The $P_2^1$ Margolis homology of connective topological modular forms

Abstract

The element $P_2^1$ of the mod 2 Steenrod algebra has the property $(P_2^1{)}^2=0$. This property allows one to view $P_2^1$ as a differential on $H_{\ast }(X,{\mathbb{F}}_2)$ for any spectrum $X$. Homology with respect to this differential, $\mathcal{M}(X,P_2^1)$, is called the $P_2^1$ Margolis homology of $X$. In this paper we give a complete calculation of the $P_2^1$ Margolis homology of the 2-local spectrum of topological modular forms $tmf$ and identify its ${\mathbb{F}}_2$ basis via an iterated algorithm. We apply the same techniques to calculate $P_2^1$ Margolis homology for any smash power of $tmf$.

Cited by in corpus (1)

• The 2-primary Hurewicz image of tmf