Homology of real projective plane with two points identified

Question

What is the delta homology of the real projective plane with two points identified. I can't think of any space that it retracts that I can find a delta structure does anyone have any idea

Answer 1

$X$ with two points $a,b$ identified is homotopy equivalent to $X$ plus a line segment between $a,b$. If $a,b$ are in the same path component of $X$ then you can slide one to the other and your segment becomes a loop, so you get $X \vee S^1$.

All this might depend on $a,b$ being nondegenerate in some way, but that shouldn't be a problem in a cell complex.