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# Homology, Homotopy and Applications

## Volume 23 (2021)

### Number 2

### The $\mathsf{P}^1_2$ margolis homology of connective topological modular forms

Pages: 379 – 402

DOI: https://dx.doi.org/10.4310/HHA.2021.v23.n2.a21

#### Authors

#### Abstract

The element $\mathsf{P}^1_2$ of the $\operatorname{mod}2$ Steenrod algebra $\mathcal{A}$ has the property $(\mathsf{P}^1_2)^2 = 0$. This property allows one to view $\mathsf{P}^1_2$ as a differential on $H_\ast (X, \mathbb{F}_2)$ for any spectrum $X$. Homology with respect to this differential, $\mathcal{M} (X, \mathsf{P}^1_2)$, is called the $\mathsf{P}^1_2$ Margolis homology of $X$. In this paper we give a complete calculation of the $\mathsf{P}^1_2$ 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 $\mathsf{P}^1_2$ Margolis homology for any smash power of *tmf*.

#### Keywords

Steenrod algebra, Margolis homology, topological modular forms

#### 2010 Mathematics Subject Classification

55N35, 55P42, 55S10, 55S20

Copyright © 2021, Prasit Bhattacharya, Irina Bobkova and Brian Thomas. Permission to copy for private use granted.

Received 7 May 2019

Received revised 25 December 2020

Accepted 26 December 2020

Published 29 September 2021