Research of Min Yan
نویسنده
چکیده
Eulerian poset is a basic and widely used combinatorial concept. It is the combinatorial version of manifolds and has the analogue of the Poincaré duality in the form of the DehnSommerville equations (linear relations among f -vectors). The concept has always been defined in the combinatorial language. As a topologist, my main insight is that a poset is Eulerian if and only if its underlying polyhedron is a manifold from the viewpoint of the Euler characteristic (i.e., the link of any point has the same Euler number as the sphere of the right dimension). The condition is purely topological and independent of the triangulation. The insight leads to the application of the usual topological approach to the combinatorial concept. Naturally, we may define the singularities of a polyhedron as those points whose links have the “wrong” Euler number. In [2], I showed that the number of rational linear relations among the f -vectors is n−s−1 2 , where n and s are the dimensions of the polyhedron and its Eulerian singular part. This illustrates there is less “Poincaré duality” for more singular space. In [4], we applied the stratified technique familiar to the topologists. We defined Eulerian stratified space, introduced the weighted combinations of f -vectors of strata, and developed the full Dehn-Sommerville theory. The theory provides a general framework for treating f -vectors for non-Eulerian posets. We also interpreted the theory as a boundary
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