The homology of $\mathrm{tmf}$

TL;DR

This paper computes the mod 2 homology of the topological modular forms spectrum (tmf) by establishing a key equivalence with a known spectrum, using stack language to describe elliptic homology and discussing odd prime analogs.

Contribution

It provides a new modular stack description of tmf's homology and proves a 2-local equivalence with a known spectrum, advancing understanding of tmf's structure.

Findings

Computed mod 2 homology of tmf

Established 2-local equivalence with BP⟨2⟩

Provided stack-theoretic description of elliptic homology

Abstract

We compute the mod $2$ homology of the spectrum $\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\mathrm{tmf} \wedge DA(1) \simeq \mathrm{tmf}_1(3) \simeq BP\left \langle 2\right\rangle$, where $DA(1)$ is an eight cell complex whose cohomology "doubles" the subalgebra $\mathcal{A}(1)$ of the Steenrod algebra generated by $\mathrm{Sq}^1$ and $\mathrm{Sq}^2$. To do so, we give, using the language of stacks, a modular description of the elliptic homology of $DA(1)$ via level three structures. We briefly discuss analogs at odd primes and recover the stack-theoretic description of the Adams-Novikov spectral sequence for $\mathrm{tmf}$.