Arrangements of codimension-one submanifolds

I. N. Shnurnikov

Sbornik. Mathematics, Tome 203 (2012) no. 9, pp. 1357-1382

Résumé

We study the number $f$ of connected components in the complement to a finite set (arrangement) of closed submanifolds of codimension 1 in a closed manifold $M$. In the case of arrangements of closed geodesics on an isohedral tetrahedron, we find all possible values for the number $f$ of connected components. We prove that the set of numbers that cannot be realized by the number $f$ of an arrangement of $n\geqslant 71$ projective planes in the three-dimensional real projective space is contained in the similar known set of numbers that are not realizable by arrangements of $n$ lines on the projective plane. For Riemannian surfaces $M$ we express the number $f$ via a regular neighbourhood of a union of immersed circles and the multiplicities of their intersection points. For $m$-dimensional Lobachevskiǐ space we find the set of all possible numbers $f$ for hyperplane arrangements. Bibliography: 18 titles.

Détail

Keywords: hyperplane arrangements, closed geodesics
Mots-clés : partition of a surface.

Bibliographie

[1] J. Steiner, “Einige Gesetze über die Theilung der Ebene und des Raumes”, J. Reine Angew. Math., 1826, no. 1, 349–364 | DOI | Zbl

[2] L. Schläfli, Theorie der vielfachen Kontinuität, Historical Math. Monographs, Zürcher und Furrer, Zürich, 1901 | Zbl

[3] P. Orlic, L. Solomon, “Combinatorics and topology of complements of hyperplanes”, Invent. Math., 56:2 (1980), 167–189 | DOI | MR | Zbl

[4] T. Zaslavsky, Facing up to arrangements: face-count formulas for partitions of space by hyperplanes, Mem. Amer. Math. Soc., 1, no. 154, Amer. Math. Soc., Providence, RI, 1975 | MR | Zbl

[5] R. Ehrenborg, M. Readdy, M. Slone, “Affine and toric hyperplane arrangements”, Discrete Comput. Geom., 41:4 (2009), 481–512 | DOI | MR | Zbl

[6] P. Deshpande, Arrangements of submanifolds and the tangent bundle complement, Electronic thesis and dissertation repository, paper 154, 2011 http:// ir.lib.uwo.ca/etd/154/

[7] B. Grünbaum, Arrangements and spreads, Amer. Math. Soc., Providence, RI, 1972 | MR | Zbl

[8] R. Cordovil, “Sur l'évaluation $t(M;2,0)$ du polynôme de Tutte d'un matro\"{i}de et une conjecture de B. Grünbaum rélative aux arrangements du droites du plan”, European J. Combin., 1:4 (1980), 317–322 | MR | Zbl

[9] G. Purdy, “On the number of regions determined by $n$ lines in the projective plane”, Geom. Dedicata, 9:1 (1980), 107–109 | DOI | MR | Zbl

[10] N. Martinov, “Classification of arrangements by the number of their cells”, Discrete Comput. Geom., 9:1 (1993), 39–46 | DOI | MR | Zbl

[11] V. I. Arnold, “Na skolko chastei delyat ploskost $n$ pryamykh?”, Matem. prosv., ser. 3, 12, Izd-vo MTsNMO, M., 2008, 95–104

[12] I. N. Shnurnikov, “Into how many regions do $n$ lines divide the plane if at most $n-k$ of them are concurrent?”, Moscow Univ. Math. Bull., 65:5 (2010), 208–212 | DOI | MR

[13] I. N. Shnurnikov, “On the number of regions formed by arrangements of closed geodesics on flat surfaces”, Math. Notes, 90:3–4 (2011), 619–622 | DOI

[14] V. Yu. Protasov, “Closed geodesics on the surface of a simplex”, Sb. Math., 198:2 (2007), 243–260 | DOI | MR | Zbl

[15] M. Mirzakhani, “Growth of the number of simple closed geodesics on hyperbolic surfaces”, Ann. of Math. (2), 168:2 (2008), 97–125 | DOI | MR | Zbl

[16] H. Huber, “Zur analytischen Theorie hyperbolischen Raumformen und Bewegungsgruppen”, Math. Ann., 138:1 (1959), 1–26 | DOI | MR | Zbl

[17] V. V. Prasolov, I. F. Sharygin, Zadachi po stereometrii, Nauka, M., 1989 | MR | Zbl

[18] E. Melchior, “Über Vielseite der projektiven Ebene”, Deutsche Math., 5 (1941), 461–475 | MR | Zbl