\(C_2\)-equivariant topological modular forms (Q2681342)

In this paper the author computes the homotopy groups of the \(C_2\) fixed points \(TMF^{BC_2}\) of equivariant topological modular forms at \(2\). The main conclusion is that \[ TMF^{BC_2}\cong TMF \oplus TMF \oplus TMF\otimes DL \] where \(DL\) is a split spectrum playing a role in the computation of the Tate spectrum of TMF in Bailey and Ricka's paper. The \(E_2\) page of the descent spectral sequence is computed using the 2-Bockstein spectral sequence and the differentials of the descent spectral sequence are computed in term of norm maps. Most computation is carried on via sage. The isomorphism in the main theorem is obtained by obstruction theory in the last step. Moreover, in the appendix, the author constructs a hypothetical connective version of \(C_2\)-equivariant \(TMF\), via which a dual of \(TMF^{BC_2}\) in the category of \(TMF\)-modules is formulated.