Lie algebras and $v_n$-periodic spaces

TL;DR

This paper develops a $v_n$-periodic homotopy theory for pointed spaces, establishing its equivalence to Lie algebras in T(n)-local spectra and comparing it to commutative coalgebras, extending rational homotopy results.

Contribution

It generalizes rational homotopy theory to $v_n$-periodic settings, proving an equivalence with Lie algebras in T(n)-local spectra and analyzing related coalgebra structures.

Findings

$v_n$-periodic homotopy theory is equivalent to Lie algebras in T(n)-local spectra

Comparison with commutative coalgebras shows convergence issues in Goodwillie tower

Extension of rational homotopy theory to higher chromatic levels

Abstract

We consider a homotopy theory obtained from that of pointed spaces by inverting the maps inducing isomorphisms in $v_n$-periodic homotopy groups. The case n = 0 corresponds to rational homotopy theory. In analogy with Quillen's results in the rational case, we prove that this $v_n$-periodic homotopy theory is equivalent to the homotopy theory of Lie algebras in T(n)-local spectra. We also compare it to the homotopy theory of commutative coalgebras in T(n)-local spectra, where it turns out there is only an equivalence up to a certain convergence issue of the Goodwillie tower of the identity.