Let n = (n1, . . . , nr). The quotient space Pn := S n1× · · ·×Snr/(x ∼ −x) is what we call a projective product space. We determine the integral cohomology ring H∗(Pn) and the action of the Steenrod algebra on H∗(Pn;Z2). We give a splitting of ΣPn in terms of stunted real projective spaces, and determine the ring K∗(Pn). We relate the immersion dimension and span of Pn to the much-studied sect...

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 vario...

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of...

In the early 1970s, Dennis Sullivan conjectured that for any finite-dimensional space K, the space of pointed maps map∗(BZ/p,K) must be weakly contractible: all of its homotopy groups must be zero. In the early 1980s, Haynes Miller proved this conjecture by developing, essentially from scratch, huge bunches of very difficult pure algebra. The algebraic techniques led to the development of the T...

We say that a space X admits a homology exponent if there exists an exponent for the torsion subgroup of H * (X; Z). Our main result states if an H-space of finite type admits a homology exponent, then either it is, up to 2-completion, a product of spaces of the form BZ/2 r , S 1 , CP ∞ , and K(Z, 3), or it has infinitely many non-trivial homotopy groups and k-invariants. We then show with the ...

The spectral sequence associated to the Arone-Goodwillie tower for the n-fold loop space functor is used to show that the first two non-trivial layers of the nilpotent filtration of the reduced mod 2 cohomology of a (sufficiently connected) space with nilpotent cohomology are comparable. This relies upon the theory of unstable modules over the mod 2 Steenrod algebra, together with properties of...

We present some data on the cohomology of the motivic Steenrod algebra over an algebraically closed field of characteristic 0. Our results are based on computer calculations and a motivic version of the May spectral sequence. We discuss features of the associated Adams spectral sequence, and use these tools to give new proofs of some results in classical algebraic topology. We also consider a m...