CLASSIFYING SPACES OF SPORADIC SIMPLE GROUPS FOR ODD PRIMES

Abstract

Let P be a fixed p-group for an odd prime. We are interested in the localized classifying spaces BG(_p) (hereafter denoted simply by BG) for various groups G sharing the same Sylow p-subgroup P. If such groups G become bigger then BG becomes smaller, because a transfer argument shows that BG is a stable summand of BP and of intermediate subgroups. For this reason, C. B. Thomas [16] first found that, in many cases, the odd component of the cohomology of sporadic simple groups should be simple even if the cohomology of P is quite complicated. In particular, when G is the biggest Janko group J₄, D. J. Green [6] showed that H^even(G)/3 is the Dickson algebra of rank two. The Janko group has a Sylow 3-subgroup 3¹⁺²₊ = E, the extra special 3-group of order 3³ and of exponent 3. Tezuka and Yagita [14] studied the even-dimensional cohomology of all sporadic simple groups with [mid ]P[mid ] = p³; indeed, in these cases P is isomorphic to the extra-special group p¹⁺²₊ = E.

In this paper, we study BG for simple groups G with a Sylow p-subgroup E. We note the importance of G-conjugacy classes of p-pure elementary abelian p-subgroups of rank two. Here ‘p-pure’ means that all nonzero elements in the subgroup are G-conjugate. In Section 1, we see that BG is expressed as a homotopy pushout of B(N_G(E)) and of B(p-pure subgroup), when N_G(E) = N_G(Z(E)). The last condition is always satisfied if p>3. In Section 2, the case p = 3 is studied; for example, we see the homotopy equivalence BJ₄ ≅ BRu. In Section 3, we show that stable homotopy splitting is given from the dominant summands of E and of non-p-pure subgroups. When p = 3 the splitting is given explicitly; for example, BJ₄ is the summand induced from the trivial representation in F₃[Out(E)]. In the last section, we add the list of sporadic simple groups (due to Yoshiara) and cohomology H*(G)/√0 for p = 3 from the paper [14] for the reader's convenience. The author thanks Satoshi Yoshiara, who pointed out to him the paper of Benson and taught him the importance of p-pure subgroups. He also thanks Michishige Tezuka; indeed, most of the results in this paper are natural consequences of results in the joint paper [14] with Tezuka.