Representation of extension groups. Applications. (Représentation des groupes d’extension. Applications.) (French) Zbl 0809.55006

Hayat-Legrand, Claude

Compos. Math. 90, No. 3, 351-366 (1994).

This paper is concerned with the homotopy classification of spaces \(X\) having two nonvanishing homotopy groups, \(\Pi\) and \(\Pi'\), possibly admitting module structures for some discrete group \(G\). (The model here is the action of the fundamental group on the universal covering space.) In order to do this it is necessary to describe the \(n\)th group of \(G\)- extensions \(\text{Ext}^ n_ G (\Pi, \Pi')\) as the space of homotopy classes of sections of a fibration constructed via the classifying space for \(G\), Theorem 1.1. Intuitively the obstruction to geometric realisation by a twisted product of \(K(\Pi, p)\) and \(K(\Pi, p + 1)\) is an element \(\theta_ G\) in \(\text{Ext}^ 3_ G (\Pi, \Pi')\). The method used is patient exploitation of known techniques, and the paper ends with the following application:
Let \(\widetilde X\) be simply-connected with \(\pi_ 2 = \Pi\), \(\pi_ 3 = \Pi'\), and assume that \(\Pi\) is free. In the presence of a \(G\)-action the set of possible homotopy types is in (1-1) correspondence with \(\text{Ext}^ 2_ G (\Pi, \Pi')\).

Reviewer: C.B.Thomas (Cambridge)

MSC:

55P10	Homotopy equivalences in algebraic topology
55S45	Postnikov systems, \(k\)-invariants

Keywords:

Postnikov systems; spaces having two non-vanishing homotopy groups; \(G\)- extensions; homotopy classification; action of the fundamental group

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