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.