TITLE: Topology of the set of Nash equilibria of a normal form game
QUESTION [6 upvotes]: 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 $A_j$).
General 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, to ask a specific question, is there an example where it has non-trivial $H_1$ (first homology group): can it be homeomorphic to a circle or an annulus?
[I asked this some time ago on MSE but received no response.]

REPLY [4 votes]: Nash-equilibria satisfy a universality theorem. This is due to Ruchira Datta's paper "Universality Of Nash Equilibria", https://arxiv.org/pdf/math/0301001.pdf. From her abstract: 

Every real algebraic variety is isomorphic to the set of totally mixed Nash equilibria of some three-person game, and also to the set of totally mixed Nash equilibria of an N-person game in which each player has two pure strategies.