## Document Type

Article

## Abstract

Let Ω*B* → *PB* → *B* be the path fibration over the simply-connected space *B*, let Ω*B* → *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 Τοτ_{H*B}(*Z*₂,*H***X*), where cohomology is with *Z*₂ coefficients. Since the Elienberg-Moore spectral sequence which converges to *H***E* has *E*₂ = Τοτ_{H*B}(*Z*₂,*H***X*), the conjecture is equivalent to saying *E*₂ = *E*_{∞}. In the present paper we set *X* = *K*(*Z*₂ + *Z*₂,2),*B* = *K*(*Z*₂,4) and ƒ**i* = the product of the two fundamental classes, and we prove that *E*₂ ≠ *E*_{₃}, disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms *C***X*→ᷛ*C̄*(*H**Ω*X*)→ͪ*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**Ω*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.

## Disciplines

Mathematics

## Recommended Citation

C. Schochet, A two-stage Postnikov system where *E*₂ ≠ *E*_{∞} in the Eilenberg-Moore Spectral sequence, *Transactions of the American Mathematical Society* **157** (1971) 113-118.

## Comments

First published in the

Transactions of the American Mathematical Society157(1971, http://dx.doi.org/10.1090/S0002-9947-1971-0307242-0), published by the American Mathematical Society.