[Version of 02 September, 2005] The Osgood-Schoenflies Theorem Revisited _ in honor of Ludmila Vsevolodovna Keldysh and her students on the centenary of* * her birth _ by Laurent Siebenmann(*) The very first unknotting theorem of a purely topological character est* *ablished that every compact subset of the euclidean plane that is homeomorphic to * *a circle can be moved onto a round circle by a globally defined self-homeomorphism of * *the plane. This difficult hundred year old theorem is here celebrated with a partly * *new elemen- tary proof, and a first but tentative account of its history. Some quite * *fundamental corollaries of the proof are sketched, and some generalizations are menti* *oned. 1 Introduction This retrospective article presents an elementary, and hopefully direct a* *nd clear, geo- metric proof of what is usually called the (classical planar) Schoenflies T* *heorem; it is stated as (ST) in x4 below _ with mention of its early history, including W.F. Osg* *ood's rarely cited contributions. This (ST) is essentially the fact _ surprising in view* * of known fractal curves _ that every compact subset of the cartesian plane R2 that is homeom* *orphic to the circle S1, is necessarily the frontier in R2 of a set homeomorphic to t* *he 2-disk. Be- ware that the `Generalized Schoenflies theorem' of B. Mazur [Maz] and M. Br* *own [Brow1] _ proved five decades later and valid in all dimensions _ does not imply (S* *T) since it assumes a condition of flatness (or local flatness [Brow2]). The Schoenflies Theorem (ST) is, in three respects, more awkward than oth* *er equally crucial and famous theorems of plane topology, notably the Jordan Curve The* *orem and Invariance of Domain, which are stated as (JCT) and (IOD) in x3. Indeed, mo* *st extant proofs of (ST) lack three features of some well-known proofs for (JCT) and * *(IOD): (i) to be essentially homological. (ii)to apply in all dimensions to prove an important result. (iii)to be easily motivated, remembered, and explained. I have encountered exceptions to the above dismissive judgements. R.H. Bing* *'s exposition [Bing5] 1983 can claim feature (iii); it is similar to an earlier one in [N* *ewm], but clearer. Moreover, A. Chernavsky's highly original proof [Cher] of the natural exten* *sions of (ST) to dimensions > 4 could perhaps be adapted to claim both features (ii) and * *(iii); however, I find his argument unnecessarily mysterious when adapted to dimension 2. A* *lso worth special attention is the proof of (ST) by S. Cairns [Cair] (with contributi* *ons from John Nash, then a graduate student). Likewise, the proofs based on the theory o* *f conformal mappings [Osg3][OsgT][Cara1-3][Koeb1-3][Stud][Ker'e ][Cour].(y) As for non* *-conformal proofs, those in textbooks include [Newm][HallS][Why2][Mois2][WhyD][MohT] w* *hile those in research monographs include [Ker'e ][Moor2][Wild][Why1][Kura][Keld][Bing* *5]. The proof of (ST) to follow achieves neither (i) nor (ii); but it will, p* *erhaps, be most fully credited with feature (iii). It is exceptional in using a striking co* *mbinatorial design (Figure 6-a) well known in plane hyperbolic geometry; this simple design la* *ys bare the crucial proof certifying the homeomorphism establishing (ST). Additionally,* * our policy is to classify all pl (= piecewise linear) surfaces encountered; hopefully thi* *s will add insight at slight extra cost. On the other hand, at a low level, our techniques are* * fairly typical of textbook proofs _ in using a mixture of some general topology and some p* *l topology ____________ (*) Math'ematique, B^at. 425, Universit'e de Paris-Sud, 91405-Orsay, France Email the author via http://topo.math.u-psud.fr/~lcs/contact (y) Not to mention conformal proofs of (ST) in treatises on complex analysis,* * for example the mid-century ones by A.I. Markushevich or E. Hille. Furthermore, some s* *olutions of the famous Plateau problem (of finding an area minimizing and well behaved * *disk map spanning a given embedded continuous closed curve in R3) are conformal in c* *haracter, and imply (ST) when applied to planar curves; see the books by R. Courant (1950* *), M. Struwe (1988), and