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Extra resources for Topics in K-Theory: The Equivariant Ka1/4nneth Theorem in K-Theory. Dyer-Lashof Operations in K-Theory

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O A Kunneth embedding for h .. -- -- - - ,uW --~u (in C ) is a mapping W -- tO @ : X § Z such that Z is - _a right Kunneth space and h (~) : h (Z) + h*(X) is an epimorphism. such a ~ exists, (c) C If for a @ire n X ,u~ I say X admits a Kunneth embeddin5 fo__~rh . has enough Kunneth spaces for h if for every X in C some susl0ension smx admits a Kunneth embeddin~ for h , and m is bounded as X runs over C . . . ~ o Notes I. Trivially, e @ sTM is a Kunneth space for any h ; and the category of K~nneth spaces is closed under suspensions and wedges.

Of the maps involved; and < is the one for h , The d i a g r a m i s c o m m u t a t i v e f r o m t h e so by definitions in the top row we are using the fact that h (X x B) is not merely isomorphic to h (X) @ h (B), but the h (B)-module structure is identified h u n d e r K @ 1 w i t h t h e a c t i o n on t h e r i g h t f a c t o r i n t h e t e n s o r p r o d u c t ; hence the isomorphism ~ in the top line. We deduce that

2. < : h (smx /X +l ) @ P P h is an isomorphism. ;h ), au@mented by c, e" over ~*(X), in w D. ;h ~*(Y) respectively; and the discussion which precedes it they are actually resolutions. 1. h-chain complexes, but replacing < by <), is therefore an isomorphism from each E1 p @ E1q ElP-q, Hence : Cp = ElP ), as by lemma (Here and etc. ). ey : Y + Y. 5 (i) tells us that the diagram 41 l@s" (1) s@l C, _8 h*(Y)< h C, _8 D, h EI(X . @ Y; ~*)< is commutative, >h EI(X , 8 Y,;~*) where the columns are the E these are isomorphlsms of chain complexes, resolutions.

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