{"paper":{"title":"The homology of $\\mathrm{tmf}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.AT","authors_text":"Akhil Mathew","submitted_at":"2013-05-27T03:34:47Z","abstract_excerpt":"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 des"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.6100","kind":"arxiv","version":5},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}