The homology of $\mathrm{tmf}$

Abstract

We compute the mod $2$ homology of the spectrum $\mathrm{tmf}$ of topological modular forms by proving a 2-local equivalence $\mathrm{tmf}\wedge DA(1)\simeq {\mathrm{tmf}}_1(3)\simeq BP⟨2⟩$, 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 $\mathrm{tmf}$.

arXiv comment25 pages. Substantially revised. To appear in Homology, Homotopy, and Applications