Let E_2 be the Morava E-theory of height 2 at prime 2 and S_2 be the automorphism group of the Honda formal group law of height 2 over the finite field with 4 elements. There is a norm map on S_2 whose kernel is normal subgroup denoted S^1_2. In her PhD thesis advised by Paul Goerss, Irina Bobkova constructed a topological duality resolution of (E_2)^{hS^1_2} at prime 2 into variants of the homotopy fixed points spectra of E_2 by finite subgroups of S_2. This resolution is quite important in studying the homotopy type of the K(2)-local sphere at prime 2, and it has key applications in Agnes Beaudry's disproof of Hopkins' strong form of the chromatic splitting conjecture at height 2 and prime 2.

 

One key spectra in this resolution is (E_2)^{hC6}. In this work, we will introduce the appropriate contexts as above and compute the homotopy groups of (E_2)^{hC6} ^ RP^2 and (E_2)^{hC6} ^ (RP^2 ^ CP^2) respectively (which chromatic homotopy theorists like to call V(0) and Y). This work is developed from the eCHT Summer 2024 REU and is joint work with the fellow participants and mentors on the project.