A two-stage Postnikov system where 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 be the path fibration over the simply-connected space , let be the induced fibration via the map , and let and be generalized Eilenberg-Mac Lane spaces. G. Hirsch has conjectured that is additively isomorphic to , where cohomology is with coefficients. Since the Eilenberg-Moore spectral sequence which converges to has , the conjecture is equivalent to saying . In the present paper we set and the product of the two fundamental classes, and we prove that , disproving Hirsch's conjecture. The proof involves the use of homology isomorphisms developed by J. P. May, where is the reduced cobar construction. The map commutes with cup- products. Since the cup- product in is well known, and since differentials in the spectral sequence correspond to certain cup- products, we may compute on specific elements of .

**[1]**J. F. Adams,*On the non-existence of elements of Hopf invariant one*, Ann. of Math. (2)**72**(1960), 20–104. MR**0141119****[2]**H. Cartan et al.,*Algèbres d'Eilenberg-Mac Lane et homotopie*, Séminaire Henri Cartan École Normale Supérieure 1954/55, Secrétariat mathématique, Paris, 1956. MR**19**, 439.**[3]**G. Hirsch,*Cohomologie d'un espace de Postnikov (cas non stable)*(preprint).**[4]**J. P. May,*The algebraic Eilenberg-Moore special sequence*(preprint).**[5]**J. P. May,*The cohomology of principal bundles, homogeneous spaces, and two-stage Postnikov systems*, Bull. Amer. Math. Soc.**74**(1968), 334–339. MR**0239596**, 10.1090/S0002-9904-1968-11947-0**[6]**J. Peter May,*Simplicial objects in algebraic topology*, Van Nostrand Mathematical Studies, No. 11, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR**0222892****[7]**-,*The structure and applications of the Eilenberg-Moore spectral sequence*(to appear).**[8]**C. Schochet,*Unstable two-stage Postnikov systems*, Thesis, University of Chicago, Chicago, Ill., 1969.

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1971-0307242-0

Keywords:
Cup- products,
Eilenberg-Moore spectral sequence,
secondary cohomology operations,
two-stage Postnikov system

Article copyright:
© Copyright 1971
American Mathematical Society