Secondary cohomology and \(k\)-invariants
type: Item
instance of: scholarly article
title: Secondary cohomology and \(k\)-invariants
zbMATH Open document ID: 1259.55008
author: Mihai D. Staic
published in: Bulletin of the Belgian Mathematical Society - Simon Stevin
publication date: 2012-11-29T00:00:00Z
full work available at URL: https://arxiv.org/abs/0909.1086
full work available at URL: https://projecteuclid.org/euclid.bbms/1347642383
review text: Let \((X,x_0)\) be a pointed topological space. The realization problem asks when a family of isomorphisms \(\phi_i : \pi_i(X) \to \pi_i(Y)\) is induced by a morphism \(f: X\to Y\). \textit{S. Eilenberg} and \textit{S. MacLane} [Proc. Natl. Acad. Sci. USA, 32, 277--280 (1946; Zbl 0061.40707)] gave a solution to this problem in terms of the \(k\)-invariant in the case \(\pi_i(X)=0\) for \(i>2\). Inspired by their work, the author constructs an invariant called the secondary \(k\)-invariant. This is an element of the secondary cohomology group. The body of the paper is devoted to showing that the secondary \(k\)-invariant is indeed an invariant, first in the case when \(X\) is simply connected and then in the general case. After describing the third cohomology group of \(X\) in terms that may be amenable to defining higher order \(k\)-invariants, the author concludes with a discussion of potential generalizations as well as how the invariant may be used to study other problems of interest.
reviewed by: Nicholas A. Scoville
Mathematics Subject Classification ID: 55S45
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 6111044
zbMATH Keywords: \(k\)-invariant
zbMATH Keywords: group cohomology
MaRDI profile type: Publication
arXiv ID: 0909.1086
description: scientific article