Nonevasive abstract simplicial complex

On the numerical side, if X ne Y, then the Euler characteristics of X and Y are the same. In particular, a nonevasive abstract simplicial complex has reduced Euler characteristic equal to 0. 

To see statement (2) we can first prove that a nonevasive abstract simplicial complex is collapsible. Perhaps the simplest argument is to use induction on the number of vertices. Indeed, if our abstract simplicial complex has one vertex, then it must be nonempty and is therefore collapsible. For the induction step we notice that for any abstract simplicial complex X and any vertex V of X, if both lkx( ) and dlx(v) are collapsible, then so is X. To find a collapsing sequence for X, start by collapsing away all the simplices that contain n, following some collapsing sequence for lkx( ). This will remove the open star of n, and we can finish off by continuing with any collapsing sequence for dlx(r ). 

By a simple induction on the number of vertices, it will follow that if Q is a nonevasive graph property, then the abstract simplicial complex A Q) is collapsible see Proposition 13.7 and Proposition 13.9. 

Since the graph property Q is nonevasive, we know that the abstract simplicial complex A Q) is collapsible see Proposition 13.7(2). In particular, it has to be Zp-acyclic. We can now use Theorem 13.5 to conclude that x A Q)r) = 1- On the other hand, the group F acts doubly transitively on the set [n], i.e., any ordered pair of points can mapped to any other ordered pair of points by a transformation from F. This means that the group F acts transitively on the set of vertices of the abstract simplicial complex A Q). Therefore, the only point of A G) that can possibly be fixed by every element in F is the barycenter of the simplex on all (") vertices, which of course can be the case only when A(Q) is a full simplex. This contradicts our assumption that the graph property Q is nontrivial. ... 

A finite nonempty abstract simplicial complex X is called nonevasive if either X is a point, or, inductively, there exists a vertex v ofX such that both X ti andlkxv are nonevasive. Otherwise, the complex X is called evasive. 

If X and X2 are abstract simplicial complexes such that X ne X2 and Y is an arbitrary abstract simplicial complex, then X Y ne X2 Y. A cone over any abstract simplicial complex is nonevasive. 

In this section we mention some other classes of abstract simplicial complexes that are studied in Combinatorial Algebraic Topology and are defined in a recursive way. These families are less prominent than the nonevasive complexes, so we shall keep our presentation brief. 

---