William Browder, Lecture 3

A homological approach to problems of compact group actions:
Elementary abelian p-groups.

Speaker: William Browder. Lecture 3 

Abstract: For G = (Z/p)^r, where H^2(G) = linear space over F_p with basis y_1,..., y_r, we define
the nilpotent index of C (nilp-index) as the rank of the kernel of the map H^2(G) --- >
lim over p-th powers of H*(C/G) where C is a connected G-chain complex and the map
above is defined using the generator of H_0(C/G) to map H*(G) to H*(C/G). Nil-index(C)
= r, the rank of G, implies C is H*(G)-nilpotent.
We show that for a space X, a G action is H*(G) nilpotent if and only if , for
every cyclic subgroup K in G, the restriction of the action to K is H*(K) nilpotent. The implication from G to K is obvioius since the cohomology maps surjectively, and is true for G chain complexes, but the implication in the other direction seems to need a space and the proof uses the Evens norm. As a consequence one can show for instance, that G of arbitrarily high rank can act H*(G)-nilpotently on an Eilenberg-MacLane space K(Z,3), so that the homotopy analog of Smith's theorem is very false.