The homology of tmf

2013/05/27 by Mathew, Akhil
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics

paper · doi:10.48550/arxiv.1305.6100

Abstract

We compute the mod 2 homology of the spectrum tmf of topological modular forms by proving a 2-local equivalence tmf \wedge DA(1) ≃ tmf₁(3) ≃ BP ⟨ 2⟩, where DA(1) is an eight cell complex whose cohomology "doubles" the subalgebra A(1) of the Steenrod algebra generated by Sq¹ and Sq². 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 tmf.

Related

• The homotopy groups of the spectrum Tmf
• Topological modular forms with level structure
• The \mathbb C-motivic Adams-Novikov spectral sequence for topological modular forms
• Connective versions of TMF(3)
• The algebraic Novikov spectral sequence for topological modular forms