J. Jost (1994) _ which were kindly pointed out by J"urgen Jost. 1 of Euclidean space; such methods have more than didactic merit, since they domi* *nate the study of topological embeddings of manifolds in dimensions > 2, cf. [Keld][Dave* *r2]. Interestingly, in dimension 2 itself, conformal methods dominated for a good * *part of the last century, thanks to early initiatives of Osgood [Osg3] 1903, and Carath* *'eodory [Cara3] 1913. The latter established a theory of "prime ends", cf. [Stud][Koeb* *3][Ker'e], which generalizes (ST) to an analysis of the frontier of any connected open sub* *set U with connected complement in a closed 2-manifold M _ a theory having since had impor* *tant applications to dynamics, cf. [Mat1][BarK][Epst]. In 1982, J. Mather [Mat2] provided a bootstrapping purely topological treatme* *nt of the topological aspects of "prime ends" complementing earlier (but hard to loca* *te) ones of P. Urysohn [Ury4] (cf. [Ury1-3]) of the 1920s, and of M.H.A. Newman [Newm] 1* *939. Mather writes: "It does not seem possible to give a brief account of Carath'eo* *dory's theory which does not rely on some [other] deep theory." This should be a warn* *ing to some readers, but a challenge to others. Mather includes a remarkable conse* *quence, cf. [Newm][BreB], generalizing (ST):_ any contractible open subset U in M , has* * a nat- ural "prime end" compactification Ub that is a 2-disk, and, if the frontier ffi* *U in M is locally connected, then the closure U~ of U in M is naturally the quotient of* * Ub by a continuous map of the circle boundary @Ub onto the frontier ffiU . In high dim* *ensions, there is related current research concerning `mapping cylinder neighborhoods', * *see [Quinn] and [Daver2, x47]. Purely topological methods have led to other deep results related to (ST), no* *tably several characterizations of 2-manifolds, cf. [Moor2][Wild][Bing1], for example* *, L. Zippin's characterization [Zipp][HallS] of the 2-sphere, which states roughly this: A Pe* *ano contin- uum B , in which there is at least one embedded circle and in which the stateme* *nt of the Jordan Curve Theorem (JCT) holds true, is necessarily homeomorphic to the 2-sph* *ere. Compare R.D. Edwards' characterization of manifolds of dimension > 4 [Lat][Dave* *r2]. Leaving aside such difficult extensions of (ST), the last section x9 gives, m* *ostly as ex- ercises, a few important corollaries of (ST) and its easy pl (= piecewise linea* *r) analog. One is the approximability by plhomeomorphisms of merely continuous homeomorphi* *sms between pl manifolds. I have collected many references to the early literature concerning (ST) with* * the help of the recently constituted electronic version of the 1868-1942 review journal * *[JFM]. This literature is curiously disconnected _ for example: the earliest explicit refer* *ence I have thus far encountered to Schoeflies' original (but partial) proof of (ST) in 190* *6 [Scho3] is in L.V. Keldysh's 1966 monograph [Keld]! I am indebted to Alexey Chernavsky and Jean Cerf for encouragements and criti* *cism that helped to improve my exposition at several points. Lucien Guillou alerted * *me to the fascinating and relevant histories of (JCT) in [DosT][Gugg] and their bibliogra* *phies. In translating this article for its Russian edition, Prof. Chernavsky provided, no* *t just wel- come collections of errata, but several interesting references, notably P.S. Al* *exandroff's reminiscences [Alxf], and Urysohn's cited work on prime ends. 2 Notions and notations All spaces are by assumption metrizable topological spaces unless the contrar* *y is stated. A space is connected if it cannot be expressed as the disjoint union of two non* *-empty subsets that are both open and closed. A component of X , will, in this articl* *e, be understood to mean a connected subset that is both open and closed. A subset A * *of X is bounded if its closure in X is compact. If X itself is compact, then a subse* *t A of X is closed if and only if it is compact. The following easy "Frontier Crossing Lemma" of general topology (dating from* * Brouwer [Brou1] if not earlier) will often be used without mention: (FCL) Let Y be a s* *ubspace of space X . If a connected set C in X contains points of both Y and X - Y , then* * C also contains points of the frontier ffiY of Y in X . Beware that ffiY depends on* * X although our notation does not indicate this; the frontier of Y in Y itself is always * *empty. 