Multi-connected sum decomposition of $n$-manifolds

Question

A connected sum decomposition of a closed $n$-manifold $M^n$: $M^n = M_1^n \# M_2^n$, is to view $M^n$ as two closed $M_1^n$ and $M_2^n$, joined by a neck $I\times S^{n-1}$.

Similarly, a $k$-connected sum decomposition of a closed $n$-manifold $M^n$: $M^n = M_1^n \#_k M_2^n$, is to view $M^n$ as two closed $M_1^n$ and $M_2^n$, joined by $k$ necks, each has a form $I\times S^{n-1}$.

I like to know the reference is this "$k$-connected sum decomposition". Is there a classification of $n$-manifolds via the $k$-connected sum decomposition? Is there a classification of 3-manifolds via the $k$-connected sum decomposition?

Answer 1

I don't have a reference, but I think it's not too hard to see that $M_1 \#_k M_2 \approx M_1 \# X_k \# M_2$, where $X_k = (S^1 \times S^{n-1})^{\# (k-1)}$. (Connected sum depends on choices of embedded disks, and so does $k$-connected sums; I'm assuming you want all $k$ pairs of disks to be isotopic.) So decomposing using $k$-connected sum is not so different from decomposing using usual connected sum, except for some bookkeeping about extra summands of $S^1 \times S^{n-1}$.

The idea of this formula $M_1 \#_k M_2 \approx M_1 \# X_k \# M_2$ is that gluing the first neck gives $M_1 \# M_2 \approx M_1 \# S^n \# M_2$; the remaining $k-1$ necks may then be attached to the middle $S^n$ instead of between $M_1$ and $M_2$, each of which creates an $(S^1 \times S^{n-1})$-summand.