The recent introduction of differential topology into economics was brought about by the study of several basic questions that arise in any mathematical theory of a social system centered on a concept of equilibrium. The purpose of this paper is to present a detailed discussion of two of those questions, and then to make a rapid survey of some related developments of the last five years.

Let e be a complete mathematical description of the economy to be studied (e.g., for an exchange economy, e might be a list of the demand functions and of the initial endowments of the consumers). Assumptions made a priori about e (e.g., assumptions of continuity on the demand functions) define the space ℰ of economies to which the study is restricted. By a state of an economy we mean a list of specific values of all the relevant endogenous variables (e.g., prices and quantities of all the commodities consumed by the various consumers). We denote by S the set of conceivable states. Now a given equilibrium theory associates with each economy e in ℰ, the set E(e) of equilibrium states of e, a subset of S (see Figure 1).