Homeotopy Groups for Nonsingular Foliations of the Plane

Abstract

Let F be a non-singular foliation on the plane with all leaves being closed subsets, H + (F ) be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and H + 0 (F ) be the identity path component of H + (F ). The quotient π 0 H + (F ) = H + (F )/H + 0 (F ) is an analogue of a mapping class group for foliated homeomorphisms. We will describe the algebraic structure of π 0 H + (F ) under an assumption that the corresponding space of leaves of F has a structure similar to a rooted tree of finite diameter.2010 Mathematics Subject Classification. Primary 57S05; Secondary 57R30, 55Q05 .

Fundamental groupoids and homotopy types of non-compact surfaces

“…Polulyakh [8,7,9,10] and Yu. Soroka [14,15] it was studied a class of foliations on non-compact surfaces Z (called striped) glued from open strips in a certain "canonical" way. In a joint paper [11] of the above three authors it was also described an analogue of mapping class group for foliated homeomorphisms (sending leaves into leaves) of such foliations F and proved that it is isomorphic with an automorphism group of a certain graph (one-dimensional CW-complex) G encoding an information about gluing a surface from strips.…”

Homeotopy groups of one-dimensional foliations on surfaces

“…I" [12] a"d [13] the third aАth"r stАdied a special class "f s"-called r""ted tree like striped sАrfaces, c"mpletelД described algebraic strАctАre "f h"me"t"pД gr"Аps "f their f"liati""s, a"d als" related th"se gr"Аps Вith the h"me"t"pД gr"Аps "f the space "f leaБes Z/∆.…”

“…NamelД, Вe sh"В that π 0 H(∆) is is"m"rphic t" a gr"Аp "f aАt"m"rphism "f a certai" graph Вith additi""al strАctАre, see The"rem 8.1. I" particАlar, these resАlts h"ld f"r all f"liati""s c""sidered i" [9], [12], [13].…”

“…We Вill ""В ass"ciate t" q a certai" graph G Вhich e"c"des a c"mbi"at"rial i"f"rmati"" ab"Аt glАi"g strips Бia q. It Вas rstlД c""sidered i" [12] f"r a special class "f r""ted tree like striped sАrfaces. That graph maД haБe mАltiple edges a"d l""ps a"d als" half-"pe" edges.…”

Одновимірні Шарування На Топологічних Многовидах

“…HomotopД properties of foliations on sАrfaces glАed from strips similarlД to (2) are stАdied in S. MaksДmenko and E. PolАlДakh [25] and [26] and YА. Soroka [44].…”