Homeotopy groups of leaf spaces of one-dimensional foliations on non-compact surfaces with non-compact leaves

Let $Z$ be a non-compact two-dimensional manifold obtained from a family of open strips $\mathbb{R}\times(0,1)$ with boundary intervals by gluing those strips along some pairs of their boundary intervals. Every such strip has a natural foliation into parallel lines $\mathbb{R}\times t$, $t\in(0,1)$, and boundary intervals which gives a foliation $\Delta$ on all of $Z$. Denote by $\mathcal{H}(Z,\Delta)$ the group of all homeomorphisms of $Z$ that maps leaves of $\Delta$ onto leaves and by $\mathcal{H}(Z/\Delta)$ the group of homeomorphisms of the space of leaves endowed with the corresponding compact open topologies. Recently, the authors identified the homeotopy group $\pi_0\mathcal{H}(Z,\Delta)$ with a group of automorphisms of a certain graph $G$ with the additional structure which encodes the combinatorics of gluing $Z$ from strips. That graph is in a certain sense dual to the space of leaves $Z/\Delta$. On the other hand, for every $h\in\mathcal{H}(Z,\Delta)$ the induced permutation $k$ of leaves of $\Delta$ is in fact a homeomorphism of $Z/\Delta$ and the correspondence $h\mapsto k$ is a homomorphism $\psi:\mathcal{H}(\Delta)\to\mathcal{H}(Z/\Delta)$. The aim of the present paper is to show that $\psi$ induces a homomorphism of the corresponding homeotopy groups $\psi_0:\pi_0\mathcal{H}(Z,\Delta)\to\pi_0\mathcal{H}(Z/\Delta)$ which turns out to be either injective or having a kernel $\mathbb{Z}_2$. This gives a dual description of $\pi_0\mathcal{H}(Z,\Delta)$ in terms of the space of leaves.


Introduction
In the present paper we study foliations on non-compact surfaces similar to foliations on the plane (considered by W. Kaplan [12,13], see also [6,7]) with "sufficiently regular behavior". Those foliations are studied in a series of papers by S. Maksymenko, Ye. Polulyakh, and Yu. Soroka, see [16-19, 21, 23, 24]. Our main result (Theorem 1.4) relates the groups of isotopy classes of foliated (i.e. sending leaves to leaves) self-homeomorphisms of such surfaces with the groups of isotopy classes of homeomorphisms of the corresponding space of leaves (being in those cases non-Hausdorff one-dimensional manifolds). This extends results by Yu. Soroka [24].
Let Z be a surface endowed with a one-dimensional foliation ∆. It will be convenient to say that the pair (Z, ∆) is a foliated surface. For an open subset U ⊂ Z denote by ∆| U the induced foliation on U consisting of connected components of non-empty intersections ω ∩ U for all ω ∈ ∆.
Let Y = Z/∆ be the set of all leaves, and p : Z → Y be the natural projection associating to each x ∈ Z the leaf of ∆ containing x. Endow Y with a quotient topology, so a subset By sthe saturation Sat(U ) of a subset U ⊂ Z we mean the union of all leaves of ∆ intersecting U . Evidently Sat(U ) = p −1 (p(U )). A subset U ⊂ Z is called saturated (with respect to a foliation ∆) whenever U = Sat(U ).
Let K be a one-dimensional manifold. Then by a trivial foliation on R × K we will mean a foliation by lines {R × y} y∈K .
Let Notice that by definition each h ∈ H(Z, ∆) induces a permutation of leaves of ∆, and therefore it induces a bijection k : Y → Y making commutative the following diagram: One easily checks that k is a homeomorphism of Y and that the correspondence h → k is a homomorphism of groups Definition 1.1. We will say that a foliated surface (Z, ∆) belongs to the class F, whenever it satisfies the following conditions: (A1) Z has a countable base; (A2) every leaf ω of ∆ is a non-compact closed subset of Z, so it is an image of R under some topological embedding R ⊂ Z; (A3) each boundary component of Z is a leaf of ∆.
Our aim is to show (Theorem 1.4) that for foliations from class F satisfying certain additional "local finiteness" condition we have that This will imply that the image of ψ is a union of path components of H(Y ) and that ψ induces a homomorphism of the corresponding homeotopy groups: Moreover, we also show that in this case for connected Z the kernel of ψ 0 is either trivial or is isomorphic with Z 2 .
Definition 1.2. Let (Z, ∆) be a foliated surface from class F. 1) Say that a leaf ω ∈ ∆ is regular if there exists an open saturated neighborhood U of ω such that the pair (U , U ) is "trivially foliated" in the sense that (U , U ) is foliated homeomorphic if ω ⊂ ∂Z, with trivial foliation into lines R × t, via a homeomorphism sending ω to R × 0.
2) A leaf which is not regular will be called singular 1 .
