Fixed point sets of isotopies on surfaces

Fixed point sets of isotopies on surfaces

François Béguin, Sylvain Crovisier et Frédéric Le Roux
Abstract

We consider a self-homeomorphism of some surface . A subset of the fixed point set of is said to be unlinked if there is an isotopy from the identity to that fixes every point of . With Le Calvez’ transverse foliations theory in mind, we prove the existence of unlinked sets that are maximal with respect to inclusion. As a byproduct, we prove the arcwise connectedness of the space of homeomorphisms of the -sphere that preserves the orientation and pointwise fix some given closed connected set .

Résumé

Considérons un homéomorphisme d’une surface . Nous dirons qu’un ensemble de points fixes de est non enlacé s’il existe une isotopie de l’identité à qui fixe tous les points de . Motivés par la théorie des feuilletages transverses de Patrice Le Calvez, nous montrons l’existence d’ensemble non-enlacés qui sont maximaux au sens de l’inclusion. Notre critère de non-enlacement conduit aussi au corollaire suivant : pour toute partie fermée et connexe de la sphère, l’espace des homéomorphismes de la sphère qui préservent l’orientation et qui fixent tous les points de est connexe par arcs.

Contents

1 Introduction

1.1 Main result

Throughout this text, we consider a connected surface without boundary, not necessarily compact nor orientable. Let be the space of homeomorphisms of , equipped with the topology of uniform convergence on compact subsets of . An isotopy is a continuous path from a compact interval to ; this isotopy is denoted by or, more simply, by .

Let be a closed subset of . We denote by the subgroup of consisting of elements that fix every point of . A homeomorphism is isotopic to the identity relatively to if it belongs to the arcwise connected component of the identity in , in other words if there is an isotopy such that , , and for every in and . In this case, the set is said to be unlinked for . Our main technical result is the following criterium of unlinkedness.

Theorem 1.

Let . Assume there exists a dense subset of such that every finite subset of is unlinked for . Then is unlinked for .

This criterium has applications to the existence of transverse foliations and to the topology of spaces of homeomorphisms. We explain this in the next two subsections.

1.2 Unlinked continua

Let be the closed unit disk in the plane, and let be a homeomorphism of the plane that fixes every point of . The celebrated Alexander trick (Proposition A.3) provides an isotopy from the identity to relative to ; more generally, the space of homeomorphisms that fix every point of is easily seen to be contractible. What happens if we replace by any closed connected non-empty subset of the plane? Suppose for simplicity that the complement of is also connected. When is locally connected, one may use the Rieman-Caratheodory conformal mapping theorem to transport the Alexander trick, by conjugacy, from the outside of the unit disk to the outside of , and again get an isotopy that fixes every point of . Our first corollary solves the general (non locally connected) case.

Corollary 1.1.

Let be a closed connected subset of the plane. Then the space

of homeomorphisms that preserve the orientation and fix every point of is arcwise connected. In other words, every closed connected subset of the fixed point set of an orientation preserving homeomorphism of the plane is unlinked.

We conjecture that the space is always contractible, as far as contains more than one point. The generalization of Corollary 1.1 to arbitrary surfaces will be discussed in section 9.

1.3 Maximal unlinked sets and transverse foliations

We describe now the application of Theorem 1 that was our main motivation.

Corollary 1.2.

Let . Assume is a closed unlinked set for . Then there exists a closed unlinked set , containing , and maximal for the inclusion among unlinked sets.

The maximality amounts to saying that for every isotopy from the identity to , relative to , and for every fixed point of which is not in , the loop is not contractible in (see Lemma A.8 below).

Proof of Corollary 1.2.

In order to apply Zorn’s lemma, we consider a totally ordered family of closed unlinked sets containing . Let be the union of the ’s, and be the closure of . A subset of an unlinked set is clearly unlinked, thus every finite subset of , being included in some , is unlinked. Theorem 1 entails that is unlinked. Thus every totally ordered family of closed unlinked sets is included in a closed unlinked set: we may apply Zorn’s lemma, and we get some closed maximally unlinked set containing , as required by the corollary. ∎

