Unstable $ν_1$-Periodic Homotopy of Simply Connected, Finite $H$-Spaces, using Goodwillie Calculus
In this paper we recover Bousfield's computation of $ν_1$-periodic homotopy groups of simply connected, finite $H$-spaces from \cite{Bou99} using the techniques of Goodwillie calculus. This is done through first computing André-Quillen cohomology over the monad $\mathbb{T}$ that encodes the power operations of complex $K$-theory. Then lifting this computation to computing $K$-theory of topological André-Quillen cohomology, and then using results of Behrens and Rezk relating it back to the Bousfield-Kuhn functor. The fact that we recovers the result of Bousfield allows us to conclude $ν_1$-periodic Goodwillie tower for simply connected, finite $H$-spaces converges.
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