Homotopy-commutative $H$-spaces

Abstract

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.