Connected sum of a smooth manifold with itself is the boundary of another manifold

Question

While searching connected sum, I got this page. I have a doubt with the notation $\natural$ that I don't know how to define formally. It says, given a smooth manifold $M$ one has $\partial(W\natural W)=M\sharp M$, here $W$ is the total space of the disk bundle of $M$. Also, if $\bullet$ denotes deleting a small embedded disk from the manifold then $(M_0 \sharp M_1)^\bullet = (M_0^\bullet) \natural (M_1^\bullet) \simeq M_0^\bullet \vee M_1^\bullet. $

Any reference for the notation $\natural$ will be helpful.

Answer 1

The symbol $\natural$ denotes the boundary connected sum in this case. This is defined as follows.

Let $M$ and $N$ be $d$-manifolds with nonempty boundaries $\partial M$ and $\partial N$. Fix two $(d-1)$-disks embedded in the boundaries $D^{d-1}\subset \partial M \subset M$ and $D^{d-1} \subset \partial N \subset N$. Then $M\natural N$ is obtained by gluing the embedded disks of $M$ and $N$ together. In the oriented setting, this should be done such that they are glued together along an orientation-reversing homeomorphism.