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On numerical homotopy invariants and homotopy functors Chen, Dien Wen

Abstract

The main object of study in this paper is the "Smash Functor" BΛ-, which associates to a space X the smash product BX = BΛX. We find that various numerical homotopy invariants, such as strong category, weak category, Lusternik-Schnirelmann category, and in the case where X is a co-group, the nil-potency do not increase under the smash functor, i.e. we have CatBX ≤ CatX, catBX ≤ catX, WcatBX ≤ WcatX and nilBX ≤ nilX. We then consider the particular case of the smash functor where B = A⁺ (disjoint union of A with a point) in which case BX = [sup AxX]/A. This functor actually preserves L-S category. Furthermore we show that (in the category of based spaces of the based homotopy type of CW-complexes) when X is a co-H space, then the spaces [sup AxX]/A and XV(AX) are homotopy equivalent; this is a generalization of [sup A x ΣB]/A ≃ ΣBv(AΣB) . We also investigate conditions a functor F has to satisfy in order to have the properties we found for BΛ- . Finally, we collect a few counterexamples to show that the duals of some of our results are false.

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