Cobordism and h-cobordism

Question

Is there a way to simply explain cobordism and h-cobordism? I am not looking for a math based explanation, but rather, just the main ideas behind the concepts.

Answer 1

A cobordism between two manifolds $M$ and $N$ of the same dimension $n$ is a manifold $P$ of dimension $n+1$ so that the boundary of $P$ is diffeomorphic to the disjoint union of $M$ and $N$. So, a cobordism between $M$ and $N$ can be thought of as saying that $M$ and $N$ are two $n$-dimensional puzzle pieces, that when put together form the border of a completed $n+1$-dimensional picture.

It helps to think of some examples: Let's take $M=\{a\}$ and $N=\{b\}$ for real numbers $a<b$. Then $P=[a,b]$ is a cobordism of $M$ and $N$ because the boundary of $[a,b]$ is $\{a,b\}=\{a\}\cup\{b\}$. An example with $M$ and $N$ different is to take $M$ to be a circle and $N$ to be two disjoint circles. Then a cobordism $P$ is a pair of pants with $M$ the waist of the pants and $N$ the two leg-holes.

An $h$-cobordism contains even more information: it says that not only are $M$ and $N$ puzzle pieces that fit together to form the border of a completed picture, but the way that they fit into that completed picture is the same, up to maybe a little squishing and squeezing. More specifically, the images of $M$ and $N$ in the cobordism $P$ must be homotopic: this means that one can be "continuously deformed" into the other. Maybe we could stretch part of $M$ out and squeeze part of $N$ together, and in doing so, we realize that we have made $M$ and $N$ look the same inside of $P$. It's like our puzzle pieces are made out of rubber once we complete the picture, so we can stretch them until they look the same.

In our $[a,b]$ example above, we actually have an $h$-cobordism, because $\{a\}$ and $\{b\}$ are both included in $[a,b]$ in pretty much the same way: define $i_1:\{a\}\to[a,b]$ by $i_1(a)=a$ and $i_2:\{b\}\to[a,b]$ by $i_2(b)=b$. Both of these maps don't really move anything around, and the result looks the same: the image is just a point.