Some problems in algebraic topology : polynomial algebras over the Steenrod algebra
We prove two theorems concerning the action of the Steenrod algebra in cohomology and homology. (i) Let A denote a finitely generated graded Fp polynomial algebra over the Steenrod algebra whose generators have dimensions not divisible by p. The possible sets of dimensions of the generators for such A are known. It was conjectured that if we replaced the polynomial algebra A by a polynomial algebra truncated at some height greater than p over the Steenrod algebras, the sets of all possible dimensions would coincide with the former list. We show that the conjecture is false. For example F11[x6,x10]12 truncated at height 12 supports an action of the Steenrod algebra but F11[x6,x10] does not. (ii) Let V be an elementary abelian 2-group of rank 3. The problem of determining a minimal set of generators for H*(BV,F2) over the Steenrod algebra was an unresolved problem for many years. (A solution was announced by Kameko in June 1990, but is not yet published.) A dual problem is to determine the subring M of the Pontrjagin ring H*(BV,F2). We determine this ring completely and in particular give a verification that the minimum number of generators needed in each dimension in cohomology is as announced by Kameko, but by using completely different techniques. Let v ε V - (0) and denote by a_5(v) ε H*(BV,F2) the image of the non-zero class in H2s-1(RP∞,F2) imeq F2 under the homomorphism induced by the inclusion of F2 → V onto (0,v). We show that M is isomorphic to the ring generated by (a_s(v),s ≥ 1, v ε V - (0)) except in dimensions of the form 2^r+3 + 2^r+1 + 2^r - 3, r ≥ 0, where we need to adjoin our additional generator.