On orientability of singular foliations of surfaces in closed braid complements

L.P.Plachta

Every essential closed oriented and smooth of class $C^r$ surface in standard position in a closed braid complement admits a singular foliation in a natural way. We show that each such foliation of the surface is orientable, i.e. there is a flow ${\cal F}$ of the class $C^{r-1}$ on this surface such that the trajectory of ${\cal F}$ yields the given singular foliation.

1. Birman J.S., Finkelstein E. Studying surfaces via closed braids, J. of Knot Theory Ramifications, 7 (1998), 267-334.

2. Birman J.S., Menasco W.W. Special positions for essential tori in link complement, Topology, 33 (1994), 525-556.

3. Bronstein I., Nikolaev I. Structurally stable fields of line elements on surfaces, Nonlinear Anal., 34 (1998), 461--477.

4. Bronstein I., Nikolaev I. Peixoto graphs of Morse-Smale foliations on surfaces, Topology Appl., 77 (1997), 19--36.

5. Тамура И., Топология слоений.-- М.: Мир, 1979, 317~с.

	On orientability of singular foliations of surfaces in closed braid complements
Author	L.P.Plachta Institute of Applied Problems of Mechanics and Mathematics of NAS,of Ukraine, Lviv
Abstract	Every essential closed oriented and smooth of class $C^r$ surface in standard position in a closed braid complement admits a singular foliation in a natural way. We show that each such foliation of the surface is orientable, i.e. there is a flow ${\cal F}$ of the class $C^{r-1}$ on this surface such that the trajectory of ${\cal F}$ yields the given singular foliation.
Keywords	singular foliation, closed braid complement, surface
DOI	doi:10.30970/ms.24.2.192-196
Reference	1. Birman J.S., Finkelstein E. Studying surfaces via closed braids, J. of Knot Theory Ramifications, 7 (1998), 267-334. 2. Birman J.S., Menasco W.W. Special positions for essential tori in link complement, Topology, 33 (1994), 525-556. 3. Bronstein I., Nikolaev I. Structurally stable fields of line elements on surfaces, Nonlinear Anal., 34 (1998), 461--477. 4. Bronstein I., Nikolaev I. Peixoto graphs of Morse-Smale foliations on surfaces, Topology Appl., 77 (1997), 19--36. 5. Тамура И., Топология слоений.-- М.: Мир, 1979, 317~с.
Pages	192-196
Volume	24
Issue	2
Year	2005
Journal	Matematychni Studii
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