Quotients of Simplicial Actions

Before we proceed with the general setup, let us see what makes things complicated in the simphcial situation. To do that, let us consider a simplicial 

G-action on X, where G is a finite group and X is a finite abstract simplicial complex. What can in general be said about the quotient X/G It is easy to take the topological quotient, but the simplicial (or cell) structure on it may not be derived from the simplicial structure of X in a very nice way. 

Consider, for instance, the reflection action of the additive group Z2 on an interval. The topological quotient is again an interval. One of its vertices is an orbit of the original Z2-action on the vertices of the interval, whereas the other one is not. This is not a very pleasing situation therefore we would like to require that the action satisfy further properties. 

In words. Condition (SI) says that if a simplex is preserved by a group element, then it is fixed by that group element pointwise. Note that this condition is not satisfied in the example of the Z2-action above. 

Clearly Condition (SI) is a special case of Condition (S2), when g a) = a = g a )C a. Here is an example of a situation in which Condition (SI) holds, whereas Condition (S2) does not. 

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