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Let ΩBPBB be the path fibration over the simply-connected space B, let ΩBEX be the induced fibration via the map ƒ : XB, 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 ≠ E, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms C*X→ᷛ(HX)→ͪH*X developed by J. P. May, where 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 (HX) 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.




First published in the Transactions of the American Mathematical Society 157 (1971,, published by the American Mathematical Society.

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