Homotopy exponents of mod \(2^r\) Moore spaces
In their two ground-breaking papers ``Torsion in homotopy groups'' [Ann. Math. (2) 109, 121--168 (1979; Zbl 0405.55018)] and ``The double suspension and exponents of the homotopy groups of spheres'' [Ann. Math. (2) 110, 549--565 (1979; Zbl 0443.55009)], \textit{F. R. Cohen, J. C. Moore}, and \textit{J. A. Neisendorfer} advanced our knowledge of the possible orders of elements in the homotopy groups of spheres and Moore spaces. If we write \(P^m(k)\) for the Moore space, with homology a cyclic group of order \(k\) in dimension \(m\), then when \(p\) is an odd prime, \(r\geq 1\), and \(m\geq 3\), Neisendorfer showed that \(P^m (r^p)\) has homotopy exponent \(p^{r+1}\). The present paper is a real tour de force. The author shows that if \(m\geq 4\) and \(r\geq 6\), then \(P^m(2^r)\) has homotopy exponent \(2^{r+1}\). There are also some results for smaller values of \(r\). The methods include the study of certain complexes with three cells, which come from unit tangent bundles of spheres, Whitehead products, and the James-Hopf invariants. The details are much too complex to permit a short summary here.
scientific article; zbMATH DE number 5597575
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