Realization Spaces of Polytopes
The book collects results about realization spaces of polytopes. It gives a presentation of the author's "Universality Theorem for 4-polytopes". It is a comprehensive survey of the important results that have been obtained in that direction. The approache
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1643
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg F. Takens, Groningen
1643
Springer Berlin Heidelberg New York Barcelona Budapest Hong Kong London Milan Paris Santa Clara Singapore Tokyo
Jtirgen Richter-Gebert
Realization Spaces of Polytopes
Springer
Author Jiirgen Richter-Gebert Technische Universitat Berlin Fachbereich Mathematik Diskrete Mathematik, Sekr. MA 6-1 StraBe des 17. Juni 136 D-I0623 Berlin, Germany e-mail: [email protected] Cataloging-in-Publication Data applied for
Die Deutsche Bibliothek - CIP-Einheitsaufnahme Richter-Gebert, Jorgen: Realization spaces of polytopes / Jiirgen Richter-Gebert. - Berlin; Heidelberg; New York; Barcelona; Budapest; Hong Kong; London; Milan; Paris; Santa Clara; Singapore; Tokyo: Springer, 1996 (Lecture notes in mathematics; 1643) ISBN 3-540-62084-2 NE:GT
Mathematics Subject Classification (1991): 52B11,52B40, 14PIO, 51A25, 52BIO, 52B30, 68QI5 ISSN 0075-8434 ISBN 3-540-62084-2 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1996 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the author SPIN: 10561503 46/3142-543210 - Printed on acid-free paper
For Ingrid and Angela-Sophia, who helped me so much.
PREFACE
Steinitz's Theorem (proved in 1922) is one of the oldest and most prominent results in polytope theory. It gives a completely combinatorial characterization of the face lattices of 3-dimensional polytopes. Steinitz observed that the technique of proving his theorem also implies that for any 3-dimensional polytope the set of all its realizations is a trivial topological set. In other words: realization spaces of 3-dimensional polytopes are contractible. For a long time it was an open problem whether there exist similar results in spaces of dimension greater than three. It was proved by Mnev in 1986 that the contrary is the case. As a consequence of his famous Universality Theorem for oriented matroids he showed that realization spaces of polytopes with dimension-plus-four vertices can have arbitrary homotopy type. The present research monograph studies the structure of realization spaces of polytopes in fixed dimension. The main result that is obtai
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