2 Given (possibly non-continuous)Smap f : X ! Y and also a finite collection X* *i of closed subsets of X , with X = iXi, the map f is continuous if and only if a* *ll the restrictions f|Xi: Xi! Y are continuous. A map f : X ! Y is said to be surjective or onto (respectively injective) if* *, for every point y in Y , there exists at least one (respectively, at most one) poin* *t x in X such that f(x) = y . This f is said to be bijective or one-to-one if it is both* * injective and surjective. An embedding f : X ! Y is an injective and continuous map of topological spa* *ces that gives a homeomorphism onto its image f(X). A map f : X ! Y of topological spaces is said to be an open map if, for ever* *y open subset A of X , the image f(A) is open in Y . Replacing `open' by `closed' in t* *his sentence yields the definition of a closed map. The image of a compact set under a continuous map is necessarily compact. Hen* *ce a continuous map f : X ! Y of compact spaces is closed. And if this f is bijecti* *ve, then f is also open and a homeomorphism. If there is no contrary indication, a map of spaces will normally be supposed* * continuous. Concerning pl objects and maps, see for example the first few pages of [RourS* *]. Every (metrizable) pl object is pl homeomorphic to a locally finite simplicial comple* *x. The symbol ~= will denote pl homeomorphism, whilst denotes ordinary homeomorphism. The boundary of a manifold M is denoted @M and its (manifold) interior is M - @* *M , denoted IntM . An n-simplex will always be identified with the n-dimensional * *convex hull of (n + 1) points in a real linear space. Those points and/or the dimensio* *n may be specified as arguments, as for the 1-simplex 1(p, q). The formal boundary @ * *is the union of all its faces of dimension < n, and coincides with the boundary @ of * * as a pl manifold. For dimensions n 1, the standard pl(= piecewise linear) n-disk Bn is [-1, 1* *]n Rn , and its frontier and boundary is the pl (n - 1)-sphere Sn-1 . The interior of * *Bn is IntBn = (-1, 1)n = Bn - Sn . We shall also encounter the smooth n-ball Bn (respectively the smooth n-s* *phere @Bn = Sn-1 ) consisting of all points in Rn at Euclidean distance 1 (respecti* *vely exactly 1) from the origin of Rn . These do not have a natural pl structure. Ho* *wever, there is a homeomorphism ae : Bn ! Bn sending Sn-1 to Sn-1 ; and with some effo* *rt, it can be chosen to be pl on Bn - Sn-1 . It would be quite possible to use C1 smooth manifolds and maps instead of pl* * objects and maps; see [Miln]; the required techniques are slightly less elementary, but* * perhaps more important to undergraduates. We are chiefly interested in embeddings into S2 and R2. Since R2 is pl homeom* *orphic to the complement of any point in S2, we can study any compact set X in S2 that* * omits at least one point of S2, by regarding it as a compact set in R2, and conversel* *y. A Jordan curve (respectively arc) C is a compact subset of S2 or R2 that is h* *ome- omorphic to S1 (respectively to B1). These notions also make good sense in any * *space homeomorphic to S2 or R2. If dn , n = 1, 2, 3, . . .is a sequence of real numbers, the unorthodox phras* *e "dn converges O(n)" will mean that dn converges to 0 as n converges to infinity. 