To explain the interest of the maximal unlinked sets provided by Corollary 1.2, let us recall Jaulent’s and Le Calvez’s results ([Jau14, LC05]). Let be a maximally unlinked set for , and be an isotopy from the identity to that fixes every point of . In this context, Le Calvez has proved that there exists an oriented foliation on which is transverse to the isotopy : this means that the trajectory of every point outside is homotopic in , relative to its endpoints, to a curve which crosses every oriented leaf of the foliation from left to right. Le Calvez’s theorem is a extremely powerful tool to study surface dynamics (see for example the introduction of [LCT15] and the references within). The general existence of maximally unlinked sets is a crucial auxiliary tool for Le Calvez’s theorem: it extends its range of action from some specific cases, when the existence of maximally unlinked sets is obvious (for instance when the fixed point set is assumed to be finite, as in Le Calvez’s proof of Arnol’d’s conjecture in [LC05]) or easy (for instance for diffeomorphisms, see the appendix of [HLRS15]), to the case of every homeomorphism which is isotopic to the identity.

We have to point out that Jaulent has already provided an auxiliary tool which is enough for most applications of Le Calvez’s theorem, if less satisfactory ([Jau14]). Namely, he has proved the existence of “weakly maximally unlinked sets” for , which are closed subsets with the following property: on , there is an isotopy from the identity to the restriction of , and no trajectory for this isotopy is a contractible loop. Corollary 1.2 above is a strengthening of Jaulent’s result, since we get an isotopy which extends continuously to the identity on . It answers Question 0.2 of Jaulent’s paper. This stronger version is needed for example in the work of Yan on torsion-low isotopie ([Yan14]). Another motivation for the present work was to clarify the links between various natural notions of “unlinkedness”; we will discuss this in the next subsection.


We end the discussion of maximal unlinked sets with a version of Corollary 1.2 that takes into account a given isotopy. Let . We consider couples where is a closed set of fixed points of , and is an isotopy from the identity to relatively to . The set of such couples is endowed with the order relation defined by when and for every point , the trajectories of under both isotopies and are homotopic in . Note that it suffices to check the compatibility between and on one point in each connected component of (for example by requiring that the trajectory under of some point of is a contractible loop in ). Actually, in most cases, the compatibility between isotopies is automatic: for instance, for every couples and of such that has at least three points, if is included in then . This “uniqueness of homotopies” will be a key ingredient in the construction of isotopies. It also allows the following improvement on Corollary 1.2 (see the end of section 2.3 for the proof).

Corollary 1.3.

For every there exists a maximal element such that .

1.4 Fifty ways to be unlinked

Consider as before a homeomorphism on a connected surface , and some closed set of fixed points of . If is unlinked for , then is isotopic to the identity as a homeomorphism of . The converse is not true, as shown by the example of a Dehn twist in the closed annulus with equal to the boundary. We will say that is weakly unlinked if is isotopic to the identity on . Both weak and strong unlinkedness turn out to have many equivalent formulations. Our main task in this text will be to prove the most difficult implications between these formulations. We begin by defining the various formulations of weak unlinkedness in paragraph (a), and of strong unlinkedness in paragraph (b) below. In paragraph (a) we will think of as .

(a) Homotopies and isotopies on the complement of

Let be a connected surface without boundary, and be a homeomorphism of . Remember that is said to be homotopic to the identity if there exists a homotopy, i. e. a continuous map , such that and . If the map can be chosen to be proper (inverse images of compact subsets are compact), then is said to be properly homotopic to the identity. A loop is a continuous map from the circle to ; two loops are said to be freely homotopic if there exists a map such that and . We consider the following properties on .

  1. The map is isotopic to the identity.

  2. The map is properly homotopic to the identity.

  3. The map is homotopic to the identity.

  4. The map lifts to a homeomorphism of the universal cover of that commutes with the automorphisms of the universal cover. Equivalently, acts trivially on .111More formally, the outer automorphism of induced by is trivial, see subsection 2.2.

  5. Every loop is freely homotopic to its image under .

Theorem 2 (mostly due to D. Epstein).

Let be a connected surface without boundary, and be a homeomorphism of . Then properties (W1), (W2), (W3), (W4), (W5) are equivalent, with the following exceptions (see Remark 1.4):

  • (W3) does not implies (W2) when is the plane or the open annulus.

  • (W4) does not implies (W3) when is the sphere.

