Beyond the Borsuk–Ulam Theorem: The Topological Tverberg Story

Barany’s “topological Tverberg conjecture” from 1976 states that any continuous map of an N-simplex \(\Delta _{N}\) to \(\mathbb{R}^{d}\), for N ≥ (d + 1)(r − 1), maps points from r disjoint faces in \(\Delta _{N}\) to the same point in \(\mathbb{R}^{d}\). The proof of this result for the case when r is a prime, as well as some colored version of the same result, using the results of Borsuk–Ulam and Dold on the non-existence of equivariant maps between spaces with a free group action, were main topics of Matousek’s 2003 book “Using the Borsuk–Ulam theorem.”

[1]  I. Bárány,et al.  A Colored Version of Tverberg's Theorem , 1992 .

[2]  Florian Frick,et al.  Counterexamples to the topological Tverberg conjecture , 2015 .

[3]  R. Živaljević,et al.  User's guide to equivariant methods in combinatorics. II. , 1998 .

[4]  H. Tverberg.  A Generalization of Radon's Theorem , 1966 .

[5]  A. Dold,et al.  Simple proofs of some Borsuk - Ulam results , 1983 .

[6]  Petra Ostermann.  Using The Borsuk Ulam Theorem Lectures On Topological Methods In Combinatorics And Geometry , 2016 .

[7]  H. Whitney.  The Self-Intersections of a Smooth n-Manifold in 2n-Space , 1944 .

[8]  Pavle V. M. Blagojevi'c,et al.  A tight colored Tverberg theorem for maps to manifolds (extended abstract) , 2011, 1107.1904.

[9]  Florian Frick,et al.  Tverberg plus constraints , 2014, 1401.0690.

[10]  I. Bárány,et al.  On a common generalization of Borsuk's and Radon's theorem , 1979 .

[11]  Pablo Soberón.  Equal coefficients and tolerance in coloured Tverberg partitions , 2015, Comb..

[12]  A. Volovikov,et al.  On the van Kampen-Flores theorem , 1996 .

[13]  Pavle V. M. Blagojevi'c,et al.  Optimal bounds for the colored Tverberg problem , 2009, 0910.4987.

[14]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, III. Codimension 2 , 2015, Israel Journal of Mathematics.

[15]  K. S. Sarkaria.  A generalized van Kampen-Flores theorem , 1991 .

[16]  Uli Wagner,et al.  Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems , 2015, ArXiv.

[17]  B. J. Birch,et al.  On 3N points in a plane , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  G. Ziegler,et al.  Equivariant topology of configuration spaces , 2012, 1207.2852.

[19]  G. C. Shephard,et al.  Convex Polytopes , 1969, The Mathematical Gazette.

[20]  E. Fadell,et al.  An ideal-valued cohomological index theory with applications to Borsuk—Ulam and Bourgin—Yang theorems , 1988, Ergodic Theory and Dynamical Systems.

[21]  Rolf Schneider,et al.  Problems in Geometric Convexity , 1979 .

[22]  J. Matousek,et al.  Using The Borsuk-Ulam Theorem , 2007 .

[23]  Murad Ozaydin,et al.  Equivariant Maps for the Symmetric Group , 1987 .

[24]  Kenneth S. Brown,et al.  Cohomology of Groups , 1982 .

[25]  J. Radon.  Mengen konvexer Körper, die einen gemeinsamen Punkt enthalten , 1921 .

[26]  Imre Bárány,et al.  Tverberg’s Theorem at 50: Extensions and Counterexamples , 2016 .

[27]  Imre Bárány,et al.  On a Topological Generalization of a Theorem of Tverberg , 1981 .

[28]  M. Gromov.  Singularities, Expanders and Topology of Maps. Part 2: from Combinatorics to Topology Via Algebraic Isoperimetry , 2010 .

[29]  Stephen Smale,et al.  A Vietoris mapping theorem for homotopy , 1957 .

[30]  Sinisa T. Vrecica,et al.  The Colored Tverberg's Problem and Complexes of Injective Functions , 1992, J. Comb. Theory A.

[31]  Glen E. Bredon,et al.  Equivariant cohomology theories , 1967 .

[32]  Benjamin Matschke,et al.  Optimal bounds for a colorful Tverberg--Vrecica type problem , 2009, 0911.2692.

[33]  Günter M. Ziegler.  3N Colored Points in a Plane , 2011 .

[34]  G. E. Bredon.  Topology and geometry , 1993 .

[35]  A. Björner.  Topological methods , 1996 .

[36]  Uli Wagner,et al.  Eliminating Tverberg Points, I. An Analogue of the Whitney Trick , 2014, SoCG.

[37]  R. James Milgram,et al.  Cohomology of finite groups , 1994 .

[38]  R. James Milgram,et al.  On the Chern classes of the regular representations of some finite groups , 1982 .

[39]  E. R. Kampen.  Komplexe in euklidischen Räumen , 1933 .

[40]  J. Jonsson.  Simplicial complexes of graphs , 2007 .

[41]  John McCleary,et al.  User's Guide to Spectral Sequences , 1985 .

[42]  Florian Frick,et al.  Barycenters of polytope skeleta and counterexamples to the Topological Tverberg Conjecture, via constraints , 2015, Journal of the European Mathematical Society.

[43]  W. Hsiang,et al.  Cohomology Theory of Topological Transformation Groups , 1975 .

[44]  Arkadiy Skopenkov,et al.  A user's guide to disproof of topological Tverberg conjecture , 2016, ArXiv.

[45]  A. Volovikov,et al.  On a topological generalization of the Tverberg theorem , 1996 .

[46]  László Lovász,et al.  Chessboard Complexes and Matching Complexes , 1994 .

[47]  Zoltán Füredi,et al.  On the number of halving planes , 1989, SCG '89.

[48]  Michelle L. Wachs,et al.  Torsion in the matching complex and chessboard complex , 2004 .