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THE TRIANGULATION CONJECTURE

In topology, a basic building block for spaces is the n-simplex. A 0-simplex is a point, a 1-simplex is a closed interval, a 2-simplex is a triangle, and a 3simplex is a tetrahedron. In general, an n-simplex is the convex hull of n+1 vertices in n-dimensional space. One constructs more complicated spaces by gluing together several simplices along their faces, and a space constructed in this fashion is called a simplicial complex. For example, the surface of a cube can be built out of twelve triangles—two for each face. Topologically, the cube is indistinguishable from a sphere (and also from a tetrahedron, or from an octahedron), since all these surfaces can be deformed into each other without tearing them apart; we say that they are homeomorphic. Apart from simplicial complexes, manifolds form another fundamental class of spaces studied in topology. An n-dimensional manifold is a space that looks locally like the n-dimensional Euclidean space. Manifolds are ubiquitous in many parts of mathematics; for instance, they can appear as spaces of solutions to systems of polynomial equations, or to systems of differential equations. However, knowing that a space is a manifold does not tell us much about its global structure. To study the properties of a manifold, it is helpful to triangulate it, that is, to construct a homeomorphism to a simplicial complex. For example, the surface of a sphere is a twodimensional manifold, and it admits a triangulation with twelve triangles, in the form of the cube. (Of course, it also admits many other triangulations.) A triangulation yields a combinatorial description for the manifold. Furthermore, if we have two manifolds and we try to tell them apart, the first thing to do is to check if their topological invariants (such as their homology groups) are the same. If we are able to triangulate the manifolds, it is straightforward to compute their homology groups in terms of the two triangulations. The triangulation conjecture—first formulated by Kneser in 1924—claimed that every manifold was triangulable. The conjecture turned out to be false in general, although it is true for manifolds of dimension up to 3, and also for all differentiable manifolds (those that are “smoothly” like Euclidean space, so that one can do calculus on them). In Kneser’s time, it was already known that every two-dimensional surface is triangulable, due to the work of Radó. The case of differentiable manifolds was settled in the 1940’s by Cairns and Whitehead. In 1952, Moise showed that any three-dimensional manifold is differentiable, and thus triangulable.