Topology and Invertible Maps

I study connected manifolds and prove that a proper map f: M → M is globally invertible when it has a nonvanishing Jacobian and the fundamental group п1 (M) is finite. This includes finite and infinite dimensional manifolds. Reciprocally, if п1 (M) is infinite, there exist locally invertible maps which are not globally invertible. The results provide simple conditions for unique solutions to systems of simultaneous equations and for unique market equilibrium. Under standard desirability conditions, it is shown that a competitive market has a unique equilibrium if its reduced excess demand has a nonvanishing Jacobian. The applications are sharpest in markets with limited arbitrage and strictly convex preferences: a nonvanishing Jacobian ensures the existence of a unique equilibrium in finite of infinite dimensions, even when the excess demand is not defined for some prices, and with or without short sales