2-connected map between a connected sum and a gluing along one point.

Question

I'm working with closed 3-dimensional manifolds $M_1$ and $M_2$. Consider their connected sum $M_1\#M_2$ and their gluing at only one point $M_1\vee M_2$. Intuitively I think that the map that contracts the boundary $\mathbb{S}^2$ of the ball used to make the connected sum to one point should induce an isomorphism in the fundamental groups but I don't really know how to prove this rigorously. Also, is the induced homomorphism in the $\pi_2$'s surjective? Basically my question is how can I prove that this map is 2-connected.

Answer 1

I think an application of Van Kampen's theorem should suffice to prove their fundamental groups are isomorphic. Glue in a ball $B$ to $M_1\# M_2$ on the boundary $S^2$ which you glued $M_1$ and $M_2$ along. Call this space $X$. Because the intersection of $B$ and $M_1\# M_2$ is $S^2$ (simply connected), and one of the spaces is contractible, this will not change the fundamental group.

[on second thought, the fact that $S^2$ is simply connected doesn't matter.]

Contracting the ball to a point is then a homotopy to the wedge sum and so their fundamental groups are isomorphic.

I'll think some more on the $\pi_2$ calculation and get back to you. Just some thoughts though. One would expect that pulling everything up in to the universal covers would be the best way to go. The universal cover of a connect sum of manifolds is just the universal covers of the punctured individual manifolds glued appropriately. It is also useful to know that the higher homotopy groups of a universal cover of a space are isomorphic to the homotopy groups of the space. Given that universal covers are simply connected, the Hurewicz theorem then implies that we can look at the second homology group instead of homotopy group which would make life easier as we could use Mayer-Vietoris.