Topology and Economics: from Sard’s Theorem to Social Choice

This paper provides a synthetic survey of applications of differential, algebraic, and general topology in mathematical economics, with particular emphasis on empirically verifiable results. We argue that topological machinery is intrinsically necessary on three levels. First, Smale’s proof of Walrasian equilibrium existence without convexity assumptions, and the local uniqueness result for generic economies, require Sard’s theorem and the preimage theorem; the classical Arrow–Debreu existence proof uses fixed-point methods (Kakutani) but not the full differential apparatus. Second, identifiability of structural models — including optimal-transport labour market models, partially identified models, and nonparametric IV regression — is equivalent to specific topological properties of the parameter space (connectedness, contractibility, vanishing of higher homotopy groups). Third, applying persistent homology to financial time series yields leading crisis indicators: the persistence landscape norm of H₁ barcodes rises 3–8 weeks before market crashes. Together, these results trace a path from Brouwer’s theorem and Chichilnisky–Heal homotopy groups through the outer Hausdorff metric in location theory to the topology of interbank liability networks in systemic risk analysis.