Geometric Applications of Homotopy Theory I: Proceedings, by J. C. Becker, R. E. Schultz (auth.), M. G. Barratt, M. E.

By J. C. Becker, R. E. Schultz (auth.), M. G. Barratt, M. E. Mahowald (eds.)

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By J. C. Becker, R. E. Schultz (auth.), M. G. Barratt, M. E. Mahowald (eds.)

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Read or Download Geometric Applications of Homotopy Theory I: Proceedings, Evanston, March 21–26, 1977 PDF

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Extra info for Geometric Applications of Homotopy Theory I: Proceedings, Evanston, March 21–26, 1977

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P r o c e e d i n g as i n ( 4 . 2 ) , e4(j1~) the fact that S 16 i s no_~t p a r a l l e l i s a b l e means = 0. g. see [2]). h*~r 3 evaluates non zero mod 2 on to 1 mod 2 on S 15. S 15, whence so does h*~p 02" In conclusion, e02 = <~*h*~P02, We can now assemble g16> =~1 # 0. E] our results into the table whose implications we described in the introduction. 4) Table 16 stem 8 stem manifold manifold [Xl6,~+~*f] J8 e2 1 i e4 i e01 0 I e02 ? invariant For those who like universal manifolds, X 8n merely by twisting we see that X 8n is a ~ ~ on the top cell.

K2k(~t;~/2 ). = {x) [Quillen] K2k_1(Ft). y c K2(~t;~/2 ) which generates ~{y] p~ k, p~+1 yk K2k_1(Ft;~/p) 9 c K2(~t;~/4). +1). 1)) = kyk-lx ~ 0 for K](Ft;~/p), = ~/t k - I are discussed, of facts we use. +1) and Br of the = ~ ~/2 is the mod 2 reduction so that yk of the proof proceeds generates as in the case of 4 1 t - 1, K . ( ~ t ; ~ 1 2 ) = A ( x ) ~ l e [ y ] , dim x + 1 = dim y = 2. Now suppose reduce ~t d plt d - ] this to the previous of ~t" and p # td-1 - I, p odd, d > I. , consider is a p r o j e c t i v e be d e f i n e d extending T : GL(m,A I ) ~ GL(mq,A) a ring the by A and a homomorphism A-module tensoring the m a t r i x r i n g of d e f i n i t i o n ) be defined by treating of r a n k q.

Zp Also, f is smooth. 35 (7 the standard projection) has degree Proof n = i ±I on the top cell. is essentially in [4]. For both cases, choose a regular value of and count its preimage. For example, when f'(~, i), where (½, i) a D 8 x S 7~- S(~2). 3) Corollary In H8n-l(S([n )) ~ 7 ~ , n = 2 f' a regular value is given by [] we have f*~mgn = x, where x is the generator. Proof We have f*z*g8n-I = x, and we know that generates HSn-l(s 8n-l) ~7/_2). [] We can now define our manifolds. q*$n over S 4n+l.

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