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

In this paper, we study the elliptic spectral sequence computing $tmf_*(\mathbb{R} P^2)$ and $tmf_* (\mathbb{R} P^2 \wedge \mathbb{C} P^2)$. Specifically, compute all differentials resolve exotic extensions by 2, $\eta$, $\nu$. For P^2)$, also effect of $v_1$-self maps $\mathbb{R} P^2$ on $tmf$-homology.