The topological modular forms of RP2$\mathbb {R}P^2$ and RP2∧CP2$\mathbb {R}P^2 \wedge \mathbb {C}P^2$

Abstract

We study the elliptic spectral sequence computing tmf∗(RP2)$tmf_*(\mathbb {R}P^2)$ and tmf∗(RP2∧CP2)$tmf_* (\mathbb {R} P^2 \wedge \mathbb {C} P^2)$. Specifically, we compute all differentials and resolve exotic extensions by 2, η$\eta$, and ν$\nu$. For tmf∗(RP2∧CP2)$tmf_* (\mathbb {R} P^2 \wedge \mathbb {C} P^2)$, we also compute the effect of the v1$v_1$‐self maps of double-struckRP2∧double-struckCP2$\mathbb {R} P^2 \wedge \mathbb {C} P^2$ on tmf$tmf$‐homology.