The \(v_n\)-periodic Goodwillie tower on Wedges and Cofibres

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.