The implication (W5) (W4) is purely algebraic, and due to Grossman and Allenby, Kim, and Tang ([Gro75, AKT01]). The implication (W4) (W3) is standard in homotopy theory, see for example [Hat02, Proposition 1.B.9]. The most difficult part of the Theorem, namely that property (W2) implies property (W1), is proved by Epstein in his classical work [Eps66]. We will provide a complete proof (spread in Propositions 2.5, 3.3, 3.6, 8.8 and Section 8.5).

Remark 1.4. The exceptional cases may be further analyzed by considering the ends of and the orientation (see sections A.1 and A.2). Propositions 3.3 and 8.7 will imply:

  • When is the plane or the sphere, any orientation-preserving homeomorphism of is properly homotopic to the identity (hence, (W4), (W3) and (W2) are equivalent when one restricts to orientation-preserving homeomorphisms).

  • When is the open annulus, (W3) implies (W2) if one restricts to homeomorphisms fixing each of the two ends of .

An orientation-reversing homeomorphism of the plane is homotopic to the identity but is not properly homotopic to the identity. Similarly, a homeomorphism of the open annulus which interchanges the two ends is homotopic, but not properly homotopic, to the identity. Finally, an orientation-reversing homeomorphism of the sphere is not homotopic to the identity, although it acts trivially on the (trivial!) fundamental group.

(b) Homotopies and isotopies relative to

Let be a surface with a homeomorphism , and be a closed set of fixed points of . We will say that is strongly homotopic to the identity relatively to if there exists a stron homotopy relative to , i. e. a map such that

  • , ,

  • for every and every ,

  • and for every and every .

In other words, the restriction is homotopic to the identity, and the homotopy extends continuously to as a map that fixes every point of . We consider the following properties on and .

  1. is isotopic to the identity relatively to .

  2. is strongly homotopic to the identity relatively to .

  3. (the “finitely homotopic” criterion) There exists a dense subset of such that for every finite subset of , is strongly homotopic to the identity relatively to .

Note that, clearly, Property (S1) implies Property (S2), and Property (S2) implies Property (S3). The main task of the paper is to prove the converse implications. The following statement clearly implies Theorem 1.


Theorem 3.

Let be a connected surface without boundary, a closed non-empty subset of , and . Then Properties (S1), (S2), (S3) are equivalent.

(c) The totally discontinuous case

In the case when is totally discontinuous, every isotopy on the complement of extends to an isotopy of fixing every point of . Thus in this case weak and strong unlinkedness are equivalent, as expressed by the following proposition.

Proposition 1.5.

Let be a connected surface, a closed non-empty subset of , and . Let , and be the restriction of to .

Assume that is totally discontinuous. Then property (S1) for is equivalent to property (W1) for .

It is not true in general that property (W1) for implies property (S1) for . The most elementary counterexample is given by a twist map on an annulus, with equal to the boundary of the annulus: the restriction of to the interior of the annulus is isotopic to the identity, but is not isotopic to the identity relative to its boundary. It is plausible that every counter-example is a variation on this one.

(d) The braid viewpoint

Property (S3), which appears in Theorem 1, reduces the problem of unlinkedness to the case of finite sets. Unlinkedness of finite sets can be characterized using braids, as follows (this will not be used anywhere else in the text). Let be a closed subset of a surface . A geometric pure braid based on is a continuous map with for every in , and such that is injective for every . A geometric pure braid represents the trivial braid if there exists a continuous map such that , is the constant braid , and is a geometric pure braid for every . Now consider a homeomorphism of that fixes every point of . Assume is isotopic to the identity, and choose any isotopy from the identity to . This isotopy generates the geometric pure braid . The following criterium is an easy consequence of the fact that for any given point , the map that takes a homeomorphism to the image of under is a fiber bundle (see Lemma A.8).

Proposition 1.6.

Assume that is a finite set. If the geometric pure braid represents the trivial braid, then is isotopic to the identity relatively to .

1.5 Strategy for Theorem 3

Our main task is to prove the non-trivial implications of Theorem 3, namely that “finitely homotopic” implies strongly homotopic, i. e. (S3) implies (S2), and that trongly homotopic implies isotopic, i. e. (S2) implies (S1).

To prove (S3) implies (S2), we need to build homotopies. In order to build a homotopy between and the identity, one chooses continuously for each point in the surface a path between and . This is done in two steps: first identify the homotopy class of paths in which will contain and then select continuously the path in the class . The hypothesis (S3) provides an increasing family of finite subsets of , whose union is dense in , and for each a homotopy relative to . For the first step, fix a point in the complement of . Then we prove that up to homotopy, there exists a unique paths from to in the complement of which for each is homotopic in the complement of to the trajectory of under the homotopy (Lemma 4.8); for this the essential ingredient is the “uniqueness of homotopies” which is discussed in section 2.3. For the second step, we apply a selection technique due to E. Michael, which provides a continuous selection for a multivalued map under some very general assumptions.


