Let ΩB → PB → B be the path fibration over the simply-connected space B, let ΩB → E → X be the induced fibration via the map ƒ : X → B, and let X and B be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that H*E is additively isomorphic to ΤοτH*B(Z₂,H*X), where cohomology is with Z₂ coefficients. Since the Elienberg-Moore spectral sequence which converges to H*E has E₂ = ΤοτH*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 ƒ*i = the product of the two fundamental classes, and we prove that E₂ ≠ E₃, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms C*X→ᷛC̄(H*ΩX)→ͪH*X developed by J. P. May, where C̄ is the reduced cobar construction. The map g [represented above in uppercase due to html limitations - Ed.] commutes with cup-¹ products. Since the cup-¹ product in C̄(H*ΩX) 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.
C. Schochet, A two-stage Postnikov system where E₂ ≠ E∞ in the Eilenberg-Moore Spectral sequence, Transactions of the American Mathematical Society 157 (1971) 113-118.