Simplicial complexes and the evasiveness conjecture

We review the topological approach to evasiveness in [5], study the sizes of automorphism groups of graphs, and use this to estimate the Euler characteristic of the simplicial complex associated to a nontrivial graph property  on 6 vertices. The condition of being nonevasive implies that the Euler characteristic of these simplicial complexes is 1, and our estimate of  obligates  to contain some classes of special graphs. The use of Oliver groups (as in [5]) is essential to deal with the different cases that appear.