By V. Srinivas (auth.)
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E. the product is formed in the category of compactly generated spaces). g. ln. 9)). Finally, we note that the homology of Let 'bCF) IFI can be computed as follows. denote the free abelian group on n be the map induced by F(ai). If chain complex. 1)). I: groups. x0 . for all S(x 0 )C S(X) n) such that f : jS(X)j ~x. e. _) S(X) Corollary(22. 3)). b)). 4): Let (Sn) be the simplicial set described in the geometric realisation sn. 2). Then is naturally homeomorphic to then-sphere 2 nondegenerate simplices of (Sn) are is a CW complex with exactly 2 cells.
Commute, so the homomorphism iM = ioiM' , We note that from the sequence (E) above, and the facts that q~ 1 = q~ 1 , o q in diagram ~;here 1:; , giving notice 2 more arrows 0 2-simplices in ---;. I I i! oqir• q·. i u : 0 __,. ~~ in Q y. M in cc B Q~. == q~qMo in Q'( , ~;e have a From the diagram, we also I i! 0 q~l ' and namely By definition the composite M''! corresponding to the diagram I q~1 i~l! o iM'! and I the shaded triangles commute: The shaded triangles give that in fact I I qM' . ,M in I q.
BGL(R), :ZZ). :> (x) = x ® 1 + 1 ®x). Hn(X, Ill) are This result has been used by A. 7(1974)235272) to =npute tl:eJallks• of the higher K-groups of the ring of algebraic integers in a number field. The Plus Construction 30 Let The following discussion is based closely on (L]. 8): RIM\RK : Let * 2R. * ••• J, • • • • = [X 8 0 X 0 X 0 ··:1 ... ·j X X .. .. . 0 0 X 2k, otherwise. . 0 X = = 2i-l •j • ••• X X . .. . EGL(R). is a homomorphism. Q (a, 8) ~----+ Ys, are conjugate by an element of since the two maps factor through tion by A E GLd(R) for any a E GL ( R), aea /1 (a , 8) Ys, but the ~as(£) 8s, GL(R) (even by one of E(R), GLd ( R), d = 2 max(m, n), on which conjuga- equals conjugation by -1 (R), and as@ Then GL(R).