3) For a leaf ω define its Hausdorff closure by so it is the intersection of closures of all saturated neighborhoods of ω. Evidently, hcl(ω) is saturated and ω ⊂ hcl(ω). We will say that ω is special whenever hcl(ω) \ ω = ∅. Denote by ∆ reg , ∆ spec , and ∆ sing respectively the families of all regular, special and singular leaves of ∆. Then we have the following relations: Notice that by definition each one-dimensional foliation ∆ on a surface Z is "locally regular", i.e. every point x ∈ Z has a neighborhood U such that the leaf of ∆| U containing x is regular in the sense of Definition 1.2. Thus Definition 1.2 puts restriction to a global structure of the foliation near a leaf. Let us also mention that it is possible that a leaf might have a trivially foliated saturated neighborhood U , however for any such neighborhood the pair (U , U ) will never be trivially foliated. For example, let ∆ = {R × y} y∈R be a trivial foliation on R 2 into horizontal lines and Z = R 2 \ 0. Then it is easy to see that ∆| Z has two singular leaves ω 1 = (−∞; 0) × 0 and ω 2 = (0; +∞) × 0. Indeed, any open saturated neighborhood U of either of ω 1 or ω 2 contains R × (−ε, ε) \ (0, 0) for some small ε > 0, and therefore (U , U ) can not be foliated homeomorphic with R × [−1, 1], R × (−1, 1) . This implies hcl(ω 1 ) = hcl(ω 2 ) = ω 1 ∪ ω 2 , whence both ω 1 and ω 2 are special.
The main result of the present paper is the following theorem relating the groups π 0 H(Z, ∆) and π 0 H(Y ). In fact in a series of papers by the authors it was obtained a characterization of foliated surfaces (Z, ∆) from class F satisfying also the assumption (A4) of Theorem 1.4. It is shown in [20,Theorem 4.4] that every such surface is glued from certain "model strips" foliated by parallel lines. Such surfaces were called striped , and the proof of Theorem 1.4 essentially exploits an existence of that gluing. This description is motivated by results of W. Kaplan, see Theorem 3.1.5 below, describing the structure of foliations on R 2 .
Furthermore, in the joint paper with Yu. Soroka [21] the homeotopy group π 0 H(Z, ∆) of canonical foliation ∆ of a striped surface Z is identified with the group of automorphisms of a certain graph G with additional structure encoding the gluing of Z from model strips. In fact, the vertices of G are strips and the edges are "seams" (i.e. leaves along which we glue the strips), so this graph is in a certain sense dual to the space of leaves Y . It is also proved in S. Maksymenko and O. Nikitchenko [15] that G (as a one-dimensional CW-complex) is homotopy equivalent to the surface Z.
Thus Theorem 1.4 also gives a dual description of the homeotopy group π 0 H(Z, ∆) in terms of the space of leaves.
Structure of the paper. In Section 2 we consider several properties of non-Hausdorff T 1 -spaces. In particular, we recall the exponential law for them as well as describe the relations between isotopies and paths in the groups of homeomorphisms for such spaces, see Corollary 2.1.2. We also present a characterization of the identity path component of the group of homeomorphisms of second countable T 1 -manifolds that are not necessarily Hausdorff, see Lemma 2.4.1. In Section 3 we recall the results about striped surfaces, and prove Theorem 1.4 in Section 4.

T 1 -spaces
We will recall here several elementary properties of T 1 -spaces which are not necessarily Hausdorff. They are well-known and rather trivial for T 2 -spaces, while for T 1 -ones they should also be known for specialists, thought we did not find their explicit exposition in the literature. Therefore to make the paper self-contained and for future references we will collect them together and present their short proofs. These resutls will be applied to the spaces of leaves of foliations.
Recall that a topological space Y is compact if every open cover of Y contains a finite subcover. In this paper a (not necessarily Hausdorff) topological space Y will be called locally compact 2 whenever for each y ∈ Y and an open neighborhood U of y there exists a compact subset B ⊂ Y such that y ∈ Int(B) ⊂ B ⊂ U . called an exponential map and defined as follows: if F : This theorem is well known and usually formulated for Hausdorff spaces [2]. For non-Hausdorff spaces its proof is presented in [14,Lemma 3.3].
If B is locally compact, and C is an arbitrary topological space, then the exponential map is a homeomorphism. In particular, it induces a bijection between the corresponding sets of path components.
Moreover, for groups of homeomorphisms one can say more. Let Evidently, z ∈ hcl(z). We say that z is special 3 whenever hcl(z) \ z = ∅. Denote by S the set of all special points of Y .
Further, let L ⊂ Y be a subset, y ∈ L, and hcl L (y) be Hausdorff closure of z in L with respect to the induced topology from Y . It is straightforward that however the opposite inclusion can fail.
