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Let $X^n$ and $Y^n$ be manifolds without boundary of dimension $n$.

Let $W$ be a cobordism between $X$ and $Y$ such that $W$ is homotopy-equivalent to $X \times \left[{0 \,.\,.\, 1}\right]$.

(formally, $\exists \phi: W \to X$ such that $\phi$ is a retract, which for $X$ and $Y$ simply connected is equivalent to $H_* \left({W, M; \Z}\right) = 0$)

Then W is said to be an h-cobordism.