The Goodwillie tower of the identity functor and the unstable periodic homotopy of spheres
The authors investigate the Goodwillie tower \(\cdots \rightarrow P_{3}(X) \rightarrow P_{2}(X) \rightarrow P_{1}(X) \simeq Q(X)\) in the case where \(X\) is a sphere localized at a prime \(p\). They show that, if \(X\) is an odd-dimensional sphere localized at \(p\), then the map \(X \rightarrow P_{p^{k}}(X)\) is a \(v_{j}\)-periodic equivalence for all \(k \geq 0\) and all \(0 \leq j \leq k\). Along the way, they show that there are equivalences \(D_{n}(X) \simeq \ast\) for the homotopy fibres \(D_{n}(X)\) of the maps \(P_{n}(X) \rightarrow P_{n-1}(X)\) if \(n\) is not a power of a prime, and that \(D_{p^{k}}(X)\) has only \(p\)-primary torsion. The main result is a substantial generalization of a result of \textit{M. Mahowald} and \textit{R. D. Thompson}, which asserts that \(X \rightarrow P_{2}(X)\) induces an isomorphism in \(v_{1}\)-periodic homotopy if \(X\) is a odd sphere localized at 2 [Topology 31, No. 1, 133-141 (1992; Zbl 0759.55010)]. This paper is beautifully written, and the reviewer will not attempt to paraphrase anything beyond a statement of the main results. The introduction of the paper contains an illuminating account of the genesis and scope of the theory.
scientific article; zbMATH DE number 1337386