Definition:Cobordism

Definitions
Let $$X^n \ $$ and $$Y^n \ $$ be manifolds without boundary of dimension n. A cobordism $$W^{n+1} \ $$ between X and Y is an (n+1) dimensional manifold such that $$\partial W = X \cup Y$$. An oriented cobordism W can be said to exist between two orientable manifolds X and Y where W is a cobordism such that $$\partial W = X \cup \overline{Y}$$, where this final symbol means Y taken with reverse orientation.

If W is homotopy-equivalent to $$X \times [0,1]$$ (formally, $$\exists \phi:W \rightarrow X$$ such that $$\phi$$ is a retract, which for X and Y simply connected is equivalent to $$H_*(W,M;\mathbb{Z})=0$$), then W is said to be an h-cobordism.