94c:57060 57T10 20J06 22E40 55R40

Broto, Carlos ^{(}E-BARA^{)}; Henn, Hans-Werner ^{(}D-HDBG^{)} Some remarks on central elementary abelian p-subgroups and cohomology of classifying spaces. Quart. J. Math. Oxford Ser. (2) 44 (1993), no. 174, 155–163. Let G be a compact Lie group, C a central elementary abelian psubgroup of G, and X a G-space on which C acts trivially (for example, a point). The authors observe that there is then a map BC × (EG ×_{G} X) → EG ×_{G} X which induces an H^{∗}BC-comodule structure on H^{∗}(EG ×_{G} X) (here we use H^{∗} to denote cohomology with Z/p coefficients). This leads to a short proof of a result of Duflot and Stong: If H^{∗}X is finite-dimensional over Z/p then the depth of H^{∗}(EG ×_{G} X) is at least as big as the rank of C. On the other hand, choosing any point in X gives a map BC → EG ×_{G} X, and the authors show that the image of the induced map H^{∗}(EG ×_{G} X) → H^{∗}BC is a sub-Hopf algebra of H^{∗}BC. This in fact gives a very precise hold on the structure of that image.

The authors then use this result to look at the following question: When does a compact Lie group G have a central elementary abelian p-subgroup C that is detecting, in the sense that the restriction H^{∗}BG → H^{∗}BC is a monomorphism? Their answer is that G has such a subgroup precisely when G ≅ E × N × S, where E is an elementary abelian p-group, N is a central extension of a torus by a finite group of order prime to p, S is trivial if p > 2, and S is a product of S^{3}’s if p = 2. In particular, if G is finite then G ≅ C × N where N is a group of order prime to p. Steven R. Costenoble (1-HOFS)

\begin{matrix}c \\ ◯ \end{matrix} Copyright American Mathematical Society 1994, 1996