Induced subcomplex

Definition 2.40. Let X be a polyhedral complex, and let S he the set of some of its vertices. We let X[S ] denote the polyhedral complex that consists of all cells whose set of vertices is a subset of S. This complex is called the induced subcomplex. 

The next theorem provides a handy criterion for being able to conclude that an induced subcomplex (see Definition 2.40) of a shellable complex is shellable as well. 

Proof. First we notice that any maximal simplex r of /1[S] can be obtained as an intersection of / [S] with some maximal simplex of A just take any one that contains t, and use the fact that we are dealing with an induced subcomplex. Since the converse of that statement is the assumption of the theorem, we see that the list of intersections of maximal simplices of A with. 4[S] consists of all maximal simplices of /lliS], though each maximal simplex will tj ically appear several times on that list. 

Theorem 12.5 can frequently be applied in this situation. When the acyclic category C is graded, one can take any rank selection R. Clearly, the nerve of R is an induced subcomplex of the nerve of C. Furthermore, one can see that any maximal chain of C will intersect f in a maximal chain hence the conditions of Theorem 12.5 are satisfied. We can therefore conclude the following proposition. 

Remark 11. If - The converse of Theorem 11.13(a) is clearly true in the following sense if Ac is a subcomplex of A and if there exists a sequence of collapses from Z to. 4c, then the matching on the cells of 4 4c induced by this sequence of collapses is acyclic. In particular, a polyhedral complex 4 is collapsible if and only if the poset 1F A) 6 allows a complete acyclic matching. 

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