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Suspension of the Lusternik-Schnirelmann Category

Published online by Cambridge University Press:  20 November 2018

William J. Gilbert
William J. Gilbert*
Affiliation:
University of Waterloo, Waterloo, Ontario
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Let cat be the Lusternik-Schnirelmann category structure as defined by Whitehead [6] and let be the category structure as defined by Ganea [2],

We prove that

and

It is known that w ∑ cat X = conil X for connected X. Dually, if X is simply connected,

1. We work in the category of based topological spaces with the based homotopy type of CW-complexes and based homotopy classes of maps. We do not distinguish between a map and its homotopy class. Constant maps are denoted by 0 and identity maps by 1.

We recall some notions from Peterson's theory of structures [5; 1] which unify the definitions of the numerical homotopy invariants akin to the Lusternik-Schnirelmann category.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 6 , 01 December 1970 , pp. 1129 - 1132
Copyright
Copyright © Canadian Mathematical Society 1970

References

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2. Ganea, T., A generalization of the homology and homotopy suspension, Comment. Math. Helv. 39 (1965), 295–322.Google Scholar
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6. Whitehead, G. W., The homology suspension, pp. 89–95 in Colloque de topologie algébrique, Louvain, 1956 (Masson et Cie, Paris, 1957).Google Scholar