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](https://math.stackexchange.com/questions/1732012/topology-of-the-set-of-nash-equilibria) on MSE but received no response.]



Topology of the set of Nash equilibria of a normal form game
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: 



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> 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. 
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