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Curves and Isotopies Appendix Curves and Isotopies Teichmuller theory hinges strongly on the fact that homotopies of simple curves can be carried out by surface homeomorphisms. The necessary theo­ rems have been studied by Baer [1], Epstein [1], Zieschang [1] and others. We reprove these theorems for completeness. The proofs are similar to those in the literature but are not based on piecewise linear mappings. At certain points we use hyperbolic geometry. In the final part we give an application to the mapping class group of the 3-holed sphere and to length decreasing homotopies. The Theorems of Baer-Epstein-Zieschang Throughout the appendix we make the following assumption. A.O Assumption. S is a smooth compact connected orientable two dimen­ sional manifold with possibly empty piecewise smooth boundary. At any boundary point p of S where the boundary is not smooth, we assume that the adjacent arcs do not have parallel tangent vectors. Since all compact surfaces of the same signature are homeomorphic, this is no restriction of generality. Moreover, all compact hyperbolic surfaces satisfy this assumption. If two surfaces S and S' satisfy Assumption A.O, and if both have the same signature, then they are piecewise diffeomorphic (see for instance Massey [1] or Hirsch [1]). We may therefore assume, when this is conve­ nient, that S carries a metric of constant curvature in which the boundary is P. Buser, Geometry and Spectra of Compact Riemann Surfaces, Modern Birkhäuser Classics, 409 DOI 10.1007/978-0-8176-4992-0, © Springer Science+Business Media, LLC 2010 410 Curves and Isotopies [Appendix piecewise geodesic with all interior angles < n. Thus, in most cases S is a surface with a hyperbolic structure and we shall use results of Chapter 1. A.l Definition. Let S and S' be given as above. Two homeomorphisms (p0, <px: S -> S' are called isotopic if there exists a continuous mapping /: [0, 1] x S —» S" with the following properties. (i) /(0, ) = <Po, /(l, ) = 9v (ii) J(p, ): S -» S' is a homeomorphism for any p G [0, 1]. Now assume that S = S'. If A c S is a subset, then the homeomorphisms <p0, <Pi'. S —> S are called isotopic with A pointwise fixed if ^0(x) = x and ^(x) = x for any * e A, and if there exists a mapping / with the above prop­ erties such that in addition (iii) J(p, x) = x for any x e A and any p e [0, 1], A homeomorphism h : S —> 5 is called a 1-homeomorphism if it is isotopic to the identity with dS pointwise fixed, h is an A-1-homeomorphism if it is iso­ topic to the identity with A u dS pointwise fixed. The mapping / in Definition A.l is sometimes called an isotopy or a level- preserving isotopy. A.2 Theorem. (Alexander-Tietze). LetD c5 be a domain homeomorphic to the compact unit disk in R2 and let A = S - int D. If <p : S —> S is a homeo­ morphism which is the identity on A, then (p is an A-1-homeomorphism. Proof. It suffices to prove the theorem for the case (p: E —> E, where E = {x e R2 | U| < 1}, and where <p(x) = x for all x e dE. We define, for x e E and p e [0, 1], f* if \x\ > P P' * [p<Kx/p) if x < p . Then /:£x[0, l]-^£isa continuous mapping and satisfies the conditions (i) -(iii) of Definition A. 1 with <p0 = id and <px = (p. O We now state the theorems of Baer, Epstein and Zieschang in a form which is adapted to the canonical curve systems as in Chapter 6. Afterwards we shall look at some examples to explain the lengthy hypotheses of the theo­ rems. The proofs are be based on the Jordan Schoenflies theorem and will begin after Corollary A.7. We introduce the following notation. Curves a and p are said to be Appendix] The Theorems of Baer-Epstein-Zieschang 411 non± homo topic if a is neither homotopic to (5 nor to fi~l, where /T1 is the curve parametrized in the opposite sense. (If non-oriented curves are consid­ ered then "non± homotopic" may be replaced by "non-homotopic".) In the first theorem, homotopy means free homotopy. A.3 Theorem. (Baer-Zieschang). Let cv ..., cm be pairwise disjoint homo- topically non-trivial simple closed curves in the interior ofS which are pair- wise non±homotopic. Ifyly ..., ym in the interior of S is another set of curves with these properties, and if c^ is freely homotopic to y^ for p, - 1,..., m, then there exists a l-homeomorphism (f>: S —> S such that <f> ° cfl = ytl,p=l,...,m. The next theorem concerns simple loops that is, simple closed curves with a given base point. In this theorem, "homotopy" means