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A polyhedral model for Cartan's Hypersurface in S4

Published online by Cambridge University Press:  26 February 2010

Ulrich Brehm  and
Wolfgang Kühnel
Ulrich Brehm
Affiliation:
Fachbereich Mathematik der Techn., Universität Berlin, Str. des 17. Juni 136, D-1000 Berlin 12, Federal Republic of Germany.
Wolfgang Kühnel
Affiliation:
Fachbereich Mathematik der Universität Duisburg, Lotharstr. 65, D-4100 Duisburg 1, Federal Republic of Germany.
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Abstract

E. Cartan's famous isoparametric hypersurface in S4 with three distinct constant principal curvatures is geometrically a parallel hypersurface of the Veronese surface, and topologically it is an 8-fold quotient of the 3-sphere. In the present paper we describe a polyhedral analogue with only 15 vertices. Combinatorially this is an 8-fold quotient of the boundary complex of the 600-cell, and geometrically it is a quite regular subcomplex of a certain almost convex simplicial 4-sphere in E5. The euclidean symmetry group of this embedding is isomorphic to the icosahedral group A5 acting transitively on the 15 vertices.

Type
Research Article
Information
Mathematika , Volume 33 , Issue 1 , June 1986 , pp. 55 - 61
Copyright
Copyright © University College London 1986

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