Face enumeration-from spheres to manifolds
For a convex \(d\)-dimensional polytope, let \(f_i\) denote the number of its \(i\)-dimensional faces. The tuple \((f_0,\ldots,f_d)\) is called the \(f\)-vector. One of the greatest achievements in the theory of convex polytopes is the \(g\)-theorem that characterizes all possible \(f\)-vectors of convex simplicial polytopes. A long standing open question, known as \(g\)-conjecture, is whether the same characterization holds for arbitrary simplicial spheres. As the title of the present paper indicates, the author studies a similar question for a wider class of objects. He uses combinatorial and algebraic techniques developed in the study of convex polytopes. The author formulates a \(g\)-conjecture (and its algebraic version) for arbitrary homology manifolds (triangulations in which the links of all faces are homology spheres). Also he shows that Kühnel's triangulations of sphere bundles over the circle minimize the \(f\)-vector over all homology manifolds with nonzero first Betti number. By adapting Walkup's construction, the author gives a complete characterization of \(f\)-vectors for \(S^1 \times S^3\), \(\mathbb{C}\mathbb{P}^2\), and some other \(4\)-dimensional manifolds. Finally, he shows that every homology manifold can be triangulated in a \(2\)-neighborly fashion.
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