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On the Criterion of Stasheff

Published online by Cambridge University Press:  20 November 2018

C. S. Hoo
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In (9), Stasheff showed that a loop space ΩX is homotopy-commutative if and only if the map e∇ : ΣΩX ∨ ΣΩX → X may be extended to ΣΩX × ΣΩX, where ∇ is the folding map and e:ΣΩX → X is the map whose adjoint is the identity map of ΩX. In (5), Ganea, Hilton, and Peterson showed that this criterion does not dualize. Our aim in this paper is to give a reformulation of Stasheff's criterion, which is equivalent to it, but in a form which does dualize. In the course of the paper, we shall discuss why Stasheff's criterion, in its original form, does not dualize. In (5), the authors have, of course, given an explicit counterexample to the dual of StashefFs criterion in its original form.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 250 - 255
Copyright
Copyright © Canadian Mathematical Society 1969

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