The $P_2^1$ Margolis homology of connective topological modular forms
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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 $H_*(X, mathbb{F}_2)$ 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 $tmf$.
