I am trying to compute the homology of the mapping torus of $X = \mathbb{R}P^n \vee \mathbb{R}P^n$, where $f:X \rightarrow X$ is a map that exchanges the two copies of $\mathbb{R}P^n$; that is, $f(x,y) = (y,x)$. I can do this with exact sequences except that I don't know what the induced map $f_*$ is. What if the space is instead $X = \mathbb{C}P^3 \vee \mathbb{C}P^3$. Help?
Note: By mapping torus for a map $f:X \rightarrow X$, we mean the space $X \times [0,1]/\sim$ where $\sim$ is the equivalence $(x,0) \sim (f(x),1)$.