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 fundam ental group pi(1)(M) is finite. This includes finite and infinite dime nsional manifolds. Reciprocally, if pi(1)(M) is infinite, there exist locally invertible maps that are not globally invertible. The results provide simple conditions for unique solutions to systems of simultane ous equations and for unique market equilibrium Under standard desirab ility conditions, it is shown that a competitive market has a unique e quilibrium if its reduced excess demand has a nonvanishing Jacobian. T he applications are sharpest in markets with limited arbitrage and str ictly convex preferences: a nonvanishing Jacobian ensures the existenc e 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. (C) 1998 Academic Press.