To prove (S2) implies (S1), we need to build isotopies. More precisely, we have a non-empty closed set of fixed points of and we want to build an isotopy relative to between the identity and , under the hypothesis that there is a strong homotopy. To explain the strategy, let us assume for simplicity that is the closed unit square, and is any closed subset of . We take the equivalent viewpoint of constructing an isotopy, relative to , from the identity to the inverse of . Then the strategy will go as follows (see Figure 1). Take the middle vertical segment , and we find a first isotopy , that does not move the points of , from the identity to a homeomorphism that brings back into place, namely such that fixes . Now take the middle horizontal segment and find a second isotopy from to a homeomorphism , that does not move the points of , and brings back into place. Go on this way, bringing successively back into place a sequence of segments that cut the square into pieces with smaller and smaller diameters. The infinite concatenation of all these isotopies is easily seen to converge to the inverse of , as wanted.

Figure 1: Proof of (S2)(S1): the first steps in the construction of the isotopy.

With this strategy in mind, it is clear that the key step in the above explanation is the existence of each isotopy . More precisely, we need the two following properties:

  • Given that is strongly homotopic to the identity, and given any arc , there is an isotopy relative to such that . We call this the “Arc Staightening Lemma” (Proposition 7.1). The proof of this lemma is probably the novelty of this paper in terms of techniques.

  • With the same notations, the map is strongly homotopic to the identity relatively to . This will be the content of Lemma 8.2.

Finally, let us give a hint about the “Arc Straightening Lemma”. Again the required isotopy will be obtained as an infinite concatenation of more elementary isotopies. As a first rough approximation, let us pretend that each elementary isotopy consists in bringing back a subarc of which is (the closure of) a connected component of . Before doing this there is a preliminary technical step to get the additional property that coincides with in a neighborhood of its endpoints, and is transverse everywhere else. With this additional property, the isotopy that sends back to is constructed by “pushing bigons” to successively remove all the transverse intersection points between and . This technique goes back at least to Epstein’s paper. We have some constraints on the support of the isotopy, in order to ensure the convergence of the infinite concatenation of our elementary isotopies (the precise statement is Proposition 5.11). One key issue is to ensure the convergence of the infinite concatenation of all these elementary isotopies, each bringing back into place one connected component of . If no precaution is taken, then the trajectory of a point outside could be pushed closer and closer to by each elementary isotopy, thus converging to a point in , and then the concatenation would converge to a non-injective map. In the proof the elementary isotopies are concatenated in a carefully chosen order (using the above additional property), so that each point travels only under a finite number of elementary isotopies, thus circumventing this pitfall.

1.6 Organization of the paper

The diagram on Figure 2 shows where the various implications are proven.

Strong unlinkedness ,   

Weak unlinkedness

(S1) is isotopic to the identity relatively to

(W1) is isotopic to the identity

(S2) is strongly homotopic to the identity relatively to

(W2) is properly homotopic to the identity

(S3) is strongly homotopic to the identity relatively to every finite subset of a dense subset

(W3) is homotopic to the identity

(W4) acts trivially on

(W5) every loop is freely homotopic to its image

selection of trajectories (sections 3,4)

arcs straightening (sections 5,6,7,8)

algebra of surface groups (section 2)

selection of trajectories (section 3)

selection of trajectories (section 3)

Epstein, arc straightening (section 8)

when is totally disconnected

Figure 2: Organization of the paper. The uncommented arrows denote the trivial implications

In section 2, we collect the needed algebraic properties of fundamental groups of surfaces. In particular in 2.2 we recall the main arguments for the proof that (W5) implies (W4) (Proposition 2.5). These algebraic properties are also used in 2.3 to prove the key observation that homotopies are essentially unique.


Then we start building homotopies.


In section 3, we prove a general criterion to select a curve continuously in a given homotopy class. We first use this in 3.2 to construct a homotopy, proving that (W4) implies (W3) (Proposition 3.3). Then again in 3.3 to transform a homotopy into a proper homotopy, proving that (W3) implies (W2) (Proposition 3.6). Some alternative proofs using hyperbolic geometry are provided in 3.4.

