at.algebraic topology – Are there alternate descriptions of `elementary cobordisms’?

Let $M^d$, $N^d$ be cobordant $d$-manifolds. Then $M^d sqcup bar{N}^d = partial W^{d+1}$ for some $(d+1)$-manifold $W$. This cobordism can be implemented via an elementary set of ‘moves’ called elementary cobordisms. In particular, the elementary cobordisms for dimension $d$ are labeled by $k = 0,dots,lfloor frac{d-1}{2} rfloor$ where we denote the floor of $frac{d-1}{2}$ as $lfloor frac{d-1}{2} rfloor$. An elementary cobordism is a surgery takes as input some embedding $S^k times D^{d-k} subset M$, drills it out, and glues in $D^{k+1} times S^{d-k-1}$. The ‘gluing’ means identifying the common boundaries $partial(S^k times D^{d-k}) = partial(D^{k+1} times S^{d-k-1}) = S^k times S^{d-k-1}$. Each elementary cobordism turns $M$ into some manifold $M’$, then turns $M’$ into some manifold $M”$ etc. The claim is that every $N$ cobordant to $M$ can be obtained by a sequence of such surgeries.

One can see that the elementary cobordisms corresponding to $k=0$ are all equivalent to taking the connected sum with $S^1 times S^{d-1}$. This is illustrated in dimensions 1,2 in the image below. And I think that in the $k=0$ unoriented case where we glue in the two $S^{d-1}$ copies with opposite orientations, we can think of it as taking the connected sum with $S^1 tilde{times} S^{d-1}$, i.e. the twisted $S^{d-1}$ bundle over $S^1$. (For $d=2$, this means connect summing with the Klein bottle).

My higher-dimensional visualization skills are failing me at the moment, so I was wondering if there was a way to realize any elementary cobordism as a connected sum with some appropriate manifold. Does the answer change if we’re considering unoriented bordism? Spin/Pin bordism? G-bundle bordism?

I phrased the question open-endedly since I haven’t seen many places go into depth about this concept. I believe these elementary bordisms can be thought of via putting a Morse function on the bordism geomety. Any other illuminating ways to think about elementary cobordisms would be appreciated!

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