Some free modules over the cohomology of \(K(Z/2Z,n)\): A short walk in the alps (Q2707505)

The authors consider the basic question posed by D. Arlettaz whether a \(2\)-connected Postnikov tower with finitely many non-zero, and finitely generated homotopy groups always has arbitrary high torsion in its integral cohomology groups. Motivated by this problem, the authors also investigate the existence of the example of loop maps \(\Omega f:\Omega X\to K(\mathbb Z/2\mathbb Z,n)\) with the property that \(H^*(\Omega X,\mathbb Z/2)\) is a free module over \(H^*(K(\mathbb Z/2,n),\mathbb Z/2)\). By work of H. Miller and J. Lannes, it is necessarily the case that any of these examples must satisfy \(n>1\) or otherwise sections always exist. In this paper, they give examples such that \(\Omega f\) does not admit a section. The main basic examples considered in this paper are provided by (1) the canonical multiplicative extension of Serre's map \(e:(\mathbb R \text{P}^{\infty})^n\to K(\mathbb Z/2,n)\) for \(n\geq 2\), and (2) the canonical multiplicative extension of the second Stiefel-Whitney class in the mod \(2\) cohomology of \(BSO(3)\) in the case of \(n=2\).NEWLINENEWLINEFor the entire collection see [Zbl 0955.00041].