how to calculate $H_2(S^{2}\times S^{1}\# S^{2}\times S^{1}\# S^{2}\times S^{1})$ ? When advancing in some calculations, I found the problem of computing: $H_2(S^{2}\times S^{1}\# S^{2}\times S^{1}\# S^{2}\times S^{1})$.
I found this Mayer-Vietoris sequence of which is:
$$0\to \widetilde{H_n}(M\# N)\to \widetilde{H_n}(M\vee N) \to \widetilde{H}_{n-1}(S^{n-1})\to \widetilde{H}_{n-1}(M\# N)\to \widetilde{H}_{n-1}(M\vee N) \to 0$$
this sequence is valid for n = 2? where N and M are 3-manifolts closed and compact.
someone can give me some bibliographic reference for calculating the homology groups connected sum of 3-varieddes or n-varieties. Thank you.

This exact sequence exists and is valid only for $n$ the dimension of the manifolds (determined by the gluing sphere in the sequence). Moreover this is *not * the Mayer Vietoris sequence, but instead the long exact sequence for the pair $(M\#N, S^{n-1}) $ (together with the isomorphism $H_*(A, B) =H_*A/B$ for good pairs. The zeroes in your sequence arise as homology groups of the sphere. 
Generally there are two approaches for the computation of connected sums. One is the above and the other is Mayer Vietoris for the partition into the two summands. I wouldn't know a reference because this is usually left as an exercise (a good one, you should try it yourself). And note that the answer depends on the orientability of the manifolds! 
