The connected covers of the classifying space BO induce a decreasing filtration {B,) of H, ( B O ;212) by sub-Hopf algebras over the Steenrod algebra A. We describe a multiplicative grading on H*( B O ;212) inducing a direct sum splitting of B, over A,, where {A,) is the usual (increasing) filtration of A. The pieces in the splittings are finite, and the grading extends that of H,R2S3 which spl...

Unstable modules over the Steenrod algebra with only top $k$ operations are introduced in language of ringoids. We prove category such has homological dimension at most $k$. A pratical method, which generalizes $\Lambda$ complex, to compute $\mathrm{Ext}$ group from spheres is proposed. also able establish several functors relate and unstable algebra, describe connections between groups them.

We compute the cohomology algebra of P (1), the subalgebra of the Steenrod algebra generated by P 1 and P p. This completes a partial result given by Arunas Liulevicius in 1962 and provides explicit representatives in the cobar construction for all but one of the algebra generators.

We establish a motivic version of Adams’ vanishing line of slope 1 2 in the cohomology of the motivic Steenrod algebra over C.

We establish a motivic version of Adams’ vanishing line of slope 1 2 in the cohomology of the motivic Steenrod algebra over C.

The indicated operation algebra is studied by methods dual to the usual ones for studying the Steenrod algebra. In particular, the operations are constructed using higher symmetries of the shuffle map and their "Adem relations" are computed using the transfer map in the cohomology of symmetric groups.

We characterize Hopf spaces with finitely generated cohomology as algebra over the Steenrod algebra. We “deconstruct” the original space into an Hspace Y with finite mod p cohomology and a finite number of p-torsion EilenbergMac Lane spaces. One reconstructs X from Y by taking extensions by principal H-fibrations. We give a precise description of homotopy commutative H-spaces in this setting an...