Jordan Curve Theorem (Homological Proof)

Core Idea

The Jordan curve theorem states that every simple closed curve in the plane R² divides it into exactly two connected components (a bounded "inside" and an unbounded "outside"), with the curve as their common boundary. While intuitively obvious, the theorem is notoriously hard to prove for arbitrary continuous curves. The homological proof uses the Mayer-Vietoris sequence and the homology of S² to establish the separation property, generalizing naturally to the Jordan-Brouwer separation theorem: any embedded S^n-1 in Sⁿ separates Sⁿ into exactly two components.

Explainer

The Jordan curve theorem (JCT) states that if C is a simple closed curve in R² (the image of a continuous injection gamma : S¹ -> R²), then R² \ C has exactly two connected components, one bounded and one unbounded, and C is the common boundary of both. Equivalently, working in S² = R² union {infinity} (the one-point compactification), the complement S² \ C has exactly two connected components. The theorem was stated by Jordan in 1887, and the first correct proof was given by Veblen in 1905. Modern proofs using homology are considerably cleaner.

The homological approach proves the more general Jordan-Brouwer separation theorem: if h : S^n-1 -> Sⁿ is an embedding (a homeomorphism onto its image), then Sⁿ \ h(S^n-1) has exactly two connected components. The proof uses Alexander duality, which relates the homology of the complement Sⁿ \ K to the cohomology of the compact subspace K. Specifically, H_k(Sⁿ \ K; Z) = H^n-k-1(K; Z) (with Cech cohomology for full generality). For K = h(S^n-1) = S^n-1: H_0(Sⁿ \ S^n-1) = H^n-1(S^n-1) = Z (in reduced homology, this gives exactly two components). The full Alexander duality also shows H_k(Sⁿ \ S^n-1) = 0 for k > 0, so each component is homologically trivial.

An alternative proof uses the Mayer-Vietoris sequence more directly. One version proceeds by induction, building up the simple closed curve from simpler arcs and using Mayer-Vietoris to track how the homology of the complement changes at each stage. The key step is showing that the complement of an arc (homeomorphic image of [0,1]) in Sⁿ is connected and has trivial homology — i.e., arcs do not separate Sⁿ. Then, decomposing S¹ into two arcs and applying Mayer-Vietoris to the complement of their union gives the separation result.

The theorem has several important extensions and related results. The Schoenflies theorem (in dimension 2) strengthens the JCT by saying that the closure of each component is homeomorphic to a closed disk D² — not just that there are two components, but that each component is "shaped like a disk." In higher dimensions (n >= 3), the Schoenflies theorem fails without additional hypotheses: the Alexander horned sphere is an embedding of S² in S³ whose complement has two components, but one component is not simply connected (and hence not homeomorphic to a ball). This shows that the Jordan-Brouwer separation theorem (homological statement) generalizes cleanly, but the Schoenflies theorem (topological characterization of the components) requires additional conditions (such as the embedded sphere being "locally flat").

The homological proof of the Jordan curve theorem is a triumph of the algebraic topology method: a statement that is "intuitively obvious" for smooth curves but fiendishly difficult for arbitrary continuous curves becomes a straightforward computation when translated into the language of homology and Alexander duality. The proof treats all continuous curves uniformly and generalizes to all dimensions, demonstrating the power of homological methods for establishing topological facts that resist elementary proof techniques.