Geodesic
The homology of $\mathrm{tmf}$
Homology, homotopy, and applications, Tome 18 (2016) no. 2, pp. 1-29.

Voir la notice de l'article provenant de la source International Press of Boston

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 $\mathrm{tmf}$.
DOI : 10.4310/HHA.2016.v18.n2.a1
Classification : 55P42, 55P43
Keywords: topological modular form, algebraic stack, Steenrod algebra 
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     author = {Akhil Mathew},
     title = {The homology of $\mathrm{tmf}$},
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     pages = {1--29},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.4310/HHA.2016.v18.n2.a1/}
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Akhil Mathew. The homology of $\mathrm{tmf}$. Homology, homotopy, and applications, Tome 18 (2016) no. 2, pp. 1-29. doi : 10.4310/HHA.2016.v18.n2.a1. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2016.v18.n2.a1/

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