homotopy with fixed base point. A.4 Theorem. (Epstein-Zieschang). Letp e intS and let cx, ..., cmbe homotopically non-trivial simple loops in intS with base point p which pairwise intersect each other only at p and which are pairwise non±homotopic. Ify{, ..., ymin intS with base pointp is another set of loops with these properties, and if c^ is homotopic to y^ (with p fixed) for p = 1,..., m, then there exists ap-l-homeomorphism (f>: S —» S such that 0° Cfl = ytl>p = 1,..., m. The third theorem will be needed in the induction proof of Theorem A.4. In itself, it is interesting mainly in the case m = 1 and for arcs cl,yl with two different endpoints. In this theorem, "homotopy" means homotopy with fixed endpoints. A.5 Theorem. (Epstein-Zieschang). Letpv ..., pm, qx,..., qm e dS and let for p = 1,..., m, c^be an arc from p^ to qH which, except for the end- points, lies in the interior ofS. Assume the following hypotheses. (i) IfPn * Qp> then c^ is a simple arc oriented from p^ to q^ ifp^ = q^, then c^ is a simple homotopically non-trivial loop with base point p^; p = 1,..., m. (ii) Forp^vthe curvesc^andcvare non±homotopic (in the homotopy class with p^ and q^ fixed). (iii) The curves c{,..., cm pairwise intersect each other at most at their endpoints. s V Yi> • • • > Ymi another set of curves with these properties, where y^ is oriented from p^ to q^ p = 1,..., m, and if each c^ is homotopic (with fixed 412 Curves and Isotopies [Appendix endpoints) to y^ then there exists a l-homeomorphism <j>: S —> S such that 0° ctl=y^^i= l,...,m. We first consider a few examples. GEDD QMS) Figure A.l Fig. A.l shows that Theorem A.3 fails if homotopic curves are admitted: the closed curves cx and c2 shown on the cylinder cannot be interchanged by a homeomorphism of the cylinder which keeps the boundary fixed. The second example in Fig. A. 1 shows that the theorem also fails if cx is homotopic to some c~l. Similar counterexamples can be given in the case of Theorems A.4 and A.5. The next example is more difficult and we give two figures, one showing the surface S (an annulus) and one showing its universal covering. The example shows that Theorem A.5 cannot be extended to arcs with different endpoints in the interior of 5. One checks in the universal covering that c and y are homotopic with fixed endpoints. However, any homotopy between c and y moves c over one of the endpoints. Therefore this homotopy cannot be carried out by a {p, q}-l-homeomorphism. The final example is due to Epstein-Zieschang [1] and shows that Theorem A.4 needs certain modifications if 5 is a non-orientable surface. The example is on the Mobius band. In the first figure, the Mobius band is shown cut open into a rectangle. The second figure shows again the universal covering. Note that the group of covering transformations is generated by a Appendix] The Theorems of Baer-Epstein-Zieschang 413 Figure A.3 glide reflection which maps the covering curve y of y onto f and c onto 5. In the proof of Theorem A.4 the reader will indeed find an argument which uses the fact that in the orientable case no covering transformation is a glide reflec­ tion. For the proofs of the theorems we need the Jordan-Schoenflies theorem and corollaries which we state in the following condensed form. For the proof we refer to Dostal-Tindell [1], Stillwell [1] and Moise [1]. A.6 Theorem. (Jordan-Schoenflies). (i) Let c and y be simple closed curves in the interior of the closed unit disk E. Then there exists a l-homeo- morphism 0: is —» £ such that (f><>c = yor(l)<>c = y'1. (ii) Let p, q e E (possibly withp = q) and let c and y be simple arcs con­ necting p to q which, except possibly for the endpoints, are contained in the interior ofE. Then there exists a {p, q }-l-homeomorphism 0 : E —> E such that <p°c = yor(j)<>c = y~l. We note the following corollary. A.7 Corollary. Let c be a homotopically trivial simple closed curve in S. Then c bounds a disk. Proof. We assume that S has signature (g, n) with g + n > 2 and leave the remaining cases to the reader. We may assume that S carries a hyperbolic structure and let n: S —> S be a universal covering with 5cD, where D is the unit disk model of the hyperbolic plane. Since c is simple and homotopically trivial, c has a simple closed lift c in 5cD which bounds a disk DcD.No boundary point of § is in the interior of D for otherwise dS would intersect c and then dS would have to intersect c.
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