These lecture notes are based on a series of lectures given at the Nankai Institute of Mathematics in the fall of 1998. They provide an overview of the work of the author and the late Chih-Han Sah on various aspects of Hilbert's Third Problem: Are two Euclidean polyhedra with the same volume "scissors-congruent", ie. can they be subdivided into finitely many pairwise congruent pieces? The book starts from the classical solution of this problem by M. Dehn. But generalization to higher dimensions and other geometries quickly leads to a great variety of mathematical topics, such as homology of groups, algebraic K-theory, characteristics classes for flat bundles, and invariants for hyperbolic manifolds. Some of the material, particularly in the chapters on projective configurations, is published here for the first time.
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First Sentence:
It is elementary and well-known that two polygons P and P' in the Euclidean plane have the same area if and only if they are scissors congruent (s.c.), i.e. if they can be subdivided into finitely many pieces such that each piece in P is congruent to exactly one piece in P'. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
hyperhomology spectral sequence, rational lune, hyperbolic simplex, scissors congruent, strict flag, scissors congruences, rational vector space, projective configurations, group homology, chain homotopic, chain homotopy, bar complex, exact sequence, chain complex, spherical case, homology groups, spectral sequences, boundary map Key Phrases - Capitalized Phrases (CAPs): (learn more)
Homology of Lie, Non-Euclidean Hilbert New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
It is elementary and well-known that two polygons P and P' in the Euclidean plane have the same area if and only if they are scissors congruent (s.c.), i.e. if they can be subdivided into finitely many pieces such that each piece in P is congruent to exactly one piece in P'. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
hyperhomology spectral sequence, rational lune, hyperbolic simplex, scissors congruent, strict flag, scissors congruences, rational vector space, projective configurations, group homology, chain homotopic, chain homotopy, bar complex, exact sequence, chain complex, spherical case, homology groups, spectral sequences, boundary map Key Phrases - Capitalized Phrases (CAPs): (learn more)
Homology of Lie, Non-Euclidean Hilbert New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 6
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- Gesammelte Abhandlungen. Erster Band: Zahlentheorie (AMS Chelsea Publishing) by David Hilbert in Back Matter
- Hyperbolic Geometry (London Mathematical Society Student Texts) by Birger Iversen in Back Matter
- Differential Geometry (Proceedings of Symposia in Pure Mathematics, vol. 27, pt. 2) by Shiing-Shen Chern in Back Matter
- Algebraic Geometry (Dover Books on Mathematics) by Solomon Lefschetz in Back Matter
- Algebraic Topology by Edwin H. Spanier in Back Matter
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- Contemporary Trends in Algebraic Geometry and Algebraic Topology by Shiing-Shen Chern in Front Matter
- Groups, Combinatorics & Geometry: Durham 2001 by A. A. Ivanov on page 94
- Minimal Submanifolds and Related Topics (Nankai Tracts in Mathematics) by Yuanlong Xin in Front Matter
- Least Action Principle of Crystal Formation of Dense Packing Type & the Proof of Kepler's Conjecture by Wu Yi Hsiang in Front Matter
- Iterated Integrals and Cycles on Algebraic Maniforlds (Nankai Tracts in Mathematics, Vol. 7) by Bruno Harris in Front Matter
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