Geodesic
The coloured Tverberg theorem, extensions and new results
Izvestiya. Mathematics , Tome 86 (2022) no. 2, pp. 275-290.

Voir la notice de l'article provenant de la source Math-Net.Ru

We prove a multiple coloured Tverberg theorem and a balanced coloured Tverberg theorem, applying different methods, tools and ideas. The proof of the first theorem uses a multiple chessboard complex (as configuration space) and the Eilenberg–Krasnoselskii theory of degrees of equivariant maps for non-free group actions. The proof of the second result relies on the high connectivity of the configuration space, established by using discrete Morse theory.
Keywords: Tverberg theorem, chessboard complex
Mots-clés : equivariant map. 
@article{IM2_2022_86_2_a3,
     author = {D. Jojic and G. Yu. Panina and R. \v{Z}ivaljevi\'c},
     title = {The coloured {Tverberg} theorem, extensions and new results},
     journal = {Izvestiya. Mathematics },
     pages = {275--290},
     publisher = {mathdoc},
     volume = {86},
     number = {2},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_2_a3/}
}
TY  - JOUR
AU  - D. Jojic
AU  - G. Yu. Panina
AU  - R. Živaljević
TI  - The coloured Tverberg theorem, extensions and new results
JO  - Izvestiya. Mathematics 
PY  - 2022
SP  - 275
EP  - 290
VL  - 86
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IM2_2022_86_2_a3/
LA  - en
ID  - IM2_2022_86_2_a3
ER  - 
%0 Journal Article
%A D. Jojic
%A G. Yu. Panina
%A R. Živaljević
%T The coloured Tverberg theorem, extensions and new results
%J Izvestiya. Mathematics 
%D 2022
%P 275-290
%V 86
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IM2_2022_86_2_a3/
%G en
%F IM2_2022_86_2_a3
D. Jojic; G. Yu. Panina; R. Živaljević. The coloured Tverberg theorem, extensions and new results. Izvestiya. Mathematics , Tome 86 (2022) no. 2, pp. 275-290. http://geodesic.mathdoc.fr/item/IM2_2022_86_2_a3/

[1] J. Matoušek, Using the Borsuk–Ulam theorem, Lectures on topological methods in combinatorics and geometry, Universitext, Springer-Verlag, Berlin, 2003, xii+196 pp. ; ed. 2nd corr. printing, 2008, xii+214 pp. | DOI | MR | Zbl | Zbl

[2] R. T. Živaljević, “Topological methods in discrete geometry”, Ch. 21, Handbook of discrete and computational geometry, Discrete Math. Appl. (Boca Raton), 3rd ed., CRC Press, Boca Raton, FL, 2017, 551–580 | DOI | MR | Zbl

[3] I. Bárány, S. B. Shlosman, A. Szűcs, “On a topological generalization of a theorem of Tverberg”, J. London Math. Soc. (2), 23:1 (1981), 158–164 | DOI | MR | Zbl

[4] M. Özaydin, Equivariant maps for the symmetric group, Unpublished preprint, Univ. of Wisconsin-Madison, 1987, 17 pp. http://minds.wisconsin.edu/handle/1793/63829

[5] A. Yu. Volovikov, “On a topological generalization of the Tverberg theorem”, Math. Notes, 59:3 (1996), 324–326 | DOI | DOI | MR | Zbl

[6] T. Schöneborn, On the topological Tverberg theorem, arXiv: math/0405393

[7] T. Schöneborn, G. M. Ziegler, “The topological Tverberg theorem and winding numbers”, J. Combin. Theory Ser. A, 112:1 (2005), 82–104 ; arXiv: math/0409081 | DOI | MR | Zbl

[8] A. B. Skopenkov, “A user's guide to the topological Tverberg conjecture”, Russian Math. Surveys, 73:2 (2018), 323–353 | DOI | DOI | MR | Zbl

[9] A. Vučić, R. T. Živaljević, “Note on a conjecture of Sierksma”, Discrete Comput. Geom., 9:4 (1993), 339–349 | DOI | MR | Zbl

[10] S. Hell, “On the number of Tverberg partitions in the prime power case”, European J. Combin., 28:1 (2007), 347–355 | DOI | MR | Zbl

[11] R. T. Živaljević, S. T. Vrećica, “The colored Tverberg's problem and complexes of injective functions”, J. Combin. Theory Ser. A, 61:2 (1992), 309–318 | DOI | MR | Zbl

[12] S. T. Vrećica, R. T. Živaljević, “New cases of the colored Tverberg theorem”, Jerusalem combinatorics '93, Contemp. Math., 178, Amer. Math. Soc., Providence, RI, 1994, 325–334 | DOI | MR | Zbl

