Using the Borsuk-Ulam Theorem Lectures on Topological Methods in Combinatorics and Geometry

Using the Borsuk-Ulam Theorem Lectures on Topological Methods in Combinatorics and Geometry / by Jiri Matousek.

"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. Lovász featured an unexpected use of the Borsuk-Ulam theore...

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Bibliographic Details
Main Author: Matousek, Jiri (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Berlin, Heidelberg : Springer Berlin Heidelberg : Imprint: Springer, 2003.
Edition:1st ed. 2003.
Series:Universitext,
Springer eBook Collection.
Subjects:
Combinatorics.
Algebraic topology.
Computer science—Mathematics.
Computers.
Mathematics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • 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.

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