Sub-Hopf algebras of the Steenrod algebra and the Singer transfer

Let A denote the mod 2 Steenrod algebra (see Steenrod and Epstein [28]). The problem of computing its cohomology H∗,∗(A) is of great importance in algebraic topology, for this bigraded commutative algebra is the E2 term of the Adams spectral sequence (see Adams [1]) converging to the stable homotopy groups of spheres. But despite intensive investigation for nearly half a century, the structure of this cohomology algebra remains elusive. In fact, only recently was a complete description of generators and relations in cohomological dimension 4 given, by Lin and Mahowald [12, 11]. In higher degrees, several infinite dimensional subalgebras of H∗(A) have been constructed and studied. The first such subalgebra [1], called the Adams subalgebra, is generated by the elements hi ∈ H1,2 i (A) for i ≥ 0. Mahowald and Tangora [14] constructed the so-called wedge subalgebra which consists of some basic generators, propagated by the Adams periodicity operator P1 and by multiplication with a certain element g ∈ H4,20(A)1. The wedge subalgebra was subsequently expanded by another kind of periodicity operator M , discovered by Margolis, Priddy and Tangora [16]. On the other hand, perhaps the most important result to date on the structure of H∗(A) is a beautiful theorem of Palmieri [24] which gives a version of the famous Quillen stratification theorem in group cohomology for the cohomology of the Steenrod algebra. Loosely