On the Segal Conjecture for Z 2 × Z 2

Abstract

The Segal conjecture regarding the Burnside ring and stable cohomotopy of a finite group G is reduced for the case G = Z2X Z2 to a statement about Ext groups. This statement has since been proved by H. Miller, J. F. Adams and J. H. C. Gonawardena. The Segal conjecture states that for any finite group G, there is an isomorphism (G: A (G) -> 7T(BG) from the completed Burnside ring to the zeroth stable cohomotopy group of its classifying space. The conjecture was proved for cyclic groups in [4, 2, and 9]. In this paper we reduce the conjecture for Z2 X Z2 to a statement, (1), about Ext groups. This Ext statement was conjectured by Davis in [1] based upon extensive calculations and is proved by Adams, Gunawardena, and Miller in [7]. Let P = Z2[x, x'] be made into a module over the mod 2 Steenrod algebra A as in [5], and for S < k < n < xo let Pk' be the subquotient of P which is nonzero in degree k through n, inclusive. If n = x or k = -x, they will usually be omitted from the notation. The suspension liM of a graded module M is defined by (7Jm),+j= Mi. STATEMENT 1. There is an epimorphism of A -modules EP (g) Ep_ 1 Z2 (D EPO which induces an isomorphism in ExtA (, Z2). In fact p(sxa 0 sx b) = (?~I(?~~'(+ a+,)Sa+b+ I) +(SX () SX = ((a0+l)(b0+l) (a+ b+l)SX ) As mentioned above, Statement 1 is proved in [7]. The main result of this paper is THEOREM 2. Statement 1 implies that CZ2 XZ2 is an isomorphism. The proof of Theorem 2 mimicks [4]. The main part is to use Statement 1 to calculate 'u?(RP' x RP ) via the Adams spectral sequence. We begin by deducing from (1) the Ext groups relevant to [RPI A RP,I, S0I, the group of stable homotopy classes of maps. DEFINITION 3. If M is a (left) A-module, let DM denote the dual module, made into a left A-module using the antiautomorphism X; i.e., (DM)k = HomZ2(M k, Z2) with (9(+)(m) = ((X(O)m). Received by the editors January 30, 1981. 1980 Mathematics Subject Classification Primary 55Q10; Secondary 55T15.