Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-15T21:15:10.908Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Reduced Product Construction

Published online by Cambridge University Press:  20 November 2018

Peter Fantham ,
Ioan James  and
Michael Mather
Peter Fantham
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles, OxfordEngland OX1 3LB
Ioan James
Affiliation:
Mathematical Institute, University of Oxford, 24-29 St. Giles, OxfordEngland OX1 3LB
Michael Mather
Affiliation:
16 Parkview Avenue, TorontoCanadaM4X1V9, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Under certain conditions the reduced product space JX of a space X has the same homotopy type as ΩΣX, the loop-space on the suspension of X. Several proofs can be found in the literature. The original proof [6] made unnecessarily strong assumptions. Later, in the last chapter of [3], torn Dieck, Kamps and Puppe gave a proof under much weaker conditions and showed that they could not be further weakened. The question then arose as to whether the reduced product construction could be generalized to provide combinatorial models not only of ΩΣX but also of Ω2Σ2X, Ω3Σ3X and so on. This was answered in the affirmative by May [10], using ideas of Boardman and Vogt [1], and the construction was further developed by Segal [11] and others.

The present note, however, is not concerned with these generalizations. Its purpose is to give a simple proof of the original result, without striving for maximum generality, and to show that the same method can be used to prove an equivariant version of the reduced product theorem and hence a fibrewise version. Fibrewise versions of the reduced product theorem have previously been given by Eggar [5] and, more recently, by one of us [8] but it may be useful to have a relatively simple treatment which is adequate for the majority of applications.

Keywords

55P1555P3555P91
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 4 , 01 December 1996 , pp. 385 - 389
Copyright
Copyright © Canadian Mathematical Society 1996

References

1. Boardman, J. M. and Vogt, R. M., Homotopy-every thing H-spaces, Bull. Amer. Math. Soc. 74(1968), 1117—1182.Google Scholar
2. torn Dieck, T., Transformation groups, de Gruyter, 1987.Google Scholar
3. torn Dieck, T., Kamps, K. H. and Puppe, D., Homotopietheorie, Lecture Notes in Math. 57, Springer Verlag, 1970.Google Scholar
4. Dold, A., Partitions of unity in the theory offibrations, Ann. Math. 78(1963), 223–255.Google Scholar
5. Eggar, M. G., Ex-homotopy theory, Compositio Math. 27(1973), 185–195.Google Scholar
6. James, I. M., Reduced product spaces, Ann. Math. 62(1955), 170–197.Google Scholar
7. James, I. M., Alternative homotopy theories. In: L'Enseignement Mathématique, vol. in honour of B. Eckmann, 1978.Google Scholar
8. James, I. M., Fibrewise reduced product spaces, Adams Memorial Symposium on Algebraic Topology 1, Cambridge, 1992.Google Scholar
9. Mather, M., Pull-backs in homotopy theory, Canad. J. Math. 28(1976), 225–263.Google Scholar
10. May, J. P., The geometry of iterated loop-spaces, Lecture Notes in Math. 271, Springer Verlag, 1972.Google Scholar
11. Segal, G. B., Configuration spaces and iterated loop-spaces, Invent. Math. 21(1973), 213–221.Google Scholar
12. Vogt, R. M., Homotopy limits andcolimits, Math. Z. 134(1973), 11–52.Google Scholar