In section 4, assuming property (S3), we first build a homotopy on the complement of . Then we modify this homotopy so that it extends to , thus getting a strong homotopy and proving that (S3) implies (S2) (Proposition 4.1). Here the selection criterion of the previous section is essential.


From then on, we undertake to build isotopies.


In section 5, we show how to “push bigons” in order to construct an isotopy relative to , sending an arc to another arc which is assumed to be homotopic to , in the easy case where is essentially disjoint from . In section 6 we do the same in the case where is a simple closed curve.

In section 7 we prove the Arc Straightening Lemma (see subsection 1.5 for a brief description of the content of this Lemma).

In section 8 we use the Arc Straightening Lemma to build an isotopy from a homeomorphism to the identity, thus proving that (S2) implies (S1) (Proposition 8.1). We also use it to give a proof that (W2) implies (W1) in 8.5.

In section 9 we prove Corollary 1.1 concerning unlinked continua and its generalization to every surface.

Part I Homotopies

2 Algebra of surface groups and applications

Throughout the paper, we consider curves as maps from an interval or from the circle to the surface. A (free) homotopy from a curve to a curve is a continuous maps such that and .

2.1 Surface groups

We collect some useful facts about fundamental groups of (compact and non compact, orientable and non orientable) surfaces. Fundamental groups of all surfaces are computed for instance in  [Sti12, section 4.2], including a proof that the fundamental group of a non compact surface is always a free group.

Proposition 2.1.

Let be the fundamental group of some connected surface without boundary. Then

  1. if is not the torus nor the Klein bottle, then every abelian subgroup of is cyclic.

  2. if is not the projective space, the torus, the Klein bottle, the annulus or the Möbius band, then the center of is trivial.

Proof.

Assume contains some abelian subgroup of rank greater than 1. By the covering spaces classification theorem, there is a surface which is a covering of and has fundamental group isomorphic to . By the classification of fundamental groups of surfaces, is a torus. In particular is a compact surface. Furthermore, since a covering maps multiplies the Euler characteristic by some integer, the Euler characteristic of is zero, hence is a torus or a Klein bottle.

Now assume the center of , say , is non trivial. Let be a covering of with fundamental group isomorphic to . Then is a surface with non trivial abelian fundamental group, thus a torus, a projective space, an annulus or a Möbius band. The surface is a quotient of one of these, thus it belongs to the list of the five exceptional cases of the second point of the proposition. ∎

An element in a group is called primitive if it has no root, i. e. there is no element and integer such that .

Proposition 2.2.

If is a primitive element in the fundamental group of a non compact surface, and if commutes with , then is a power of .

Proof.

The subgroup generated by and is abelian, hence it is a cyclic group according to the first point of the previous proposition. Thus and are powers of a commun element, but since is primitive, is a power of . ∎

The following proposition says that an essential simple closed curve on an orientable surface is always primitive in the fundamental group, and describes the only exceptions in the non-orientable case.

Proposition 2.3.

Let be an essential simple closed curve on a surface . Assume that there exists a closed curve and an integer such that is freely homotopic to . Then bounds a compact Möbius strip in , , and is freely homotopic to the core of the Möbius strip.

Proof.

(See also [FM11] for a proof in the orientable hyperbolic case.) The existence of implies that is not the sphere, and the existence of a root of implies that is not the projective plane. Thus the universal cover of is homeomorphic to the plane. Let be an automorphism corresponding to : there is a lift of which is invariant under . This automorphism acts freely and properly discontinuously on the plane, thus it is conjugate either (1) to a translation or (2) to the composition of a translation and a symmetry. We may lift the free homotopy between and , thus getting a lift of joining a point and its image by . If is a translation, then the quotient is an open annulus, and the projection of is a simple closed curve which makes turns around this annulus. This is impossible. Thus is the composition of a translation and a symmetry, and the quotient is an open Möbius band.

We claim that . Assume by contradiction that . Consider any point on , its image under is another point on ; both points bound a closed subarc of . Since is a simple closed curve homotopic to , this arc is disjoint from all its iterates except for . In particular, and are pairwise disjoint, but meets . This contradicts Bonino’s Corollary 3.11 in [Bon04]222One can also get a contradiction by using the uniformization theorem, the classification of the isometries of the eulidean (resp. hyperbolic) plane.. Hence

Now since is an automorphism corresponding to , we deduce that is two-sided (i. e. it has a tubular neighborhood homeomorphic to an annulus, not a Möbius band).

