On The Homotopy Type of 𝐵𝐺 for Certain Finite 2-Groups 𝐺

Abstract

Abstract. We consider the homotopy type of classifying spaces BG, where G is a finite p-group, and we study the question whether or not the mod p cohomology of BG, as an algebra over the Steenrod algebra together with the associated Bockstein spectral sequence, determine the homotopy type of BG. This article is devoted to producing some families of finite 2-groups where cohomological information determines the homotopy type of BG.

Cohomological uniqueness, Massey products and the modular isomorphism problem for $2$-groups of maximal nilpotency class

“…In [BL97], C. Broto and R. Levi initiated the study of the cohomological uniqueness of BG in terms of Steenrod operations and Bockstein spectral sequences. In this setting, they proved the cohomological uniqueness of the classifying space of every dihedral 2-groups [BL97], and every quaternion group [BL02].…”

“…In [BL97], C. Broto and R. Levi initiated the study of the cohomological uniqueness of BG in terms of Steenrod operations and Bockstein spectral sequences. In this setting, they proved the cohomological uniqueness of the classifying space of every dihedral 2-groups [BL97], and every quaternion group [BL02]. Unfortunately, the available techniques seem not to be strong enough to decide the cohomological uniqueness of the classifying space of semidihedral 2groups in terms of Steenrod operations and Bockstein spectral sequences.…”

Cohomological uniqueness of some p-groups

“…In this regard, Broto and Levi [2] suggested that mod p cohomology rings of finite p-groups should be considered objects in the category K β of unstable algebras endowed with Bockstein spectral sequences (see § 2 for precise definitions). Here we follow that line and consider the family of groups studied by Leary in [7], proving the following theorem.…”