ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

13. Combinatorics

1991 MS Classification: 05, 20, 57 Lutz, Frank Hagen, Technische Universität Berlin, Germany Combinatorial manifolds with vertex-transitive automorphism group. A d-dimensional combinatorial manifold is a simplicial complex such that the link of every vertex is a (d − 1)-dimensional combinatorial sphere. For a number of topological manifolds triangulations have been found that are combinatorial manifolds with a minimal number of vertices. We make use of a list of all transitive permutation groups of small degree and determine, up to combinatorial equivalence, all combinatorial manifolds with vertex-transitive automorphism group on n ≤ 12 vertices. With the exception of groups of small order, we extend this result to n ≤ 15 vertices. We furthermore obtain combinatorial manifolds with the homology of S^{2} × S^{3}, S^{3} × S^{3}, S^{2} × S^{5}, and S^{3} × S^{4} on 14, 16, and 18 vertices. Part of these results is joint work with E. Köhler. Reference: E. Köhler, and F. H. Lutz, Combinatorial manifolds with transitive automorphism group on few vertices, in preparation.