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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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


Author: Claude Schochet
Journal: Trans. Amer. Math. Soc. 157 (1971), 113-118
MSC: Primary 55H20
MathSciNet review: 0307242
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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\mathop \to \limits^g \bar C({H^ \ast }\Omega X)\mathop \to \limits^h {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}$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0307242-0
PII: S 0002-9947(1971)0307242-0
Keywords: Cup-$ 1$ products, Eilenberg-Moore spectral sequence, secondary cohomology operations, two-stage Postnikov system
Article copyright: © Copyright 1971 American Mathematical Society