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.