Topological Methods in Walrasian Economics

In winter 71/72 I held a seminar on general equilibrium theory for a jOint group of students in mathematics and in econo- mics at the university of Bonn, w.Germany1 The economists, how- ever, had a mathematical background well above the average - Most of the material treated in that seminar is described in these notes. The connection between smooth preferences and smooth demand func- tions [ see Debreu (1972) ] and regular economies based on agents with smooth preferences are not presented here - Some pedagogical difficulties arose from the fact that elementary knowledge of algebraic topology is not assumed although it is helpful and indeed necessary to make some arguments precise - It is only a minor restriction, at present, that functional ana- lysis is not used - But with the development of the theory more economic questions will be considered in their natural infinite dimensional setting - Economic knowledge is not required, but especially a reader without economic background will gain much by reading Debreus classic "Theory of Value" (1959) - Although the formulation of our economic problem uses a map between Euclidean spaces only, we shall also consider ma- folds - Manifolds appear in our situation because inverse images under differentiable mappings between Euclidean spaces are very often differentiable manifolds - ( Under differentiability assump- tions, for instance, the graph of the equilibrium set correspon-