On the Hopf Algebra Structure of the mod 2 Cohomology of a Finite $H$-Space

Abstract

The action of the Steenrod algebra on the cohomology of a finite //-space has placed several restrictions on the possible spaces which can be finite //"-spaces. It is now known, for example, that the loop space of a finite //-space has no homology torsion. We continue this study here by showing that in the mod 2 cohomology of a finite //-space, every generator of degree 4/e + l is in the image of Sq 2k .This result has several consequences. For example, if an integer n has dyadic expansion n=2where O^s^St+i we say n has length I. If 4& + 1 has length / and there is a generator of the mod 2 cohomology of a finite //-space in degree 4& + 1, then there is a generator in degree 2 Z -1. Hence if a finite //-space is highly connected this result implies the non-existence of generators of small length.Another implication is that there are no 4/e dimensional primitive indecomposables in the mod 2 cohomology of the loop space. Also, all generators of degree 8/2+5 in the cohomology of a finite //-space are in the image of Sq 2 . Throughout this paper it is assumed that all finite //-spaces used in this paper are simply connected and have associative mod 2 homology rings. Under this assumption it is easily shown that finite //-spaces that have no two torsion have primitively generated mod 2 cohomology rings. Hence their structure over the Steenrod algebra is restricted by several results of Thomas [4]. It is therefore more interesting to study finite //-spaces that have two torsion in their homology.Under these circumstances it was shown that the two torsion is of order at most two and arises from odd degree mod 2 cohomology generators whose cup product is non trivial. In chapter 3 of this paper a structure theorem is given for the action of the Steenrod algebra on these generators. In particular if y is Communicated by N.