A two-stage Postnikov system where $E\sb{2}\not=E\sb{\infty }$ in the Eilenberg-Moore spectral sequence

Abstract

Let Q.B -* PB -* B be the path fibration over the simply-connected space B, let Q.B—> E-+ X be the induced fibration via the map /: X —» B, and let X and B be generalized Eilenberg-MacLane spaces. G. Hirsch has conjectured that H*E is additively isomorphic to TorH.£ (Z2, H*X), where cohomology is with Z2 coefficients. Since the Eilenberg-Moore spectral sequence which converges to H*E has E2 = TorH.B(Z2, H*X), the conjecture is equivalent to saying E2 = E^. In the present paper we set X=K{Z2 + Z2, 2), B = K(Z2, 4) and /*i' = the product of the two fundamental classes, and we prove that E2J=E3, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms C*X X C(H*QX) X H*X developed by J. P. May, where C is the reduced cobar construction. The map g commutes with cup-1 products. Since the cup-1 product in C{H*Q.X) is well known, and since differentials in the spectral sequence correspond to certain cup-1 products, we may compute d2 on specific elements of E2. Introduction. A two-stage Postnikov system 3P is a diagram Q.B —> U.B