(2) In particular, for every z ∈ S the following subspace (1) is an immediate consequence of (2.4).
(2) To prove that B := (L \ S) ∪ {z} is Hausdorff it suffices to verify that hcl B (y) = {y} for all y ∈ B.
If y ∈ L \ S, then hcl(y) = {y}, whence B ∩ hcl(y) = {y}. Since z ∈ hcl(w) if and only if w ∈ hcl(z), it follows that hcl(z) ⊂ S. Therefore (3) Let y / ∈ L ∪ S. We will show that then there exists an open neighborhood W of y such that L ∩ W = ∅. This will imply that y ∈ Cl(L), whence Cl(L) ⊂ L ∪ S.
Since y is not a special point, i.e. hcl(y) = {y}, we get from (2.3) that  (1) A is closed and discrete; (2) for each y ∈ Y there exists a neighborhood V intersecting A in at most one point; (3) A is locally finite; Then we have the following implications: If Y is T 1 then we also have that It remains to prove the implication (3)⇒(1) under the assumption that Y is T 1 . Suppose A is locally finite. Then each subset B ⊂ A is locally finite as well. Moreover, as every point y ∈ Y is a closed subset, it follows that B is closed as a union of a locally finite family of its closed one-point subsets. In other words every subset of A is closed in Y . Hence A is closed and discrete.
2.4. One-dimensional T 1 manifolds. Let Y be a T 1 topological space locally homeomorphic with [0, 1). In other words, Y is a one-dimensional manifold which is T 1 but not necessarily Hausdorff. Then Y is also locally compact, and by Corollary 2.1.3 one can equally regard π 0 H(Y ) as the group of path components of H(Y ) and also as the group of isotopy classes of homeomorphisms of Y .
As usual, a point y ∈ Y is called internal , if it has an open neighborhood homeomorphic with (0, 1). Otherwise, y has an open neighborhood homeomorphic with [0, 1) and is called a boundary point. As usual, the sets of all internal and boundary points will be denoted by Int(Y ) and ∂Y respectively.
The following Lemma 2.4.1 characterizes the identity path component of Y under assumption that the set of its special points is locally finite.    (YB) k(C) = C for every connected component C of Y \ T , and the restriction map k| C : C → C is isotopic to id C (which in the case of 1-dimensional manifolds is equivalent to the assumption that k| C preserves orientation of C). If these conditions hold, then k is also fixed on ∂Y .
Proof. First notice that S and ∂Y are defined in topological terms, i.e. they are invariant under self-homeomorphisms of Y . Moreover, each z ∈ ∂Y \ S has a neighborhood U z such that U z ∩ T = {z}. Since S is locally finite, and therefore discrete due to Lemma 2.3.1, we obtain that T is discrete as well. This implies that F ([0, 1] × z) ⊂ T for any z ∈ T , and therefore F ([0, 1] × z) is contained in some connected component of T , being a one-point set since T is discrete. Hence F t (z) = F 0 (z) = z for all t ∈ [0, 1]. In particular, k is fixed on T , which proves (YA).
As Y \T is also invariant under self-homeomorphisms of Y , we obtain by similar arguments that F ([0, 1] × C) ⊂ C for every connected component C of Y \ T . Thus the restriction is an isotopy between id C and k| C . Therefore, k| C : C → C preserves orientation of C which proves (YB).
Sufficiency. Let k ∈ H(Y ) be a homeomorphism satisfying (YA) and (YB). For every connected component C of A = Y \ T fix an arbitrary isotopy F C : [0, 1] × C → C such that F C 0 = id C and F C 1 = k| C and define the map G : Then G 0 = id Y , G 1 = k, and each G t is a bijection of Y . Notice that G| A×[0,1] is an isotopy of A, and G is its extension to Y × [0, 1] fixed at special points. We need to show that G is an isotopy of Y .
For each z ∈ T let C z be a connected component of A z := A ∪ {z} containing z. Then A ∪ {C z } z∈T is an open cover of Y . Since the restriction G| A×[0,1] : A × [0, 1] → A is an isotopy, it suffices to show that for each z ∈ T , the restriction G : C z × [0, 1] → C z is an isotopy as well.
If C z is compact, i.e. it is homeomorphic either to S 1 or to [0, 1], then C z is a one-point compactification of C, and continuity of G| Cz×[0,1] follows from [1, Theorem 2.1].
Otherwise, (C z , z) is homeomorphic to one of the following pointed spaces: In this case the proof can proceed similarly to the arguments of [ [5] and many others. Also, certain variants of such results for concordancies of homeomorphisms considered in L. S. Husch and T. B. Rushing [11], applications to ambient isotopies of arcs in manifolds [4,22], and applications to Nielsen numbers are given in P. Heath and X. Zhao [9, §4].

