Nash Equilibrium Existence without Convexity

In this paper, I prove the existence of a pure-strategy Nash equilibrium for a large class of games with nonconvex strategy spaces. Specifically, if each player's strategies form a compact, connected Euclidean neighborhood retract and if all best-response correspondences are null-homotopic, then the game has a pure-strategy Nash equilibrium. As an application, I show how this result can prove the fundamental theorem of algebra.