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, $\exists \phi: W \to X$ such that $\phi$ is a retract, which for $X$ and $Y$ simply connected is equivalent to $H_*(W,M;\Z)=0$), then W is said to be an h-cobordism.