[13] D. Jojić, S. T. Vrećica, R. T. Živaljević, “Symmetric multiple chessboard complexes and a new theorem of Tverberg type”, J. Algebraic Combin., 46:1 (2017), 15–31 | DOI | MR | Zbl

[14] F. Frick, On affine Tverberg-type results without continuous generalization, arXiv: 1702.05466

[15] A. Kushkuley, Z. Balanov, Geometric methods in degree theory for equivariant maps, Lecture Notes in Math., 1632, Springer-Verlag, Berlin, 1996, vi+136 pp. | DOI | MR | Zbl

[16] D. Jojić, I. Nekrasov, G. Panina, R. Živaljević, “Alexander $r$-tuples and Bier complexes”, Publ. Inst. Math. (Beograd) (N.S.), 104:118 (2018), 1–22 | DOI | MR | Zbl

[17] D. Jojić, S. T. Vrećica, G. Panina, R. Živaljević, “Generalized chessboard complexes and discrete Morse theory”, Chebyshevskii sb., 21:2 (2020), 207–227 | DOI | MR | Zbl

[18] D. Jojić, G. Panina, R. Živaljević, “A Tverberg type theorem for collectively unavoidable complexes”, Israel J. Math., 241:1 (2021), 17–36 ; arXiv: 1812.00366 | DOI | MR | Zbl

[19] R. Forman, “A user's guide to discrete Morse theory”, Sém. Lothar. Combin., 48 (2002), B48c, 35 pp. | MR | Zbl

[20] I. Mabillard, U. Wagner, “Eliminating Tverberg points, I. An analogue of the Whitney trick”, Computational geometry (SoCG{'14}), ACM, New York, 2014, 171–180 | DOI | MR | Zbl

[21] I. Mabillard, U. Wagner, Eliminating higher-multiplicity intersections, I. A Whitney trick for Tverberg-type problems, arXiv: 1508.02349

[22] I. Mabillard, U. Wagner, “Eliminating higher-multiplicity intersections, II. The deleted product criterion in the $r$-metastable range”, 32nd International symposium on computational geometry (SoCG{'16}), LIPIcs. Leibniz Int. Proc. Inform., 51, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016, 51, 12 pp. ; arXiv: 1601.00876 | DOI | MR | Zbl

[23] A. Skopenkov, On the metastable Mabillard–Wagner conjecture, arXiv: 1702.04259

[24] A. B. Skopenkov, Eliminating higher-multiplicity intersections in the metastable dimension range, arXiv: 1704.00143

[25] A. B. Skopenkov, Invariants of graph drawings in the plane, arXiv: 1805.10237

[26] P. V. M. Blagojević, B. Matschke, G. M. Ziegler, “Optimal bounds for the colored Tverberg problem”, J. Eur. Math. Soc. (JEMS), 17:4 (2015), 739–754 ; arXiv: 0910.4987 | DOI | MR | Zbl

[27] I. Bárány, P. V. M. Blagojević, G. M. Ziegler, “Tverberg's theorem at 50: extensions and counterexamples”, Notices Amer. Math. Soc., 63:7 (2016), 732–739 | DOI | MR | Zbl

[28] I. Bárány, P. Soberon, “Tverberg's theorem is 50 years old: a survey”, Bull. Amer. Math. Soc. (N.S.), 55:4 (2018), 459–492 | DOI | MR | Zbl

[29] I. Bárány, D. G. Larman, “A colored version of Tverberg's theorem”, J. London Math. Soc. (2), 45:2 (1992), 314–320 | DOI | MR | Zbl

[30] K. S. Sarkaria, “A generalized van Kampen–Flores theorem”, Proc. Amer. Math. Soc., 111:2 (1991), 559–565 | DOI | MR | Zbl

[31] A. Y. Volovikov, “On the van Kampen–Flores theorem”, Math. Notes, 59:5 (1996), 477–481 | DOI | DOI | MR | Zbl

[32] P. V. M. Blagojević, F. Frick, G. M. Ziegler, “Tverberg plus constraints”, Bull. Lond. Math. Soc., 46:5 (2014), 953–967 | DOI | MR | Zbl

[33] R. Živaljević, “User's guide to equivariant methods in combinatorics”, Publ. Inst. Math. (Beograd) (N.S.), 59(73) (1996), 114–130 | MR | Zbl

[34] R. T. Živaljević, “User's guide to equivariant methods in combinatorics. II”, Publ. Inst. Math. (Beograd) (N.S.), 64(78) (1998), 107–132 | MR | Zbl

[35] D. Jojić, S. T. Vrećica, R. T. Živaljević, “Multiple chessboard complexes and the colored Tverberg problem”, J. Combin. Theory Ser. A, 145 (2017), 400–425 | DOI | MR | Zbl