I am looking for a proper terminology of the following. Additionally, it can be related to equivariant cohomology and I would like to know if there is any connection.
Given an action of a finite group $G$ on a manifold $M$, that is as nice as you want (ie, in a simplicial decomposition it permutes simpleces, etc.), how do you call this: $$ \chi_G(M)(g):=\sum_i (-1)^i tr (g^*:H_i(M,\mathbb{C})\rightarrow H_i(M,\mathbb{C})) $$
- Is it related to equivariant cohomology? Does it have a name there?
It is a virtual character, since it has some negative terms, but it is very specific because it involves a manifold (or simplicial complex); so I called it character of $G$ acting on $M$.
Comments on the absence of terminology are also welcomed.