James P. Lin

Loops of H-spaces with finitely generated cohomology rings

Abstract Let X be a simply connected a p -space. The mod p cohomology rings of Ω,X are studied. When these rings are finitely generated as algebras, Ω,X hasthe mod p homotopy type of a generalized Eilenberg-MacLane space. If X is just an H -space with H ∗ ( Ω,X ; Z p ) finitely generated as an algebra, H ∗ ( Ω,X ; Z p ) is still primitively generated free commutative.

On the collapse of certain Eilenberg–Moore spectral sequences

Abstract Let Z be a path connected H -space with H ∗ (Z; Z p ) concentrated in even degrees. Then the Eilenberg–Moore spectral sequences associated to the path loop fibrations ΩZ→PZ→Z,Ω 2 Z→PΩZ→ΩZ collapse at the E 2 term.

Homotopy-commutative

Let X be an H-space with H* (X; Z2) Z2[xl,..., Xd] ? A(y1, .* I Yd), where degx, = 4 and y, = Sq1 xi . In this article we prove that X cannot be homotopy-commutative. Combining this result with a theorem of Michael Slack results in the following theorem: Let X be a homotopycommutative H-space with mod 2 cohomology finitely generated as an algebra. Then H*(X; Z2) is isomorphic as an algebra over A(2) to the mod 2 cohomology of a torus producted with a finite number of CP(oo)s and K(Z2r, 1)s. 0. INTRODUCTION In this article we prove the following theorem: Theorem A. Let X be an H-space with H* (X; Z2) =Z2 [x,s *@ .. Xd] 9A(ylj SYd) where degxi = 4 and yi = SqI xi . Then X cannot be homotopy-commutative. The significance of Theorem A lies in its relationship to the following theorem, due to Michael Slack: Theorem (Slack). Let X be a homotopy-commutative H-space with mod 2 cohomology finitely generated as an algebra. Then (1) All even-degree generators have infinite height and are in degrees two and four. (2) All odd-degree generators lie in degrees one andfive. The one-dimensional generators have infinite height and the five-dimensional generators are exterior. (3) Sq1: QH4(X; Z2) _+ QH5(X; Z2) is an isomorphism. Received by the editors February 27, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 55P45, 55S40.

H

Given a multiplicative fibration F → j E → π B we study the module of indecomposables QH*(E; Z p ) for p a prime.

-spaces

It is shown that the mod 3 cohomology of a homotopy associative mod 3 H-space which is rationally equivalent to the Lie group E 7 and which has integral 3-torsion is isomorphic to that of E7 as a Hopf algebra over the mod 3 Steenrod algebra.

Indecomposables of multiplicative fibrations

We prove here that a certain 14 connected finite complex cannot admit the structure of anH-space. Thea andc invariants of Zabrodsky are used here. It was conjectured by Adams and Wilkerson that the complex described admitted anH-structure.

Characterization of the mod 3 cohomology of E~7

Abstract In this note we prove that any simply connected finite loop space that has one three-dimensional rational cohomology generator has the rational cohomology of a Lie group. Other results are obtained in the case that the rational cohomology has more than one three-dimensional generator.

On 14-connected finiteH-spaces

Abstract A new factorization of the cup product u ( Sq 8 u ) through secondary operations is used to study the existence of spaces whose mod 2 cohomology is a polynomial algebra. Criteria are developed to determine when a finite loop space has the rational cohomology of a Lie group. In a subsequent paper it is shown that the first nonvanishing homotopy group of a finite H -space must occur in degrees 1, 3 or 7.

The rational cohomology of finite loop spaces

A finite mod 3 homotopy commutative, homotopy associative simply connected H-space has mod 3 cohomology isomorphic to the cohomology of a product of Sp(2)s. This result generalizes to other odd primes.

Cup products and finite loop spaces

Abstract In this paper we prove the following theorems: Theorem 1. If X is a two-torsion free finite loop space whose type consists of integers that are divisible by 4 and are less than 60, then X is of Lie type. Theorem 2. If X is a two-torsion free finite loop space whose type consists of integers that are divisible by 4 and for which the numbers of 43-dimensional and 51-dimensional generators of H∗(X;Q) are greater than or equal to the numbers of 47-dimensional and 55-dimensional generators, respectively, then X is of Lie type.