# J. Milnor: smooth manifold triad vs cobordism

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.

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.