## On the mapping theorem for Lusternik-Schnirelmann category.(English)Zbl 0572.55002

Let all spaces be CW and 1-connected, fix an integer $$r\geq 2$$. A space X is ”r-tame”, if it is (r-1)-connected and if all $$\pi_{r+t}(X)$$ are uniquely divisible by all primes p with 2p-3$$\leq t$$. Theorem: Let $$f: X\to Y$$ be a map between $$(r+1)$$-tame spaces such that $$\pi_*(f)$$ is split injective, then cat(X)$$\leq cat(Y)$$. This generalizes the corresponding result in the rational situation given by the first author and S. Halperin. Moreover, here an easy geometric argument is given which is based on the result that an r-tame H-space is (up to homotopy) a product of Eilenberg-MacLane spaces. And a proof of this result is sketched.
Reviewer: H.Scheerer

### MSC:

 55M30 Lyusternik-Shnirel’man category of a space, topological complexity à la Farber, topological robotics (topological aspects) 55P62 Rational homotopy theory
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