Geometry and the simplex: results, questions and ideas
In \(\mathbb{R}^2\), Euler and Descartes showed: a planar graph without double edges or loops can have at most three times as many edges \(E\) (one-dimensional simplices) as it has vertices \(V\), i.e. \(E\le 3V\). The paper under review is in fact a survey telling about simplices, graphs, embedding problems, counting specific spaces, and the like. It gives a perfect illustration of the kind of things that are done in our days. Among others, the author mentions his theorem Theorem 1.1 Given a simplicial complex \(\Delta\) that piecewise-linearly embeds in \(\mathbb{R}^4\), then \(T\le 4E\), i.e. the number of triangles \(T\) (two-dimensional simplices) exceeds the number of edges \(E\) (one-dimensional faces) by a factor at most 4. The paper informs the reader about several results by the author and others. In addition, six problems and one conjecture are printed in clear sight. Let us give those! \begin{itemize} \item[1)] Do the results of Theorem 1.1 extend to topological embeddings? Do they extend to cell complexes that are strongly regular; that is, all complexes whose partially ordered set of faces is an atomistic lattice? \item[2)] Are there infinitely many combinatorially distinct types of polytopes of dimension 4 whose moduli space is a point? \item[3)] Is the number of contractible complexes with a given number of triangles \(n\) and a uniform bound on the vertex degree exponential, or superexponential? What happens if I only restrict to complexes with vanishing reduced homology? \item[4)] Given a triangulation of a connected \(d\)-manifold on \(n\) vertices, how many steps does one need from any facet to any other facet? \item[5)] Is it true that for every \(k=k(\ell)\) such that every \(d\)-polytope (or polyhedral sphere), \(d\ge k\), has a Boolean interval of length \(\ell\) in its face lattice? \item [6)] Is it true that for every \(p\ge 2\) there exists a \(k=k(p)\), such that every Riemannian metric \(R\) on \(S^d\), \(d\ge k\), has a point \(x\) and a \(p\)-dimensional subspace \(M\) of \(T_xR\), such that the average over sectional curvatures at 2-sections in \(M\) is nonnegative? \item[7)] Assume \(P\) and \(Q\) are combinatorially equivalent polytopes, such that the dihedral angles of \(P\) are bigger or equal than the corresponding dihedral angles of \(Q\). Then the polytopes are normally equivalent, that is, their normal fans are related by rotation and reflection. \end{itemize} A good and more or less complete list of references closes this interesting overview of the author's work and that of others.
scientific article; zbMATH DE number 7314412
scientific article