Topological Methods in Economics: From Equilibrium Existence to Topological Data Analysis

This paper surveys how the three branches of topology — differential, algebraic, and point-set — enter mathematical economics, and organises classical and contemporary results within a single conceptual frame. The classical layer rests on four pillars: Sard’s theorem and the preimage theorem, which underwrite Smale’s convexity-free proof of Walrasian equilibrium existence and Debreu’s local-uniqueness theorem for generic economies; the Brouwer and Kakutani fixed-point theorems behind the Arrow–Debreu existence proof; the Chichilnisky–Heal homotopy obstruction that ties continuous anonymous social choice to contractibility of preferred cones, and through it to the existence of competitive equilibrium; and the outer Hausdorff metric of Berliant and ten Raa, which makes location theory well-posed on the infinite-dimensional commodity space of land parcels. The contemporary layer turns the same apparatus toward data. The L¹ norm of H₁ persistence landscapes computed on rolling windows of asset returns rises three to eight weeks before the 2008 and 2015–2016 market crashes, outperforming the VIX in lead time; Ollivier–Ricci curvature on equity correlation networks supplies a complementary leading indicator of systemic fragility. Optimal transport recasts identifiability in matching markets as uniqueness of a dual Kantorovich plan, governed by the topology of the type-space supports. The manifold hypothesis makes nonparametric instrumental-variable convergence rates depend on intrinsic rather than ambient dimension. Together these results trace a coherent arc from fixed-point theorems to topological data analysis; a Morse-theoretic synthesis on the state manifold of an economy, linking equilibria, tâtonnement dynamics, and saddle-point crises, is stated as a research programme rather than a theorem.