3 Homologically provable results that we exploit Proofs of the following basic results using chiefly homology theory, are wide* *ly under- stood by students (even undergraduates) who have studied topology for a year or* * two. Our proof of (ST) and its complements will freely use (JCT). Jordan Curve Theorem (JCT) _ [Jord2], 1887. In R2 or S2, the complement of a Jordan curve C has exactly two components, say* * D- and D+ . Furthermore, C is the frontier of both D- and D+ . As `universal' homological proof of the first clause of (JCT), we cite Alexan* *der duality, as expounded for example in [Dold1], of which an immediate corollary is this in* *variance principle: (JA) The number of components of the complement Sn - X of any compact 3 subset X of Sn is a an intrinsic topological invariant of X itself _ indeed of * *its `Borsuk shape'. The note [Dold2] offers a simple and elegant proof of (JA) - leaving a* *side its last clause concerning shape; it relies on a homology suspension isomorphism an* *d this rudimentary unknotting lemma: The embedding of any compact set in Rn is unique * *up to (i) the `stabilization' by inclusion Rn ! R2n, and (ii) homeomorphism of R2n th* *at is the identity outside a compact set. The proof of this lemma uses Tietze's well know* *n exten- sion theorem [Tiet3] 1914. The partial proof of (JA) in [Dold2] can be generali* *zed to fully establish (JA) _ including the shape clause _ using Chapman's stable correspond* *ence [Chap] between shape and complement in Rn . For a full proof of (JA) based on a* * study maps to Sn-1 viewed up to homotopy, see [tomD] (or [Dugu] for n = 2). As for the second `frontier' clause of (JCT), we now recall a classic two par* *t argument that adapts to any dimension. (i) The complement of any Jordan arc J in S2 is c* *onnected by (JA). (ii) Given a point p on C and any Jordan arc J C not containing p, t* *he connected open set S2 - J contains the disjoint Jordan domains D- and D+ . Henc* *e, by (FCL) of x2, it meets the frontier ffiD- of D- in S2, indeed, necessarily in C * *- J . Since C - J can lie in any prescribed neighborhood of p, it follows that p lies in th* *e frontier_of D- in ffiD- . Similarly p 2 ffiD+ . * * |_| There is a strong school of thought, see [Vebl][Alex1][Scho4][DosT], that (JC* *T) was not fully proved by Jordan [Jord2]; however, I am unaware of specific objections, o* *ther than those of Schoenflies [Scho5] 1924, which Schoenflies himself considered non-fat* *al. Jordan's arguments do seem to involve less than any proof that I fully understand; compa* *re Keldysh [Keld, Chap II, Lemme 4.1]. The first complete proof of (JCT) seems to be O. Veblen's [Vebl] 1905; this i* *ntricate proof developed Schoenflies' notion [Scho3] 1902 of pl path access in R2 - C fr* *om any given point of R2 - C to (at least) a dense set of points of C . Similar proofs* * appear in most textbooks featuring (ST). Incidentally, some access path technology seems * *essential to most proofs of (ST) itself; see (SAL) in next section. Jordan's 1887 exposition assumes (JCT) for the case of pl Jordan curves; this* * case was discussed by Schoenflies in 1896 [Scho1], see [Jord1]. A pleasant inductive* * proof for pl Jordan curves largely due to N. Lennes 1903, 1911 (see [DosT]) and to M. Deh* *n (see [Gugg]) is sketched in the first remark of x7. By a Jordan domain, we will mean a bounded and connected open set D in a space homeomorphic to R2 or S2, whose frontier is a Jordan curve C . Its compact clo* *sure B = D [ C is called a sealed Jordan domain. With this language, (JCT) shows tha* *t, for every Jordan curve C in S2, the complement S2 - C consists of exactly two J* *ordan domains. The main result (ST) to be expounded reveals that D and B are homeomor* *phic to R2 and B2 respectively. Here are two easy but useful corollaries of (JCT) and (FCL). Let C and C0 be * *two Jordan curves in R2. Let D be the (unique) Jordan domain with frontier C and le* *t B be its compact closure. Define D0 and B0 similarly for C0. Jordan Subdomain Lemma If C0 B , then B0 B . __|_| Jordan Domain Disjunction Lemma Suppose C0 does not intersect D , and C does not intersect D0 (equivalently, suppose C stays outside or on C0, and reciproca* *lly)._Then_ either B = B0, or else D and D0 do not intersect (although B and B0 may interse* *ct). |_| In many proofs of the Schoenflies theorem, these lemmas are unmentioned but sil* *ently applied; we will endeavor to cite them explicitly wherever they are needed. Almost as famous and useful as (JCT) is: Invariance of Domain (IOD) _ see [Brou2]. Every embedding h : U ! R2 of an open subset U of R2 into R2 is an open mapping. L. Bieberbach 1913 (see [DosT]), attributes the first proof of (IOD) to one E. * *J"urgens in a 1879 habilitation thesis in Halle, which