Bocksteins and the nilpotent filtration on the cohomology of spaces

TL;DR

This paper proves the unbounded strong realization conjecture at the prime 2 for spaces with cohomology having trivial Bockstein action in high degrees, advancing understanding of cohomology modules over the Steenrod algebra.

Contribution

It extends the validity of the unbounded strong realization conjecture to a new class of spaces at the prime 2, where the Bockstein action is trivial in high degrees.

Findings

The unbounded strong realization conjecture holds at prime 2 for certain spaces.

Spaces with trivial Bockstein action in high degrees satisfy the conjecture.

Advances the understanding of cohomology modules over the Steenrod algebra.

Abstract

N Kuhn has given several conjectures on the special features satisfied by the singular cohomology of topological spaces with coefficients in a finite prime field, as modules over the Steenrod algebra. The so-called realization conjecture was solved in special cases in [Ann. of Math. 141 (1995) 321-347] and in complete generality by L Schwartz [Invent. Math. 134 (1998) 211-227]. The more general strong realization conjecture has been settled at the prime 2, as a consequence of the work of L Schwartz [Algebr. Geom. Topol. 1 (2001) 519-548] and the subsequent work of F-X Dehon and the author [Algebr. Geom. Topol. 3 (2003) 399-433]. We are here interested in the even more general unbounded strong realization conjecture. We prove that it holds at the prime 2 for the class of spaces whose cohomology has a trivial Bockstein action in high degrees.