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Let $X$ denote a regular cell structure on a closed (orientable) $n$-manifold (If it helps, the cells are polytopal and the attaching maps are affine). Recall that a discrete Morse function on this cell complex is same as an acyclic partial matching on the Hasse graph of the face poset of $X$.

Partial matching: is a collection of edges of the graph such that every vertex is in at most one matched edge. Acyclic matching: first orient the edges from top to bottom and then for every matched edge reverse the direction. Then one would like to have a matching such that there are no oriented cycles. Call such a matching maximal if it is not possible to add any more edges.

My question:

Under what conditions does a maximal, acyclic partial matching corresponds to a perfect Morse function? (i.e., number of critical $k$-cells is equal to $k$-th Betti)

(Robin Forman's notes has an example of a discrete Morse function on the real projective plane. The corresponding matching is maximal and there is exactly one critical cell in every dimension. So it might make sense to define perfect Morse function as the one where the number of critical $k$-cells is equal to $k$-th Betti number mod 2.)

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