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AuthorMatouลกek, Jiล{153}รญ. author
TitleUsing the Borsuk-Ulam Theorem [electronic resource] : Lectures on Topological Methods in Combinatorics and Geometry / by Jiล{153}รญ Matouลกek
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 2003
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Descript XII, 214 p. online resource


"The "Kneser conjecture" -- posed by Martin Kneser in 1955 in the Jahresbericht der DMV -- is an innocent-looking problem about partitioning the k-subsets of an n-set into intersecting subfamilies. Its striking solution by L. Lovรกsz featured an unexpected use of the Borsuk-Ulam theorem, that is, of a genuinely topological result about continuous antipodal maps of spheres. Matousek's lively little textbook now shows that Lovรกsz' insight as well as beautiful work of many others (such as Vrecica and Zivaljevic, and Sarkaria) have opened up an exciting area of mathematics that connects combinatorics, graph theory, algebraic topology and discrete geometry. What seemed like an ingenious trick in 1978 now presents itself as an instance of the "test set paradigm": to construct configuration spaces for combinatorial problems such that coloring, incidence or transversal problems may be translated into the (non-)existence of suitable equivariant maps. The vivid account of this area and its ramifications by Matousek is an exciting, a coherent account of this area of topological combinatorics. It features a collection of mathematical gems written with a broad view of the subject and still with loving care for details. Recommended reading! [โ{128}ฆ]" Gรผnter M.Ziegler (Berlin) Zbl. MATH Volume 1060 Productions-no.: 05001


Preliminaries -- 1 Simplicial Complexes: 1.1 Topological spaces; 1.2 Homotopy equivalence and homotopy; 1.3 Geometric simplicial complexes; 1.4 Triangulations; 1.5 Abstract simplicial complexes; 1.6 Dimension of geometric realizations; 1.7 Simplicial complexes and posets -- 2 The Borsuk-Ulam Theorem: 2.1 The Borsuk-Ulam theorem in various guises; 2.2 A geometric proof; 2.3 A discrete version: Tucker's lemma; 2.4 Another proof of Tucker's lemma -- 3 Direct Applications of Borsuk--Ulam: 3.1 The ham sandwich theorem; 3.2 On multicolored partitions and necklaces; 3.3 Kneser's conjecture; 3.4 More general Kneser graphs: Dolnikov's theorem; 3.5 Gale's lemma and Schrijver's theorem -- 4 A Topological Interlude: 4.1 Quotient spaces; 4.2 Joins (and products); 4.3 k-connectedness; 4.4 Recipes for showing k-connectedness; 4.5 Cell complexes -- 5 Z_2-Maps and Nonembeddability: 5.1 Nonembeddability theorems: An introduction; 5.2 Z_2-spaces and Z_2-maps; 5.3 The Z_2-index; 5.4 Deleted products good ...; 5.5 ... deleted joins better; 5.6 Bier spheres and the Van Kampen-Flores theorem; 5.7 Sarkaria's inequality; 5.8 Nonembeddability and Kneser colorings; 5.9 A general lower bound for the chromatic number -- 6 Multiple Points of Coincidence: 6.1 G-spaces; 6.2 E_nG spaces and the G-index; 6.3 Deleted joins and deleted products; 6.4 Necklace for many thieves; 6.5 The topological Tverberg theorem; 6.6 Many Tverberg partitions; 6.7 Z_p-index, Kneser colorings, and p-fold points; 6.8 The colored Tverberg theorem -- A Quick Summary -- Hints to Selected Exercises -- Bibliography -- Index

Mathematics Computers Computer science -- Mathematics Algebraic topology Combinatorics Mathematics Combinatorics Algebraic Topology Discrete Mathematics in Computer Science Theory of Computation Mathematics general


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