F. Hausdorff [Haus, p. 468] 1914 seems* * to identify as published in Leipzig 1879. For our proof of (ST), one needs (IOD) only in th* *e special case when h is pl. That case is easy to prove for any dimension by induction on* * dimen- sion, since h is then (affine) linear on the simplices of a linear triangulatio* *n of U . Here is a classic proof of (IOD) based on (JA) above, and adaptable to all dimension* *s: For 4 any 2-ball B in U , the complement R2 - h(B) is connected, by (JA). Also, R2 * *- h(@B) has two (open) components, again by (JA); clearly one is R2 - h(B) and the ot* *her is h(B - @B). Consequently h(B - @B) is open, which implies that h is an open ma* *pping._ |* *_| 4 Statement of the main result What is called the Schoenflies Theorem goes beyond (JCT) and better describ* *es the embedding of C as follows: Schoenflies Theorem (ST) Let B be a sealed Jordan domain in S2 or R2 with fro* *ntier the Jordan curve C . There exists a homeomorphism H : B2 ! B sending S1 onto * *C . Historical notes. Jordan's reputedly inconclusive arguments for (JCT) in [Jor* *d1] 1887 paradoxically made measurable progress towards (ST); they essentially prove t* *hat every complementary component D of S2 - C is homeomorphic to R2. In more detail, D * *is a nested union of sealed Jordan domains Bi D whose frontiers are a pl Jordan* * curves Ci; each Bi is B2 by [Jord1], while all successive differences are to ann* *uli, again by [Jord1]. As the 1866 date of [Jord1] might suggest, the proofs seem obscure * *to modern eyes, cf. [Hirs, Chap 9]. But see the exercise under (PLCT) of x7, which indi* *cates how to prove that D ~=R2 as directly as possible. In 1887, the fractal curves of Klein would have made any mathematician hesi* *tate to conjecture that B B2. The first clear assertion that B B2 of which I am a* *ware is by Wm. F. Osgood in 1903 [Osg3]; he had already proved in [Osg2] 1900 (cf. [S* *cho2]) that D is conformally homeomorphic to IntB2 R2 in spite of examples in [Osg1] wh* *ere C has positive Lebesgue measure. Full proofs of (ST) by Osgood and several othe* *r math- ematicians came a decade later [Cara1-3][Koeb1-3][OsgT][Stud]; all these firs* *t genera- tion proofs used complex variable theory, i.e. conformal mappings. (The proof* * in [OsgT] was accepted by Koebe in his [JFM] review and also by Courant in [Cour].) Sch* *oenflies [Scho4,x13] 1906 gave the second clear statement of (ST). He also ventured th* *e first proof; it begins by correctly establishing the easy pl version of (ST) (cf. x7), and* * concludes well by using an infinite tessellation; but he seems to make a significant blunder* * in between _ claiming to prove something impossible. Namely that, for any nested sequenc* *e of pl Jordan curves Ci D (as asserted by Jordan), a subsequence can be parametrize* *d, say by ci : S1 ! Ci R2, so that the ci converge to a topological parametrization c* * : S1 ! C of C .(*) In 1902, Schoenflies [Scho3] had discovered an interesting charact* *erization of Jordan curves in terms of access paths, one that may well have motivated Cara* *th'eodory's theory of prime ends [Cara3] 1913. The first complete proof of (ST) not based* * on conformal mappings may be H. Tietze's long argument [Tiet1, (b)][Tiet2, SatzIII] 1914. * *Or it may be that in L. Antoine's thesis [Ant1, Chap.I ] 1921, an argument of reasonable l* *ength that is detailed at the mentioned point where [Scho4,x13] seems to blunder. See also * *[Moor1] and [Keld; pp. 63-81]). The name "Schoenflies Theorem" to designate (ST) seems to* * originate with R.L. Wilder [Wild, I.6 and III.5.9]. Natural generalizations of (ST) to all dimensions n > 2 have now been prove* *d by surprisingly diverse and difficult methods _ for n = 3 by [Bing3] and [Bing4]* * 1961; for n 5 by Chernavsky [Cher] 1973, and independently by Daverman-Price-Seebeck * *[PriS] [Daver1] 1973; and finally for n = 4 by Freedman-Quinn [FreeQ] in the 1980s. * * All of these require a local fundamental group condition usually called 1-ulc, as th* *e Antoine- Alexander horned 2-sphere in S3 first revealed in 1924 [Ant2][Alex2][Alex3]. Our proof of (ST) in x6 will also prove Complement (ST+) The homeomorphism H : B2 ! B can be chosen to extend any given homeomorphism S1 ! C , and to be pl (= piecewise linear) on B2 - S1. ____________ (*) Does this blunder explain why Schoenflies' pioneering proof long went unmen* *tioned in the literature? I think not. Some 18 years later, Ker'ekj'art'o in [Ker'e, p.