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Centrally-symmetric tight surfaces and graph embeddings. (English) Zbl 0879.53045
Let \(M\) be a compact polyhedral (2-dimensional) surface (without boundary) in \(\mathbb{R}^d\), which is substantial (i.e., it is not contained in a hyperplane) and tight (i.e., for every half space \(h\subset \mathbb{R}^d\) the intersection \(M\cap h\) is connected). It is known that examples exist for arbitrarily high \(d\). In contrast, the author proves a sharp upper bound for the codimension in the case that the convex hull of \(M\) is a centrally-symmetric (i.e., invariant under \(-id_{\mathbb{R}^d})\) simplicial polytope, namely \[ 2(d-1) (d-3) \leq 3\bigl(2- \chi(M) \bigr). \] For \(d\geq 4\) equality holds if and only if \(M\) is a subcomplex of the \(d\)-octahedron with \(2d\) vertices. A second result gives a lower bound for the number of vertices of a surface triangulation admitting a fixed-point free involution.

MSC:
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
52B70 Polyhedral manifolds
05C10 Planar graphs; geometric and topological aspects of graph theory
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