Given a finite dimensional finite $CW$ complex $X$ of dimension $d$, I want to build a compact manifold $M$ (with least dimension possible) with boundary with the property that,

- $M$ has the same homotopy type as $X$.
- $M$ is inductively built and each stage is a compact manifold with boundary.
- When we go to the next stage from the previous one, there is exactly one critical point of index $\leq d$.

Point (2) can be done using a theorem related to elementary cobordism which I found in "Lectures on the h-Cobordism theorem" (theorem 3.12) by Milnor. But (1), I don't know how to start the induction process and wish to understand the mechanism.

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