0
$\begingroup$

I started studying Milnor's "Lectures on the H-cobordism theorem". On page 2 he provides the following definitions

Definition 1.3. $(W;V_0,V_1)$ is a smooth manifold triad if $W$ is a compact smooth manifold and $\partial W$ is the disjoint union of two open and closed submanifolds $V_0$ and $V_1$.

My first question is: what exactly is meant by saying "open and closed submanifolds $V_0$ and $V_1$"?

My second question: Everywhere i looked up the definition of a cobordism it was equivalent to Milnor's definition of smooth manifold triads? What am i missing?

Wikipedia says:

A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, $\partial W = M \sqcup N$

Isn't this exactly what milnor defines as a smooth manifold triad?

Milnor then continues with

Definition 1.5 Given two closed smooth $n$-manifolds $M_0$ and $M_1$ (i.e. $M_0,M_1$ compact, $\partial M_0 = \partial M_1 = \emptyset$) a cobordism from $M_0$ to $M_1$ is a $5$-tuple $(W;V_0,V_1;h_0,h_1)$ where $(W;V_0,V_1)$ is a smooth manifold triad and $h_i:V_i \to M_i$ is a diffeomorphism, $i=0,1$.

Am i correct if i conclude "a smooth manifold triad corresponds up to diffeomorphism to a cobordism"? Does this explain the equivalent definitions of smooth manifold triads and cobordisms?

Thanks for any help.

$\endgroup$
2
$\begingroup$

To answer your first question, each of $V_0,V_1$ is required to be a submanifold of $\partial W$, and an open subset of $\partial W$, and a closed subset of $\partial W$. This is equivalent to the statement that each is a union of components of $\partial W$ (because each component of $\partial W$ is itself open and closed in $\partial W$).

As for your other questions, you are not missing anything in particular. Milnor is being more careful to insist that $V_0$ and $V_1$ actually "be" part of a triad, but not that they actually be part of the cobordism. To say that the two definitions are exactly the same is not exactly correct.

$\endgroup$
1
  • $\begingroup$ hello Lee. thank you very much for your detailed answer. i will think about it and maybe come back later if a question pops up. thank you vm. $\endgroup$
    – Zest
    Oct 4 '19 at 18:36

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.