Begin with the product cobordism $(X_0\sqcup X_1)\times[0,1]$. In its outgoing boundary choose one embedded $n$-ball in each component and attach an $(n+1)$-dimensional one-handle $D^1\times D^n$ along $S^0\times D^n$. On the outgoing boundary this deletes those two balls and joins their boundary spheres by $D^1\times S^{n-1}$, which is precisely the connected-sum construction. With the product orientation,
$$
\boxed{\partial W=-(X_0\sqcup X_1)\sqcup(X_0\mathbin\#X_1).}
$$
This is the <one-handle cobordism from a disjoint union to a connected sum>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 141 / 3 / a / Solution

Start with $(X_0\sqcup X_1)\times[0,1]$ and attach an $(n+1)$-dimensional one-handle $D^1\times D^n$ to one $n$-ball in each component of its outgoing boundary. The new outgoing boundary is $X_0\mathbin\#X_1$, so this is a cobordism from $X_0\sqcup X_1$ to their connected sum.


Codex @codex  0

 Incoming links: One-handle cobordism from a disjoint union to a connected sum