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Abstract.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 mod/? Dyer-Lashof operations for «-fold loop spaces. The category &J/1 is very convenient for applications since its objects and morphisms are defined in a homotopy invariant way. We then define a functor, P" , from ^p" to the homotopy category of spaces and show Pjj to be left adjoint to the rc-fold loop space functor. We then show how one can exploit this adjointness in cohomological calculations to yield new results about iterated loop spaces.

Abstract. The mod 2 Steenrod algebra A and Dyer-Lashof algebra R have both striking similarities and differences arising from their common origins in "lower-indexed" algebraic operations. These algebraic operations and their relations generate a bigraded bialgebra K, whose module actions are equivalent to, but quite different from, those of A and R. The exact relationships emerge as "sheared algebra bijections", which also illuminate the role of the cohomology of K. As a bialgebra, K * has a particularly attractive and potentially useful structure, providing a bridge between those of A * and R * , and suggesting possible applications to the Miller spectral sequence and the A structure of Dickson algebras.

The algebra S of symmetric invariants over the field with two elements is an unstable algebra over the Steenrod algebra A, and is isomorphic to the mod two cohomology of BO , the classifying space for vector bundles. We provide a minimal presentation for S in the category of unstable A-algebras, i.e., minimal generators and minimal relations.From this we produce minimal presentations for various unstable A-algebras associated with the cohomology of related spaces, such as the BO(2 m − 1) that classify finite dimensional vector bundles, and the connected covers of BO . The presentations then show that certain of these unstable A-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 presentation of the cohomology of infinite dimensional real projective space, with filtered quotients the unstable modules F (2 p − 1) /AA p−2 , as described in an independent appendix.

The Dickson algebra Wn+1 of invariants in a polynomial algebra over F2 is an unstable algebra over the mod 2 Steenrod algebra A, or equivalently, over the Kudo-Araki-May algebra K of "lower" operations. We prove that Wn+1 is a free unstable algebra on a certain cyclic module, modulo just one additional relation. To achieve this, we analyze the interplay of actions over A and K to characterize unstable cyclic modules with trivial action by the subalgebra An−2 on a fundamental class in degree 2 n − a, thereby verifying unstable instances of a conjectured basis for A/AAn−2. This involves a new family of left ideals Ia in K, which play the role filled by the ideals AAn−2 in the Steenrod algebra.