Chichilnisky, Graciela (1997): Topology and invertible maps. Published in: Advances in Applied Mathematics , Vol. 21, (1998): pp. 113123.

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Abstract
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 pi (M) is finite. This includes finite and infinite dimensional manifolds. Reciprocally, if pi (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.
Item Type:  MPRA Paper 

Original Title:  Topology and invertible maps 
Language:  English 
Keywords:  manifolds; mathematical economics; Jacobian; supply and demand; equilibrium 
Subjects:  C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C68  Computable General Equilibrium Models C  Mathematical and Quantitative Methods > C0  General > C02  Mathematical Methods C  Mathematical and Quantitative Methods > C6  Mathematical Methods ; Programming Models ; Mathematical and Simulation Modeling > C62  Existence and Stability Conditions of Equilibrium 
Item ID:  8811 
Depositing User:  Graciela Chichilnisky 
Date Deposited:  22. May 2008 04:40 
Last Modified:  14. Feb 2013 18:02 
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URI:  https://mpra.ub.unimuenchen.de/id/eprint/8811 