Cohomological uniqueness, Massey products and the modular isomorphism problem for 2-groups of maximal nilpotency class (Q2846981)

Let \(G\) be a finite group and \(BG\) its classifying space. \textit{C.\ Broto} and \textit{R.\ Levi} [Trans. Am. Math. Soc. 349, No.4, 1487--1502 (1997; Zbl 0945.55012) and Topology 41, No. 2, 229--255 (2002; Zbl 1059.55007)] have applied Steenrod operations and Bockstein spectral sequences to prove the cohomological uniqueness of \(BG\) for dihedral and quaternion \(2\)-groups.NEWLINENEWLINEIn the present paper the authors apply iterated Massey products in \(H^\ast(BG,\mathbb{F}_2)\) to prove the cohomological uniqueness of \(BG\) for any finite \(2\)-group \(G\) of maximal nilpotency class. Then, an alternative proof (cf.\ \textit{C. Bagiński} [Commun. Algebra 20, No.5, 1229-1241 (1992; Zbl 0751.20004)] and \textit{J. F. Carlson} [J. Lond. Math. Soc., II. Ser. 15, 431--436 (1977; Zbl 0365.20015)]) of the modular isomorphism for those groups is derived.