Consider the orientation covering of , which is 2 to 1. Since is two-sided, its inverse image in has two components. Since is one-sided, its inverse image has a single component. We may lift the free homotopy between and , starting with any of the two lifts of , and this provides a homotopy between the given lift of and the lift of . Thus both lifts of in are essential simple closed curves which are homotopic and disjoint. Thus they bound an annulus (Lemma A.7). The projection of this annulus in provides a Möbius band bounded by , whose core is freely homotopic to . ∎

Lemma 2.4.

Let be a surface, be a subset of which is a surface with boundary, bounded by curves none of which is contractible in . Then the natural map induced by the inclusion is injective.

Proof.

Let be the universal cover of , and a connected component of the inverse image of in . The space is a surface bounded by lifts of the boundary curves of . Since each boundary curve is essential, its lifts are topological proper lines in . Thus , being an intersection of topological half-planes, is simply connected. This implies that is a universal cover of . The lemma follows. ∎

2.2 Free homotopies of loops and outer automorphisms

In this section, denotes a homeomorphism of a connected surface without boundary. We will prove that Property (W5) implies Property (W4). Property (W5) says that every loop is freely homotopic to its image.

Let us first explain more precisely the meaning of the phrase “ acts trivially on ” in (W4). We choose a point in . The homeomorphism induces an isomorphism between and . Similarly, any curve joining to induces isomorphism between and . We say that acts trivially in if we can choose a curve such that . Note that given any curve from to , we get the automorphism of the group . Changing amounts to composing this automorphism by an inner automorphism (a conjugacy). Thus the action of on is only defined up to a composition with a conjugacy. Formally, induces an outer automorphism of , an element of the quotient of the group of automorphisms by inner automorphisms.

Now, let us explain the equivalence between the two formulations of (W4). If is a lift of to the universal cover that commutes with the covering automorphisms, then we get a curve so that by projecting any curve joining a point to its image . Conversely, if is a curve joining to so that , then we consider a is a lift of . It joins a lift of to a lift of . Then the lift of mapping to commutes with the covering automorphisms.

Proposition 2.5.

Let be a homeomorphism of a connected surface . If every loop in is freely homotopic to its image under then acts trivially on .

Proof.

Assume every loop is freely homotopic to its image. Choose some curve from a point to its image. Let be the isomorphism induced by and respectively. Consider the automorphism of . The hypothesis entails that every element of is conjugate to its image : there is an element of such that . We want to prove that is an inner automorphism, i. e. that there exists such that for every (the same for all ’s).

This purely algebraic statement may be found in the following papers. If is a non compact surface, then is a free group, and the result is Lemma 1 of [Gro75]. If is a compact orientable surface, then the result is again proved in [Gro75]. (In this paper a group is said to have property A if every automorphims that sends each element to a conjugate is an inner automorphism; the surface groups are denoted , and the proof that has property A is exactly the proof of Theorem 3 of the paper.) If is a compact non orientable surface, then the result is Theorem 3.2 of [AKT01].


Since non compact surfaces are the interesting case for the present paper, and the argument is short, let us recall it. In this case is isomorphic to a free group. If is generated by a single element then the result is obvious. Thus in what follows we assume that is a free group of rank at least .

Lemma 2.6.

In the free group , every element which is conjugate to may be written or as a reduced word in the letters , for some word .

Proof.

Let , and write as the reduced word in . If is not a reduced word then one of and is not a reduced word either. We deal with the first case, the other being similar. If is not a reduced word then ends by the letter . Write , where a maximal ending subword of with the property that the letters of are alternatively and . Then equals either or , according to the parity of the length of . ∎

Lemma 2.7.

Let be an element in the free group , and assume that

Then there exists integers such that .

Proof.

