UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

On numerical homotopy invariants and homotopy functors Chen, Dien Wen


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.

Item Media

Item Citations and Data


For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use.