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The group of homotopy self-equivalences of non-simply-connected spaces using Postnikov decompositions*

Published online by Cambridge University Press:  14 November 2011

J. W. Rutter
J. W. Rutter
Affiliation:
Institut des Hautes Études Scientifiques, 91440 Bures sur Yvette, Franceand Department of Pure Mathematics, Liverpool University, Liverpool L69 3BX, England
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Synopsis

We give here a group extension sequence for calculating, for a non-simply-connected space X, the group of self-homotopy-equivalence classes which induce the identity automorphism of the fundamental group, that is the kernel of the representation → aut (π1(X)). This group extension sequence gives in terms of , where Xn is the n-th stage of a Postnikov decomposition. As special cases, we calculate for non-simply-connected spaces having at most two non-trivial homotopy groups, in dimensions 1 and n, as the unit group of a semigroup structure on ; and we calculate up to extension for non-simply-connected spaces having at most three non-trivial homotopy groups. The group is, for nice spaces, isomorphic to the groups and of self-homotopy-equivalence classes of X in the categories top*M and top M, respectively, where X→M = K1(π1(X)) is a top fibration which determines an isomorphism of the fundamental group; and our results are obtained initially in topM.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 120 , Issue 1-2 , 1992 , pp. 47 - 60
Copyright
Copyright © Royal Society of Edinburgh 1992

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