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

TL;DR

This paper computes the $P_2^1$ Margolis homology of the 2-local spectrum of topological modular forms (tmf), providing explicit bases and extending calculations to smash powers of tmf, advancing understanding of its algebraic structure.

Contribution

It provides a complete calculation of the $P_2^1$ Margolis homology for $tmf$ and its smash powers, introducing an iterative algorithm for basis identification.

Findings

Explicit $ extbf{F}_2$ basis for $tmf$ Margolis homology.

Algorithm for computing Margolis homology of smash powers of $tmf$.

Enhanced understanding of the algebraic structure of $tmf$.

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 $tmf$.