I have seen and worked through the explicit computation of the Morse homology of the sphere and the torus (with signs and all). But trying it for $\mathbb{R}\mathbb{P}^2$ has lead me to dead ends. Is there a fully worked out computation for this projective plane (with signs and all, for $\mathbb{Z}$-coefficients)?
As a start, the Morse function to utilize is $f(x_1,x_2,x_3)=i(|x_1|^2+|x_2|^2+|x_3|^2)$, in homogeneous coordinates on $\mathbb{R}\mathbb{P}^2$. On each neighborhood $U_1,U_2,U_3$ ($U_i$ denotes the set of coordinates $(x_1,x_2,x_3)$ where $x_i\ne 0$) there is one critical point, namely $(1,0,0)$, $(0,1,0)$, $(0,0,1)$ respectively, of Morse-index 1,2,3 respectively. Maybe the homology computation will be easier with a different function.