Topology and Invertible Maps

Abstract

I study connected manifolds and prove that a proper mapf:M→Mis globally invertible when it has a nonvanishing Jacobian and the fundamental group π1(M) is finite. This includes finite and infinite dimensional manifolds. Reciprocally, if π1(M) is infinite, there exist locally invertible maps that 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 or infinite dimensions, even when the excess demand is not defined for some prices, and with or without short sales. © 1998 Academic Press.