Infinite loop spaces with odd torsion-free homology
Let \(X\) be an infinite loop space whose integral homology is \(p\)-torsion free, \(p\) an odd prime. It is proven that the mod \(p\) cohomology of \(X\) is a free commutative algebra. This, in turn, implies that the integral homology of \(\Omega^nX\langle n-1\rangle\) is also \(p\)-torsion free, where \(X\langle n-1\rangle\) is the \(n-1\)-connected cover of \(X\). Hence the main theorem of the article: under the same assumptions, the homotopy groups of \(X\) are also torsion free. Combined with results of \textit{W. S. Wilson} [Am. J. Math. 97, 101-123 (1975; Zbl 0303.55003)], this yields a complete classification (\(p\)-locally) of such spaces. Some ingredients of the proofs are the use of the higher projective planes [\textit{N. J. Kuhn}, \textit{M. Slack} and \textit{F. Williams}, Trans. Am. Math. Soc. 347, No. 4, 1201-1238 (1995; Zbl 0851.55015)] and the results of \textit{M. Foskey} and \textit{M. Slack} [Pac. J. Math. 173, No. 1, 77-92 (1996; Zbl 0858.55016)].
scientific article; zbMATH DE number 1113438