Fundamental Group of Connected Sum of two $3$-manifolds

Question

I've been asked to find the fundamental group of $(S^1\times S^2)\#(S^1\times S^2)$. However, connected sum, as briefly defined in my class (in a vague way) was as follows: You remove a $2$-disk from each surface and concatenate them. My questions are:

1. In this case, do we remove a $3$-disk?
2. I have been told to use the Van Kampen theorem. How do I make the explicit partitions?
3. I would imagine that I would at least need the fundamental group of $(S^1\times S^2)$, which is $\mathbb{Z}$. How do I proceed from here?

Answer 1

Yes, you remove a 3-disk from each manifold.

More specifically, and to address question 2, let me set up some notation. Denote $M_1$, $M_2$ to be the two $S^1 \times S^2$ summands. Let $D_i \subset M_i$ be a subset homeomorphic to the open 3-disk $\mathbb B = \{x \in \mathbb R^3 \mid |x| < 1\}$, and let $f_i : D_i \to \mathbb B$ be a homeomorphism. Denote $\mathbb B^{1/2} = \{x \in \mathbb R^3 \mid |x| < 1/2\}$, and denote $B_i^{1/2} = f_i^{-1}(\mathbb B^{1/2})$. Let $S^i = \partial (M_i - B_i^{1/2})$ which also equals $f_i^{-1}$ of the sphere of radius $1/2$.

As a topological space, the connected sum $M_1 \# M_2$ is the quotient space of the disjoint union $M_1 - B^{1/2}_i$ and $M_2 - B^{1/2}_i$ where $S_1$ is glued to $S_2$ using the homeomorphism $f_2^{-1} \circ f_1 : S_1 \to S_2$.

The open sets needed for application of Van Kampen's theorem are $$U_1 = (M_1 - B^{1/2}_1) \cup f_2^{-1}(\mathbb B - \mathbb B^{1/2}) $$ and $$U_2 = (M_2 - B^{1/2}_2) \cup f_1^{-1}(\mathbb B - \mathbb B^{1/2}) $$

To address question 3, what you need for Van Kampen's Theorem is $\pi_1 U_1$ and $\pi_1 U_2$ and $\pi_1 (U_1 \cap U_2)$. It should be straightforward to make the following deductions:

• $\pi_1 U_i \approx \pi_1 M_i$ (this is, itself, a Van Kampen's Theorem application)
• $U_1 \cap U_2$ is homeomorphic to $S^2 \times (-1,+1)$ and hence is simply connected.