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

Abstract:

Let $\Omega B \to PB \to B$ be the path fibration over the simply-connected space $B$, let $\Omega B \to E \to X$ be the induced fibration via the map $f:X \to B$, and let $X$ and $B$ be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that ${H^ \ast }E$ is additively isomorphic to ${\text {Tor}}_{H^ \ast B}({Z_2},{H^ \ast }X)$, where cohomology is with ${Z_2}$ coefficients. Since the Eilenberg-Moore spectral sequence which converges to ${H^ \ast }E$ has ${E_2} = {\text {Tor}_{H^ \ast B}}({Z_2},{H^ \ast }X)$, the conjecture is equivalent to saying ${E_2} = {E_\infty }$. In the present paper we set $X = K({Z_2} + {Z_2},2),B = K({Z_2},4)$ and ${f^ \ast }i =$ the product of the two fundamental classes, and we prove that ${E_2} \ne {E_3}$, disproving Hirsch’s conjecture. The proof involves the use of homology isomorphisms $C^\ast X \stackrel {g}{to} \bar C({H^ \ast }\Omega X)\stackrel {h}{\to } {H^ \ast }X$ developed by J. P. May, where $\bar C$ is the reduced cobar construction. The map $g$ commutes with cup-$1$ products. Since the cup-$1$ product in $\bar C({H^ \ast }\Omega X)$ is well known, and since differentials in the spectral sequence correspond to certain cup-$1$ products, we may compute ${d_2}$ on specific elements of ${E_2}$.