Eliminating Higher-Multiplicity Intersections, I. A Whitney Trick for Tverberg-Type Problems

Motivated by topological Tverberg-type problems and by classical results about embeddings (maps without double points), we study the question whether a finite simplicial complex K can be mapped into R^d without triple, quadruple, or, more generally, r-fold points. Specifically, we are interested in maps f from K to R^d that have no r-Tverberg points, i.e., no r-fold points with preimages in r pairwise disjoint simplices of K, and we seek necessary and sufficient conditions for the existence of such maps. We present a higher-multiplicity analogue of the completeness of the Van Kampen obstruction for embeddability in twice the dimension. Specifically, we show that under suitable restrictions on the dimensions, a well-known Deleted Product Criterion (DPC) is not only necessary but also sufficient for the existence of maps without r-Tverberg points. Our main technical tool is a higher-multiplicity version of the classical Whitney trick. An important guiding idea for our work was that sufficiency of the DPC, together with an old result of Ozaydin on the existence of equivariant maps, might yield an approach to disproving the remaining open cases of the long-standing topological Tverberg conjecture. Unfortunately, our proof of the sufficiency of the DPC requires a "codimension 3" proviso, which is not satisfied for when K is the N-simplex. Recently, Frick found an extremely elegant way to overcome this last "codimension 3" obstacle and to construct counterexamples to the topological Tverberg conjecture for d at least 3r+1 (r not a prime power). Here, we present a different construction that yields counterexamples for d at least 3r (r not a prime power).

[1]  Arnold Shapiro,et al.  Obstructions to the Imbedding of a Complex in a Euclidean Space.: I. The First Obstruction , 1957 .

[2]  E. R. Kampen.  Berichtigung: zu: „Komplexe in euklidischen Räumen“ , 1933 .

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

[4]  Jirí Matousek,et al.  Hardness of embedding simplicial complexes in Rd , 2009, SODA.

[5]  J. L. Bryant.  Piecewise Linear Topology , 2001 .

[6]  J. W. Alexander.  INTERSECTIONS AND TRANSFORMATIONS OF COMPLEXES AND MANIFOLDS , 2010 .

[7]  E. C. Zeeman,et al.  Transversality for piecewise linear manifolds , 1967 .

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

[9]  Charles Terence Clegg Wall,et al.  Piecewise linear bundles in the stable range , 1965 .

[10]  K. S. Sarkaria.  A one-dimensional Whitney trick and Kuratowski’s graph planarity criterion , 1991 .

[11]  A. Skopenkov,et al.  Embeddings of homology equivalent manifolds with boundary , 2006, 1207.1326.

[12]  Mikhail Skopenkov,et al.  ON APPROXIMABILITY BY EMBEDDINGS OF CYCLES IN THE PLANE , 2003, 0808.1187.

[13]  A. Skopenkov.  Surveys in Contemporary Mathematics: Embedding and knotting of manifolds in Euclidean spaces , 2006, math/0604045.

[14]  C. Weber.  Plongements de polyèdres dans le domaine métastable , 1967 .

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

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

[17]  K. S. Sarkaria.  Tverberg partitions and Borsuk–Ulam theorems , 2000 .

[18]  Wen-tsün Wu,et al.  A theory of imbedding immersion , and isotopy of polytopes in a euclidean space , 1965 .

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

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

[21]  R. Ho.  Algebraic Topology , 2022 .

[22]  Jack Segal,et al.  Embeddings of polyhedra in Rm and the deleted product obstruction , 1998 .

[23]  Dušan Repovš,et al.  New results on embeddings of polyhedra and manifolds in Euclidean spaces , 1999 .

[24]  C. Rourke,et al.  Introduction to Piecewise-Linear Topology , 1972 .

[25]  Michael H. Freedman,et al.  Van Kampen’s embedding Obstruction is incomplete for $2$-Complexes in $\rz^{4}$ , 1994 .

[26]  Michael J. Pelsmajer,et al.  Removing even crossings , 2007, J. Comb. Theory, Ser. B.

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

[28]  C. Weber,et al.  L’élimination des points doubles dans le cas combinatoire , 1966 .

[29]  Arne Storjohann,et al.  Near optimal algorithms for computing Smith normal forms of integer matrices , 1996, ISSAC '96.

[30]  J. Hudson,et al.  Concordance, Isotopy, and Diffeotopy , 1970 .

[31]  Benjamin Matschke,et al.  Equivariant topology methods in discrete geometry , 2011 .

[32]  Jack Segal,et al.  Quasi embeddings and embeddings of polyhedra in Rm , 1992 .

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

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

[35]  J. Hudson,et al.  Extending Piecewise‐Linear Isotopies , 1966 .

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

[37]  Ch. Chojnacki,et al.  Über wesentlich unplättbare Kurven im dreidimensionalen Raume , 1934 .

[38]  E. C. Zeeman,et al.  Seminar on combinatorial topology , 1963 .

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

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

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

[42]  Jean-Guillaume Dumas,et al.  On Efficient Sparse Integer Matrix Smith Normal Form Computations , 2001, J. Symb. Comput..

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

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

[45]  Mark de Longueville.  Notes on the topological Tverberg theorem , 2001, Discret. Math..

[46]  Pavle V. M. Blagojevic,et al.  Using equivariant obstruction theory in combinatorial geometry , 2006 .

[47]  Marek Krcál,et al.  Algorithmic Solvability of the Lifting-Extension Problem , 2013, Discrete & Computational Geometry.

[48]  S. Melikhov,et al.  The van Kampen obstruction and its relatives , 2006, math/0612082.

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

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

[51]  Michael J. Pelsmajer,et al.  Removing Independently Even Crossings , 2009, SIAM J. Discret. Math..

[52]  Mark Giesbrecht,et al.  Fast computation of the Smith form of a sparse integer matrix , 2002, computational complexity.

[53]  W. T. Tutte.  Toward a theory of crossing numbers , 1970 .