Consider a normal form game with $n$ players (and finitely many options per player) defined by finite option sets $A_1,\ldots,A_n$ and payoff matrices $u_1,\ldots,u_n: \prod_{j=1}^n A_j \to \mathbb{R}$. Let $N$ be the set of Nash equilibria, which is a subset of the set $S := \prod_{j=1}^n S_j$ of mixed strategy profiles where $S_j$ is the linear simplex with vertex set $A_j$ (=set of mixed strategies on $A_j$).
Question: What interesting things can be said about this set $N$ of Nash equilibria, from a topological point of view?
Obviously it need not be connected (it can be a finite set of $>1$ points), and it's pretty clear that it's compact. But does someone have an example where it has non-trivial $H_1$ (=first homology group) or something like that (e.g., can it be homeomorphic to a circle or an annulus?