Abstract In 1947, N.E. Steenrod defined the Squares, which are mod 2 cohomology operations, using explicit cochain formulae for cup- i products of cocycles. He later recast construction in more general homological terms, group homology and acyclic model methods, rather than formulae, to define p operations all primes . Steenrod’s student J. Adem applied point view prove fundamental relations, k...

Let p be a prime, and let BV be the classifying space of an elementary abelian p-group V . J. Lannes [L1] has shown that, in many cases, the mod p cohomology of the space of continuous maps Map(BV,X) can be computed by applying an easily defined functor to H∗(X; Z/p). He lets U denote the category of unstable modules over the mod p Steenrod algebra A, and then notes that the functor H∗(BV ) ⊗ :...

We observe that the Poincaré duality isomorphism for a string manifold is an isomorphism of modules over the subalgebra A(2) of the modulo 2 Steenrod algebra. In particular, the pattern of the operations Sq, Sq, and Sq on the cohomology of a string manifold has a symmetry around the middle dimension. We characterize this kind of cohomology operation duality in term of the annihilator of the Tho...

Previous work constructed a generalized truncated Brown-Peterson spectrum of chromatic height 2 at the prime 2 as an E∞-ring spectrum, based on the study of elliptic curves with level-3 structure. We show that the natural map forgetting this level structure induces an E∞-ring map from the spectrum of topological modular forms to this truncated Brown-Peterson spectrum, and that this orientation ...

Let G be a topological group acting on a space X. We construct a family of Steenrod’s ∪i -product [Ann. of Math. (2) 48 (1947) 290] on the Bredon–Illman cochain complex of X [Quart. J. Math. Oxford Ser. (2) 47 (1996) 199]. As corollaries, we get the existence of Steenrod squares on Bredon– Illman cohomology with appropriate coefficients as well as the triviality of the Gerstenhaber bracket indu...

We present an infinite family of recursive formulas that count binary integer partitions satisfying natural divisibility conditions and show these counts are interrelated via partial sums. Moreover, we interpret the study in language graded polynomial rings apply this to mod <mml:se...