How to find homology groups of a connected sum? I need to find homology groups for the following simplicial complex:
$RP^2$ # $RP^2$ # $\Delta$ # $\Delta$
How to do it? If I am not mistaken, $RP^2$ is not orientable - so we cannot just sum the groups.
How to solve the problem?
 A: Taking the connected sum of a surface and the 2-disk is the same as puncturing the surface.  So your surface is the Klein bottle twice punctured.  Its fundamental group, by van Kampen, is free on three generators, so its first homology is $\mathbb{Z}^3$.  It has non-empty boundary, so its second homology is trivial.
A: Edit: Now that I see $\Delta$ is a disk, the surface in question has a nonempty boundary, and my answer below assumed the boundary was empty.
I'm not sure what $\Delta$ is, but if you can write your surface as an identification space of a polygon, it's easy to construct a cell complex, and therefore a chain complex that will calculate the homology. For example, $RP^2\# RP^2\# T$, where $T$ is a torus, is the quotient of a octagon where the boundary edges are glued by the pattern $aabbcdc^{-1}d^{-1}$. The chain complex has one generator in degree $0$, $4$ in degree 1 and $1$ in degree $2$. The boundary operator is zero on edges and is equal to $2a+2b$ on the unique generator in degree $2$. So $\partial_2$ is injective, implying $H_2=0$. Also $H_1=\mathbb Z^4/im(\partial_2)$, which you can verify is $\mathbb Z_2\oplus \mathbb Z\oplus \mathbb Z\oplus \mathbb Z$.
Alternatively you can appeal to the classification of surfaces, and figure out whether your surface is a connected sum of some number of projective planes (in the nonorientable case) or tori (in the orientable case), and just look up or figure out the answer for these examples. 
