# Unstable $\nu_1$-Periodic Homotopy of Simply Connected, Finite $H$-Spaces, using Goodwillie Calculus

## TL;DR

This paper uses Goodwillie calculus to recover Bousfield's $
u_1$-periodic homotopy groups of simply connected, finite $H$-spaces, demonstrating the convergence of the $
u_1$-periodic Goodwillie tower for these spaces.

## Contribution

It introduces a novel approach using Goodwillie calculus and Andre9-Quillen cohomology to recover and confirm Bousfield's results on $
u_1$-periodic homotopy groups.

## Key findings

- Successfully recovers Bousfield's $
u_1$-periodic homotopy groups
- Shows convergence of the $
u_1$-periodic Goodwillie tower for the spaces
- Establishes a new computational framework using Goodwillie calculus

## Abstract

In this paper we recover Bousfield's computation of $\nu_1$-periodic homotopy groups of simply connected, finite $H$-spaces from \cite{Bou99} using the techniques of Goodwillie calculus. This is done through first computing Andr\'{e}-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 Andr\'{e}-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 $\nu_1$-periodic Goodwillie tower for simply connected, finite $H$-spaces converges.