Computational topology of equivariant maps from spheres to complements of arrangements

Blagojević, Pavle; Vrećica, Siniša; Živaljević, Rade

doi:10.1090/S0002-9947-08-04679-5

Computational topology of equivariant maps from spheres to complements of arrangements
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by Pavle V. M. Blagojević, Siniša T. Vrećica and Rade T. Živaljević PDF

Trans. Amer. Math. Soc. 361 (2009), 1007-1038 Request permission

Abstract:

The problem of the existence of an equivariant map is a classical topological problem ubiquitous in topology and its applications. Many problems in discrete geometry and combinatorics have been reduced to such a question and many of them resolved by the use of equivariant obstruction theory. A variety of concrete techniques for evaluating equivariant obstruction classes are introduced, discussed and illustrated by explicit calculations. The emphasis is on $D_{2n}$-equivariant maps from spheres to complements of arrangements, motivated by the problem of finding a $4$-fan partition of $2$-spherical measures, where $D_{2n}$ is the dihedral group. One of the technical highlights is the determination of the $D_{2n}$-module structure of the homology of the complement of the appropriate subspace arrangement, based on the geometric interpretation for the generators of the homology groups of arrangements.

References

J. Akiyama, A. Kaneko, M. Kano, G. Nakamura, E. Rivera-Campo, S. Tokunaga, and J. Urrutia, Radial perfect partitions of convex sets in the plane, Discrete and computational geometry (Tokyo, 1998) Lecture Notes in Comput. Sci., vol. 1763, Springer, Berlin, 2000, pp. 1–13. MR 1787512, DOI 10.1007/978-3-540-46515-7_{1}
I. Bárány, Geometric and combinatorial applications of Borsuk’s theorem, János Pach, ed., Algorithms and Combinatorics 10, Springer-Verlag, Berlin, 1993.
I. Bárány and J. Matoušek, Simultaneous partitions of measures by $k$-fans, Discrete Comput. Geom. 25 (2001), no. 3, 317–334. MR 1815435, DOI 10.1007/s00454-001-0003-5
I. Bárány and J. Matoušek, Equipartition of two measures by a 4-fan, Discrete Comput. Geom. 27 (2002), no. 3, 293–301. MR 1921557, DOI 10.1007/s00454-001-0071-6
S. Bespamyatnikh, D. Kirkpatrick, and J. Snoeyink, Generalizing ham sandwich cuts to equitable subdivisions, Discrete Comput. Geom. 24 (2000), no. 4, 605–622. ACM Symposium on Computational Geometry (Miami, FL, 1999). MR 1799604, DOI 10.1145/304893.304909
A. Björner, Topological methods, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1819–1872. MR 1373690
P. Blagojević, The partition of measures by $3$-fans and computational obstruction theory, arXiv: math.CO/ 0402400, 2004
Pavle V. M. Blagojević and Aleksandra S. Dimitrijević Blagojević, Using equivariant obstruction theory in combinatorial geometry, Topology Appl. 154 (2007), no. 14, 2635–2655. MR 2340948, DOI 10.1016/j.topol.2007.04.007
Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original. MR 1700700, DOI 10.1007/978-1-4612-0647-7
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956, DOI 10.1007/978-1-4684-9327-6
Henri Cartan and Samuel Eilenberg, Homological algebra, Princeton University Press, Princeton, N. J., 1956. MR 0077480
P. E. Conner and E. E. Floyd, Differentiable periodic maps, Ergebnisse der Mathematik und ihrer Grenzgebiete, (N.F.), Band 33, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1964. MR 0176478
Tammo tom Dieck, Transformation groups, De Gruyter Studies in Mathematics, vol. 8, Walter de Gruyter & Co., Berlin, 1987. MR 889050, DOI 10.1515/9783110858372.312
Albrecht Dold, Lectures on algebraic topology, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 200, Springer-Verlag, Berlin-New York, 1980. MR 606196
Mark Goresky and Robert MacPherson, Stratified Morse theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 14, Springer-Verlag, Berlin, 1988. MR 932724, DOI 10.1007/978-3-642-71714-7
Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505, DOI 10.1007/978-3-662-02267-2
Hiro Ito, Hideyuki Uehara, and Mitsuo Yokoyama, 2-dimension ham sandwich theorem for partitioning into three convex pieces, Discrete and computational geometry (Tokyo, 1998) Lecture Notes in Comput. Sci., vol. 1763, Springer, Berlin, 2000, pp. 129–157. MR 1787521, DOI 10.1007/978-3-540-46515-7_{1}1
Atsushi Kaneko and M. Kano, Balanced partitions of two sets of points in the plane, Comput. Geom. 13 (1999), no. 4, 253–261. MR 1719059, DOI 10.1016/S0925-7721(99)00024-3
Jiří Matoušek, Using the Borsuk-Ulam theorem, Universitext, Springer-Verlag, Berlin, 2003. Lectures on topological methods in combinatorics and geometry; Written in cooperation with Anders Björner and Günter M. Ziegler. MR 1988723
James R. Munkres, Elements of algebraic topology, Addison-Wesley Publishing Company, Menlo Park, CA, 1984. MR 755006
Peter Orlik and Hiroaki Terao, Arrangements of hyperplanes, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 300, Springer-Verlag, Berlin, 1992. MR 1217488, DOI 10.1007/978-3-662-02772-1
János Pach (ed.), New trends in discrete and computational geometry, Algorithms and Combinatorics, vol. 10, Springer-Verlag, Berlin, 1993. MR 1228036, DOI 10.1007/978-3-642-58043-7
E. A. Ramos, Equipartition of mass distributions by hyperplanes, Discrete Comput. Geom. 15 (1996), no. 2, 147–167. MR 1368272, DOI 10.1007/BF02717729
Referee report
T. Sakai, Radial partitions of point sets in $R^{2}$, Manuscript, Tokoha Gakuen University, 1998.
C. Schultz, Private communications.
Sheila Sundaram and Volkmar Welker, Group actions on arrangements of linear subspaces and applications to configuration spaces, Trans. Amer. Math. Soc. 349 (1997), no. 4, 1389–1420. MR 1340186, DOI 10.1090/S0002-9947-97-01565-1
Helge Tverberg and Siniša Vrećica, On generalizations of Radon’s theorem and the ham sandwich theorem, European J. Combin. 14 (1993), no. 3, 259–264. MR 1215336, DOI 10.1006/eujc.1993.1029
Siniša T. Vrećica and Rade T. Živaljević, The ham sandwich theorem revisited, Israel J. Math. 78 (1992), no. 1, 21–32. MR 1194956, DOI 10.1007/BF02801568
S. T. Vrećica and R. T. Živaljević, Conical equipartitions of mass distributions, Discrete Comput. Geom. 25 (2001), no. 3, 335–350. MR 1815436, DOI 10.1007/s00454-001-0002-6
S. Vrećica and R. Živaljević, Arrangements, equivariant maps and partitions of measures by $4$-fans, in Discrete and Computational Geometry: The Goodman-Pollack Festschrift (J. Goodman et al., eds.), pp. 829-849, Algorithms and Combinatorics 25, Springer Verlag 2003.
C. T. C. Wall, Surgery on compact manifolds, London Mathematical Society Monographs, No. 1, Academic Press, London-New York, 1970. MR 0431216
Günter M. Ziegler and Rade T. Živaljević, Homotopy types of subspace arrangements via diagrams of spaces, Math. Ann. 295 (1993), no. 3, 527–548. MR 1204836, DOI 10.1007/BF01444901
Jacob E. Goodman and Joseph O’Rourke (eds.), Handbook of discrete and computational geometry, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004. MR 2082993, DOI 10.1201/9781420035315
Rade T. Živaljević, The Tverberg-Vrećica problem and the combinatorial geometry on vector bundles, Israel J. Math. 111 (1999), 53–76. MR 1710731, DOI 10.1007/BF02810677
Rade T. Živaljević, User’s guide to equivariant methods in combinatorics, Publ. Inst. Math. (Beograd) (N.S.) 59(73) (1996), 114–130. MR 1444570
Rade T. Živaljević, User’s guide to equivariant methods in combinatorics. II, Publ. Inst. Math. (Beograd) (N.S.) 64(78) (1998), 107–132. 50th anniversary of the Mathematical Institute, Serbian Academy of Sciences and Arts (Belgrade, 1996). MR 1668705
Rade T. Živaljević and Siniša T. Vrećica, An extension of the ham sandwich theorem, Bull. London Math. Soc. 22 (1990), no. 2, 183–186. MR 1045292, DOI 10.1112/blms/22.2.183