By Milgram R. (ed.)
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Extra resources for Algebraic and Geometric Topology, Part 2
Consequently each d2 m may be regarded as a function from the kernel of one primary operation to the cokernal of another primary operation. ing to offer you the differential dn I am go- as defining an opera- tion of the nth kind, and I ought to verify that this agrees 'Vlith onets notion of the usual procedure with universal example s • Our proc edure is given by the foIl mV' ing diagram: fft +n _ l (X, Mn _l ) \ \ ~ /JI' F = f" ~i'" E~,t J = identity (subgroup of ~t(X, Ko) That is, you realize a cohomology m-tuple by a map from stx into K: o you lift this, if you can, to a map, ~, into the universal example, • ~II" "n- I" as giving you a mt-tuple in of this m'-tuple by ..
Ce, which works under conditions: even more restrictive than I have already stated. That is, I shall assume: (III) E is a free module over the exterior algebra generated by -'(. Sql. This is equivalent to supposing that H" (Y; Z) has no elements of of order 2f 00 order, and all its elements are actually of order 2. This evidently excludes the case X =Y = So, so I must give one or two examples to show that it does not exclude all cases of interest. Ex. 1. Rp2tJRP2U. Y is the stable object corresponding to This example is relevant to the vector rie1d problem.
It is sufficient to consider the first four maps in the 35 sequence. The required constructions can be performed "on the left II and commute wi th the opera tion #Sl used in defining our category. Lemma 5. The sequence q* Map (SW,M) - ) Map (SW,SX) (Sf)* ) Map (SW,Sy) is exact. This lemma show that if we map SW into the sequence (a) of lemma 4, we have exactness at the fourth term (y UfCX) U jCY. Since all subsequent terms are also "fourth terms," we have exactness at all subsequent terms also.