We write as the reduced word in . Up to eliminating the ’s at the end of , which does not change the value of , we may assume that is reduced. The problem now boils down to proving that is a power of . We first examine the case when is reduced. Using the hypothesis and the first lemma, we may write or . Note that in these equalities all the words are reduced. The word has a ’’ in the position just after the middle, thus the second equality is impossible. Looking separately at the beginning and the end of the equality , we get and . Thus commutes with ; since the group is free, is a power of , as wanted. In the case when the word is not reduced, , then is equal to the reduced word , and we conclude using the same method as in the first case. ∎

Now consider an automorphism of the free group such that for every , , and write for . Up to composing with the inner automorphism , we may assume that . Fix some and let , , . Since is conjugate to , we are in the situation of Lemma 2.7. We get . Up to composing each with we may assume that . It remains to check that does not depend on . To see this we apply again lemma 2.7 with , with , and . We get that equals some power of , and thus since is free. ∎

2.3 Uniqueness of homotopies

Algebraic properties of fundamental groups have some important consequences on uniqueness of homotopies that we explain now. We begin by the following easy but fundamental fact (which is already in [Gra73] and [Jau14]).

Fact 2.8.

In a topological space , let be a loop based at , and be a homotopy of loops such that . Let be the loop given by the trajectory of under , namely . Then in the fondamental group , commutes with .

Proof.

Since , induces a continuous map from the quotient torus

Then the result follows from the fact that the fundamental group of the torus is abelian: indeed in the curves and induces two commuting elements of ; and their images under are nothing but the curves and . ∎

Corollary 2.9.

Let be a surface that is not the projective plane, the torus, the Klein bottle, the annulus, the Möbius band. Let be a homeomorphism of which is homotopic to the identity.

  1. If are two homotopies from the identity to , then for every point , the trajectories of under and are homotopic.

  2. There is a unique lift of to the universal cover of which is obtained by lifting a homotopy from the identity to .

  3. If is not the sphere, then the space of homotopies from the identity to is contractible.

Proof.

Let be homotopies from the identity to . Consider the homotopy from the identity to the identity obtained by concatenating and the inverse of , and let be the loop given by the trajectory of under this homotopy. Proving Property 1 amounts to showing that is homotopically trivial. Fact 2.8 implies that commutes with every other loop . In other words is in the center of . But Proposition 2.1 states that the center of is trivial. Hence, must be homotopically trivial, and Property 1 is proved.

To prove Property 2, we consider again two homotopies from the identity to . The homotopy (resp. ) can be lifted to the universal cover of as a homotopy from the identity to some homeomorphism (resp. ) of which is a lift of . We have to prove that the lifts and coincide. Given some point in , the point is the endpoint of the lift of the trajectory under of the projection of in . Likewise, is the endpoint of the lift of the trajectory of under . According to Property 1, these two trajectories are homotopic, and thus . Since is an arbitrary point, this shows that the lifts and coincide, and Property 2 is proved.

Finally we prove Property 3. According to the Uniformization Theorem (see e.g. [Rey89]), and given the restrictions on its topology, the surface admits a hyperbolic structure, the universal cover identifies with the hyperbolic plane , and the automorphisms are isometries of . For every , denote by the arclength parametrization of the unique geodesic arc from to . Let be the lift of which is given by Property 2. Then the formula provides a canonical homotopy from the identity to on , which induces a homotopy from the identity to on . Given a homotopy from the identity to , with lift , the formula

provides a continuous deformation from to which induces a deformation from to . This deformation depens continuously on . This shows that the space of homotopies from the identity to is contractible. An alternative proof can be cooked up using the selections techniques presented in Section 3. ∎


To end this section, we apply the previous results to show how Corollary 1.3 on maximal isotopies follows from Corollary 1.2.

Proof of Corollary 1.3.

We consider a closed set of a surface , a homeomorphism of that fixes every point of , and an isotopy from the identity to relative to . Let . First assume that is not one of the five exceptional cases of the first point of Corollary 2.9. Corollary 1.2 provides a maximal unlinked closed set containing . In particular, there exists an isotopy from the identity to relative to , and the couple is a maximal element of . Corollary 2.9 says that the trajectories of a point under both isotopies are homotopic in , which means that , as wanted.

Now assume is one of the five remaining cases. If is a maximal element of , then we are done. If not, we find a couple with strictly included in . Now does not belong to the excluded list anymore, and we may apply the first case to get a maximal couple such that , as wanted. ∎

3 Continuous selection of paths

In order to build a homotopy between a surface homeomorphism and the identity, we must choose continuously for each point in the surface a path between and . This is done in two steps: we first identify the homotopy class of paths in which will contain and then select continuously the path in the class . This section explains the second step. As an application we will prove the implications (W4) (W3) and (W3) (W2). These techniques will also be crucial in the next section.