Striped surfaces
In this section we will briefly recall the results by W. Kaplan, S. Maksymenko, E. Polulyakh, and Yu. Soroka.

3.1.
Quasi-striped surfaces. The following notion of a quasi-strip is a "basic block" generating foliations on the plane.  2) there exist two disjoint families X = {X γ } γ∈Γ and Y = {Y γ } γ∈Γ of mutually distinct boundary intervals of Z 0 enumerated by the same set of indexes Γ such that (a) q is injective on Z 0 \ (X ∪ Y); (b) q(X γ ) = q(Y γ ) for each γ ∈ Γ, and the restrictions The images q(X γ ) = q(Y γ ), γ ∈ Γ, will be called seams of Z (as well as of the striped atlas q). If all strips {S λ } λ∈Λ are model, then the quasi-striped atlas q will be called striped .
Condition (b) implies that for each γ ∈ Γ we have a well-defined "gluing" homeomorphism Thus a quasi-striped surface is obtained from a family of quasi-strips by gluing them along certain pairs of boundary intervals by homeomorphisms φ γ . It is allowed to glue two quasi-strips along more than one pair of boundary components, and one may also glue boundary components belonging to the same quasi-strip. Notice that the foliations ∆ S λ on strips S λ give a unique foliation ∆ 0 on Z 0 . In this case if γ is a leaf of ∆ 0 contained in a strip S λ , then we will say that S λ is a strip of γ.
Also, the images of leaves of ∆ 0 under q constitute a one-dimensional foliation ∆ on Z which we will call the canonical foliation associated with the quasi-striped atlas q.
Definition 3.1.4. A foliated surface (Z, ∆) will be called a (quasi-)striped surface whenever it admits a (quasi-)striped atlas whose canonical foliation is ∆. Now the results by W. Kaplan can be formulated as follows.
Theorem 3.1.5 (W. Kaplan [12,13]). Every one-dimensional foliation ∆ on the plane R 2 belongs to the class F and admits a quasi-striped atlas, i.e. ∆ is a canonical foliation associated with some quasi-striped atlas q on R 2 .
Remark 3.1.6. Let (Z, ∆) be any (oriented or non-oriented, compact or non-compact) connected surface distinct from S 2 and RP 2 . Then the universal covering of its interior Int(Z) is homeomorphic with R 2 , e.g. [3,Corollary 1.8]. Moreover, if p : R 2 → Int(Z) is the corresponding universal covering map, then one has a foliation F on R 2 induced from ∆: the leaves of F are connected components of the inverse images p −1 (ω), ω ∈ ∆. There is also a natural free action of the fundamental group π 1 Z on R 2 by foliated homeomorphisms so that (Int(Z), ∆| Int(Z) ) is the quotient (R 2 , F )/π 1 Z.
This observation shows that foliations on R 2 determine all foliations on all connected boundaryless surfaces distinct from S 2 and RP 2 . However, this does not help so much, since usually it is hard to explicitly describe the action of π 1 Z of R 2 by covering transformations.
Also, due to Theorem 3.1.5 the foliated surface (R 2 , F ) admits a quasi-striped atlas. However, again such a splitting into quasi-strips in general is not related with the action of π 1 Z. Therefore it is still better to work with the surface Z instead of R 2 .
In fact, the splitting of a quasi-striped surface into quasi-strips proposed by Kaplan in Theorem 3.1.5 is not unique. This led the present authors to consider quasi-striped surfaces glued from model strips only, and it turned out that splitting into such strips (if it exists) is unique, see Theorem 3.1.10.
c) In all other cases, ω ∈ ∆ spec , and hcl(ω) contains all leaves from see leaves ω 34 and ω 5 in Figure 3.1.
Notice that in the case a) one can replace S λ ∪ S λ with a single strip, and get another striped atlas for (Z, ∆) having one less strip. This leads to the following notion.
A striped atlas will be called reduced whenever The following statement is easy and we leave it for the reader. If q is reduced, then for every connected component C of Y \ ∂Y ∪ S), q −1 (p −1 (C)) is an interior of some model strip S λ ⊂ Z 0 . Define the following path γ C : (0, 1) → Int(S λ ) = R × (0, 1) by γ C (t) = (0, t), t ∈ (0, 1). Then the composition p • q • γ C : (0, 1) → C is a homeomorphism. Notice that in the case (1) all leaves of ∆ are regular. Also in the case (2) the seams are precisely the singular leaves contained in Int(Z) and thus we have a canonical decomposition of Z into strips.
But Z is connected, whence or h is constant. Therefore, one can associate to each h ∈ Q a well-defined element or h ∈ Z 2 . It is easy to checks that the correspondence h → or h is a homomorphism or : Q → Z 2 . Finally, h ∈ ker(or) iff h satisfies conditions (ZA) and (ZB), i.e. h ∈ H id (Z, ∆).