# 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!

.