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?



Multi-connected sum decomposition of $n$-manifolds
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.


