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Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.

If $P$ is connected and has one maximum element, can I conclude that the simplicial complex associated to $P$ (minus the maximal face) is the disjoint union of spheres?

I know that if one has only one 0-cell (and the some number of $(n-1)$-cells) then the complex would be a wedge of spheres, so I was wondering what would happen if one has more than one 0-cell.

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.

If $P$ is connected and has one maximum element, can I conclude that the simplicial complex associated to $P$ (minus the maximal element) is homotopy equivalent to a disjoint union of spheres?

I know that if one has only one 0-cell (and the some number of $(n-1)$-cells) then the complex would be homotopic to a wedge of spheres, so I was wondering what would happen if one has more than one 0-cell.

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.

Can I conclude that the simplicial complex associated to $P$ is the disjoint union of spheres?

I know that if one has only one 0-cell (and the some number of $(n-1)$-cells) then the complex would be a wedge of spheres, so I was wondering what would happen if one has more than one 0-cell.

Suppose one has a Morse matching (acyclic matching) of a poset $P$ of rank $n$ in which the only unmatched cells are 0-cells and $(n-1)$-cells, the same number $k$ of each one.

If $P$ is connected and has one maximum element, can I conclude that the simplicial complex associated to $P$ (minus the maximal face) is the disjoint union of spheres?

I know that if one has only one 0-cell (and the some number of $(n-1)$-cells) then the complex would be a wedge of spheres, so I was wondering what would happen if one has more than one 0-cell.