Manifold structures on abstract regular polytopes |
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Authors: | Ulrich Brehm Wolfgang Kühnel Egon Schulte |
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Institution: | (1) Fachbereich Mathematik, TU Berlin, Str. d. 17. Juni 136, 1000 Berlin 12, Germany;(2) Fachbereich Mathematik, Univ. Duisburg, 47048 Duisburg, Germany;(3) Department of Mathematics, Northeastern University, 02115 Boston, MA, USA |
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Abstract: | Summary Abstract regular polytopes are complexes which generalize the classical regular polytopes. This paper discusses the topology of abstract regular polytopes whose vertex-figures are spherical and whose facets are topologically distinct from balls. The case of toroidal facets is particularly interesting and was studied earlier by Coxeter, Shephard and Grünbaum. Ann-dimensional manifold is associated with many abstract (n + 1)-polytopes. This is decomposed inton-dimensional manifolds-with-boundary (such as solid tori). For some polytopes with few faces the topological type or certain topological invariants of these manifolds are determined. For 4-polytopes with toroidal facets the manifolds include the 3-sphereS
3, connected sums of handlesS
1
× S
2
, euclidean and spherical space forms, and other examples with non-trivial fundamental group. |
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Keywords: | Primary 51M20 Secondary 57M50 |
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