Homology, Homotopy and Applications

Volume 22 (2020)

Number 1

The $v_n$-periodic Goodwillie tower on wedges and cofibres

Pages: 167 – 184

DOI: https://dx.doi.org/10.4310/HHA.2020.v22.n1.a10


Lukas Brantner (Merton College, Oxford University, Oxford, United Kingdom)

Gijs Heuts (Mathematical Institute, Utrecht University, Utrecht, The Netherlands)


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 $\operatorname{cof}(f)$ of a map $f : X \to 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.


Goodwillie calculus, $v_n$-periodicity, Hilton–Milnor theorem

2010 Mathematics Subject Classification

55Q20, 55Q51

Received 20 August 2018

Accepted 19 February 2019

Published 13 November 2019