Frank Williams

A global structure theorem for the mod 2 Dickson algebras, and unstable cyclic modules over the Steenrod and Kudo-Araki-May algebras

The Dickson algebra W n +1 of invariants in a polynomial algebra over [ ] 2 is an unstable algebra over the mod 2 Steenrod algebra [Ascr ], or equivalently, over the Kudo–Araki–May algebra [Kscr ] of ‘lower’ operations. We prove that W n +1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyse the interplay of actions over [Ascr ] and [Kscr ] to characterize unstable cyclic modules with trivial action by the subalgebra [Ascr ] n −2 on a fundamental class in degree 2 n – a . This involves a new family of left ideals [Iscr ] a in [Kscr ], which play the role filled by the ideals [Ascr ][Ascr ] n−2 in the Steenrod algebra.

Global structure of the mod two symmetric algebra, H*(BO;F2), over the Steenrod algebra

The algebra S of symmetric invariants over the eld with two elements is an unstable algebra over the Steenrod algebraA, and is isomor- phic to the mod two cohomology of BO, the classifying space for vector bundles. We provide a minimal presentation for S in the category of un- stable A-algebras, i.e., minimal generators and minimal relations. From this we produce minimal presentations for various unstableA-algebras associated with the cohomology of related spaces, such as the BO(2 m 1) that classify nite dimensional vector bundles, and the connected covers of BO. The presentations then show that certain of these unstableA-algebras coalesce to produce the Dickson algebras of general linear group invariants, and we speculate about possible related topological realizability. Our methods also produce a related simple minimal A-module presenta- tion of the cohomology of innite dimensional real projective space, with ltered quotients the unstable modulesF (2 p 1)=AAp 2 , as described in

Hopf constructions and higher projective planes for iterated loop spaces

We define a category, fpn (for each n and p), of spaces with strong homotopy commutativity properties. These spaces have just enough structure to define the modp Dyer-Lashof operations for n-fold loop spaces. The category ffn is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, Pp, from ffn to the homotopy category of spaces and show Pp to be left adjoint to the n-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.

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

If IHI < ~0, H is said to have bounded width. Natural questions are: (1) If two maps f, g: X+ M are homotopic, are they connected by a homotopy of bounded width? (2) If so, what can be said about the bound? Clearly some restrictions on X and M are necessary in order to give a positive answer to (1). For example, letting X = R’ and M = S’, the constant map and the exponential map t +exp (it) are homotopic but not by a homotopy of bounded width. The fundamental group of M turns out to be the key. It is shown in [3] that if M is a closed Riemannian manifold with finite fundamental group, and X is a finite dimensional normal space, the answer to question (1) is yes. In fact, there is a finite bound b such that any two homotopic maps are connected by a homotopy of width less than b. Finally, if b(X, M) denotes the infimum of all such bounds, the number

-spaces

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.

The width of homotopies into spheres

We describe bialgebras of lower-indexed algebraic Steenrod operations over the field with p elements, p an odd prime. These go beyond the operations that can act nontrivially in topology, and their duals are closely related to algebras of polynomial invariants under subgroups of the general linear groups that contain the unipotent upper triangular groups. There are significant differences between these algebras and the analogous one for pD 2, in particular in the nature and consequences of the defining Adem relations.

On 14-connected finiteH-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.

The odd-primary Kudo-Araki-May algebra of algebraic Steenrod operations and invariant theory

In this paper our previous joint work with Calder on the width of homotopies into compact Riemannian manifolds with finite fundamental groups is extended to the equivariant setting. We produce necessary conditions under which the widths of equivariant homotopies are bounded and characterize these equivariant bounds in terms of the fixed point submanifolds of the action. Finally, we apply these results to covering spaces to relate the (non-equivariant) bounds for a manifold with finite fundamental group to those of its universal covering space.

The type of a torsion free finite loop space

The hit problem for a module over the Steenrod algebra