# Cell decomposition for connected sum

I am doing a past paper for a first course in algebraic topology. The question is

Calculate the homology groups with $\mathbb Z,\mathbb Z_2,\mathbb Q$ coefficients for $\mathbb RP^4$ and $\mathbb CP^2$ and their connected sum.

I am stuck at trying to calculate the homology of $\mathbb RP^4\#\mathbb CP^2$. I calculated the homology groups of the individual spaces via the standard CW complexes, which are nice and easy. Is there a way of forming a CW complex for the connected sum? Or do I need to use a homology trick for getting at the homology of the connected sum?

• Meyer-Vietoris is the obvious thing to try. – Chris Eagle Jul 7 '13 at 22:00
• Take the obvious 'puntured manifolds' as your $U$ and $V$ so that $U\cap V$ is homeomorphic to a $3$-sphere. Then, as @ChrisEagle suggest, use Mayer-Vietoris. – Dan Rust Jul 7 '13 at 22:07
• @DanielRust Thanks. Is the idea to use another Mayer-Vietoris sequence for the "punctured" manifolds, or is there another way to calculate their homology? – Earthliŋ Jul 7 '13 at 22:59