Acyclic Matchings in Hasse Diagrams of Posets

Recall from Definition 6.13 that for a generalized simplicial complex A, a sim-plicial collapse is simply a removal of interiors of two simplices cr and r such 

Sometimes such a collaprse is called an elementary collapse. Note that a simplicial collapse is possible if and only if there exists a simplex t whose link in A consists of a single vertex the simplex cr is then given by the span of r and v. For a general CW complex one has to take care of some additional technicalities see Definition 11.12. 

In any case, we see that the combinatorial encoding of a set of collapses is best provided by a matching consisting of a collection of pairs of cells (r, cr) 

It is easy to see what goes wrong in this example the prospective collapses are all hooked up with each other in a cyclic pattern, which we are unable to break by doing only single collapses. This simple observation leads to the following formalization. 

We see that Definition 11.1 allows a more general situation than just the collapses that we described above. This makes our situation quite different from the simple homotopy theory considered in Section 6.5. For example, a partial matching consisting of a single pair of simplices b y a is always 

---