* *72] 1924, after a very condensed proof of (ST), seems to assert as a corollary of that proof, t* *hat Schoen- flies' falacy above is true _ indeed even without the above-mentioned subsequ* *encing and reparametrization! 5 5 Two tools for our proof of (ST) and (ST+) The first tool collects simplest special cases of (ST) enhancing them with ex* *tra piecewise linearity. Almost pl Schoenflies Theorem (APLST) Statement (ST) holds true in case C is pl (= piecewise linear) except at a (possibly empty) finite set X of points in * *C . Further- more, in that case, a homeomorphism H can be built, from the standard pl disk B* *2, onto the sealed Jordan domain B , such that H is pl except possibly at H-1 (X). This result follows quickly from the classification of noncompact pl surface* *s, cf. Ker'ekj'art'o [Ker'e ]. For completeness, it will be proved in x7, along with * *the following complement. Complement (APLST+) If G : @B2 ! C is a homeomorphism that is pl except at G-1(X), then H offered by (APLST) can be made to extend G. The second tool (SAL) is a 2-dimensional analog of a key 3-dimensional result* * of R.H. Bing [Bing2], which was essential to his proof of the analog of (ST) in di* *mension 3 [Bing4]. (SAL) is not new, but a (partly new?) geometric proof will be provided* * in x8. Side Approximation Lemma (SAL) Let J be any Jordan arc in the Jordan curve C R2 and let B R2 be the sealed Jordan domain with frontier C . There exist* *s a Jor- dan arc J0 B with the same end points {P0, P1} as J , such that J0\ C = {P0, * *P1} and J0- C is pl. Furthermore, one can choose J0 to lie in any prescribed neighborho* *od of J . Remark. (SAL) can clearly be deduced from (ST) and its complement (ST+). * *__|_| (SAL) can be regarded as a mostly homological result. Indeed, it can be prov* *ed homologically that every open Jordan domain is 0-lc in R2 _ using a local form * *of Alexander duality valid in all dimensions and codimensions, see [Wild][Dold1], * *or [Dold2]. Then, from this 0-lcproperty, one can derive (SAL). 6 The core of the proof of (ST) and (ST+) This is the section that hopefully holds a bit of novelty for topologists. We* * prove (ST) and (ST+) assuming (APLST), (APLST+) and (SAL), plus the homologically accessib* *le result (JCT) and the trivial pl case of (IOD). Without loss, we can assume that the sealed Jordan domain B that is given for* * (ST), lies in R2. Indeed, if B is originally given in S2, let P be any point in the c* *omponent of S2 - C distinct from D = B - C which is provided by by (JCT). Then we identi* *fy R2 by a pl homeomorphism to S2 - P B . The use of R2 as ambient space rather than S2 is helpful because the (Euclide* *an) metric of R2 has the special feature: 6.1 Euclidean Metric Property The diameter of any subset X of B is realized as * * __ the Euclidean distance between two points of its frontier ffiX in R2. * * |_| The continuity of the wanted homeomorphism B2 ! B will be checked using such diameters. Fix any embedding c : S1 ! R2 of the unit circle S1 onto the Jordan curve C .* * We shall extend c to a homeomorphism h : B2 ! B . As observed in x2, there is a homeomorphism ae of the square B2 onto the smoo* *th unit disk B2 that is pl on the interior of the square. Then the composition H = h O * *ae will establish (ST) and (ST+). Thus, it will suffice now to construct h to be pl on * *IntB2 and then verify that it is a homeomorphism. 