One may obtain the family by using the general Michael’s selection theorems [Mic56]. For a two-dimensional topological space , the main assumptions for the existence of a continuous selection are: the map is lower semi-continuous; for each , the class is path-connected and simply-connected; the map is equi-locally path-connected and equi-locally simply connected. We will not use Michael’s theorem, we rather prove a selection result which is adapted to our simpler setting (Proposition 3.1). The greatest simplification comes from a strong version of lower semi-continuity.

3.1 An easy selection theorem

In this section we consider two metric spaces and . In the applications will be a surface (or more generally will contain a dense open set homeomorphic to a surface) and will be a space of paths on . We use the notation and for the balls in and .

We consider a family of subsets of indexed by points of . We will state some topological condition that guarantees the existence of a continuous selection, that is, a continuous map from to such that for every , . In this section we will often denote applications as families, e. g. the map is denoted by .

Local triviality.

We will say that the family is locally trivial if for every , there exists a neighborhood of and a continuous map

such that and is a homeomorphism from to for each (see Figure 3).

Figure 3: Local triviality and extension of continuous families

Assuming this property and these notations, let be a continuous family of points of indexed by the unit interval . Let , and denote by the two elements of such that for . Let us assume furthermore that is arcwise connected. In this case there is a continuous family , included in , that extends the family . Then the formula

produces a continuous family that extends and such that for each . Likewise, let us assume that is simply connected. Let be a continuous family of points of indexed by the standard -simplex (a triangle), and a continuous family of paths, indexed by the boundary of , such that for every . Then as above this family extends to a continuous family such that for every .

Local properties at singletons.

Let be some point for which is a singleton. We will say that the map is:

  • lower semi-continuous at if for every neighborhood of there is a neighborhood of such that, for every , the set meets ,

  • equi-locally arcwise connected at if for every neighborhood of there is a neighborhood of and a neighborhood of such that, for every and every , there exists a path from to in ,

  • equi-locally simply connected at if, for every neighborhood of , there is a neighborhood of and a neighborhood of such that, for every , every continuous family included in , extends to a continuous family included in .

Selection.

Here is the selection result that we shall need.

Proposition 3.1.

Let be a closed subset of , and assume is a surface.333Not that here is not supposed to be connected. In section 3.3, will be the end-compactification of a connected surface , and will be the set of ends of ; but in section 4.3 will be the complement of a general closed set in a surface . Assume the following hypotheses on the family :

  1. Each is arcwise connected and simply connected.

  2. On , the map is locally trivial.

  3. For every , the set is a singleton.

  4. The map is lower semi-continuous, equi-locally arcwise connected and equi-locally simply connected at every point of .

Then there exists a continuous selection .

Proof.

Since is a surface, it admits a triangulation. Any compact set of meets only finitely many triangles. By subdividing the triangulation if necessary, we may assume that each closed triangle is included in one of the open set given by the local triviality (hypothesis 2).

When , the proof is straightforward: first select for any vertex of the triangulation any element in ; then extend this selection continuously to the 1-skeleton using the local triviality; then extend the selection to the whole of using again the local triviality.

From now on we assume that is non empty. For every point in , according to hypothesis 3, is a singleton; we denote by the unique element .

For each vertex of the triangulation, we choose a point in such that , and an element of not too far from (remember that is a point in ), namely in such a way that

where the infimum is taken among all in .

For each edge of the triangulation, whose endpoints are vertices , we choose an open set given by the local triviality and containing . Since is arcwise connected, the family extends to a family , such that is in for each (as explained above after the definition of the local triviality). Again we choose this family not too far from , namely in such a way that

where the infimum is taken among all continuous families with for each in and , .

For each triangle of the triangulation, whose vertices are , we choose an open set containing . We can extend the family to a family defined for every . Again we choose the extension so that

where the infimum is taken among all continuous families with for each and when . To complete the proof it remains to show the following claim.

Claim 3.2.

The map satisfying properties is continuous.

To prove the claim, we first note that by construction the map is continuous at every point of . Given , let us prove the continuity at . From the lower semi-continuity at , we immediately get that the restriction of the map to the set is continuous.

Let . We first apply the equi-local simple connectedness to the ball , and get some balls and . We then apply the equi-local arcwise connectedness to the ball and get some balls