The homology of $\mathrm{tmf}$

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 $\mathrm{tmf}$.

Keywords

topological modular form, algebraic stack, Steenrod algebra

2010 Mathematics Subject Classification

55P42, 55P43

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Published 29 November 2016