The $v_n$-periodic Goodwillie tower on Wedges and Cofibres

TL;DR

This paper develops methods to analyze the Goodwillie tower of the identity functor on wedges and cofibre spaces, revealing differences in periodic homotopy convergence compared to spheres.

Contribution

It introduces new techniques for studying the Goodwillie tower on complex spaces and demonstrates that convergence properties differ from those on spheres.

Findings

Goodwillie tower analysis on wedges and cofibre spaces

Convergence in periodic homotopy does not hold for these spaces

Differences from the sphere case in periodic homotopy

Abstract

We introduce general methods to analyse the Goodwillie tower of the identity functor on a wedge $X \vee Y$ of spaces (using the Hilton-Milnor theorem) and on the cofibre $\mathrm{cof}(f)$ of a map $f: X \rightarrow Y$. We deduce some consequences for $v_n$-periodic homotopy groups: whereas the Goodwillie tower is finite and converges in periodic homotopy when evaluated on spheres (Arone-Mahowald), we show that neither of these statements remains true for wedges and Moore spaces.