 8 below. 5. The fundamental lemma. Suppose that P is p-adic. A connected reductive group G is said to be unramified over P if it is quasi-split and splits over an unramified extension of P. In this case, the action of wF on Gfactors through the projection of wF onto r( pun IP)' where pun is the maximal unramified extension of F. Assume that G is unramified and let T be a maximal torus contained in a Borel subgroup B of G.

### A Basic Course in Algebraic Topology (Graduate Texts in Mathematics) by William S. Massey

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