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Homotopy exponents for spaces of category two. (English) Zbl 0671.55009
Algebraic topology, Proc. Int. Conf., Arcata/Calif. 1986, Lect. Notes Math. 1370, 24-52 (1989).
[For the entire collection see Zbl 0661.00012.]
A space X is said to have an exponent at the prime p if and only if there exists an integer \(r>0\) such that for all n, multiplication by \(p^ r\) annihilates the p-torsion of \(\pi_ n(X)\). The author proves the following weakened from of Moore’s conjecture: Let X be a simply- connected finite complex of Ljusternik-Schnirelmann category two. There is a finite set P of primes such that: (a) If dim \(\pi_*(X)\otimes {\mathbb{Q}}<\infty\), then X has an exponent at every \(p\not\in P\), (b) If dim \(\pi_*(X)\otimes {\mathbb{Q}}=\infty\), then X fails to have an exponent at any \(p\not\in P\).
Reviewer: Y.Felix

MSC:
55Q05 Homotopy groups, general; sets of homotopy classes
55P62 Rational homotopy theory
55M30 Lyusternik-Shnirel’man category of a space, topological complexity √† la Farber, topological robotics (topological aspects)