6.2 Constructing a tessellation T of B2 and a map h : B2 ! B The rough idea is to gradually build up h using the tiles of an infinite tessel* *lation (=tiling) h(T ) of B that is combinatorially isomorphic by h to the tessellation of the n* *aturally compactified hyperbolic plane generated by the three reflections in the sides o* *f a triangle with all three of its vertices on the limit circle S1 R2 = C; see Figure 6-a. 6 i i = (0, 1) -1(=-1, 0) 1 -1 1 = (1, 0) -i -i = (0, -1) Figures 6-a and 6-b More precisely, we first define a standard infinite tessellation T by linea* *r simplices of the smooth unit disk B2 in R2, as illustrated in Figure 6-b. Then, using T , w* *e will build a homeomorphism h : B2 ! B that is pl on the open disk IntB2. Here, a tessellation of a space X means a triangulation of X as a (unordered* *) simpli- cial complex, also called T , except for one difference: the given bijective i* *dentification, say o : T ! X , of the simplicial complex T to the space X is assumed continuous * *but not necessarily a homeomorphism. However, o necessarily induces a topological embe* *dding into X of any (compact!) finite subcomplex T of T _ since the induced conti* *nuous bijection o| : T ! oT is a homeomorphism (x2). In particular, the infinite t* *essellation T of B2 in Figure 6-b gets a compact topology from B2 whereas the standard (w* *eak or metric) simplicial topologies of T are noncompact. (k) with k 0 denotes the convex hull of the k complex k -th roots of unit* *y in C = R2. It is the standard (solid) k -gon. Note the degenerate cases (1) = 1 * *= (1, 0) and (2) = 1(-1, 1). All other (k ) are 2-dimensional. To be quite specific (see Figure 6-b), we define the 0-simplices (=vertices)* * of the tessel- lation T to be all the continuously many points of the boundary circle S1 of B* *2; we define the 1-simplices (= edges) to be the edges of the regular convex 2n -gons (2n)* *, n 1. Finally, we define the open 2-simplices (= faces) of T to be the connected co* *mponents of (Int (2n)) - (2n-1), n 2, the closed 2-simplices being their respective clo* *sures in R2. Let Tn be (2n) with the finite triangulation inherited from T . The construction of h will mention circular arcs A(p, q) for points p and q * *in S1. Such an arc A(p, q) is always the shorter arc between p and q . Except that, when * *p and q are antipodal, A(p, q) denotes the counterclockwise arc p to q . Thus A(p, q) * *= A(q, p), except that, when q = -p, one has A(p, q) = -A(q, p) in C. Step (0) Define the restriction h|S1 to be a given Jordan curve parametrizatio* *n_c : S1 ! C B . |* *_| Recall that the points of S1 are exactly the vertices of the tessellation T . A chord of B is a Jordan arc J B such that J \ C = @J . We call it an nice* * chord if J - @J is pl. The next step of the construction uses an arbitrary sequence "n , n 1, of * *positive numbers that tends to 0, i.e. it converges O(n). Step (1) For each 1-simplex 1(p, q) of T construct a nice chord ffi(p, q) in* * B joining c(p) to c(q) in such a way that: (i) No two of these chords of B intersect except (possibly) on C . (ii)If the 1-simplex 1(p, q) of T lies in the boundary of Tn = (2n), then t* *he nice chord ffi(p, q) lies in the "n -neighborhood in B of the Jordan arc cA(p, q) * * C . Since simplices, chords etc. are unoriented, ffi(p, q) = ffi(q, p). 7 A(p, q) q X(p, q) h(q) r h 1(p, q) h(r) Figure 6-c h 2(p, q, r) 2(p, q, r) h(A(p, q)) p T (at angle jffn) n h(p) = c(p) Construction for (1) We initialize by applying (SAL) to obtain a nice * * chord ffi(1, -1) = ffi(-1, 1) of the closure B of D in R2, one that joins c(1) to c(-* *1). Suppose inductively that ffi(p, q) has been defined for every 1-simplex 1(p,* * q) in Tn , with n 1. There are 2n distinct 2-simplices S of T in Tn+1 , that are not in* * Tn . Each such S is of the form S = 2(p, q, r) where S \ Tn is 1(p, q), an edge of (2n* *), and r is the midpoint of the circular arc A(p, q) of length 2ss=2n from p to q in S1. Use (SAL) to obtain a nice chord ffi(q, r) joining c(q) to c(r) in the sealed* * Jordan do- main B(p, q) B whose frontier is the Jordan curve ffi(p, q) [ cA(q, p), requi* *ring that ffi(q, r) lie in the "n+1 neighborhood of cA(q, r). Once again, use (SAL) to obtain a nice chord ffi(r, p) joining c(r) to c(p) i* *n the sealed Jordan domain whose frontier is the Jordan curve ffi(p, q)[ffi(q, r)[cA(r, p), * *requiring that ffi(r, p) lie in the "n+1 neighborhood of cA(r, p). Then ffi(p, q), ffi(q, r) and ffi(r, p) lie in B(p, q) and meet only on C . When this has been done for each such S , the nice chord ffi(p, q) is defined* * for_every_ 1-simplex 1(p, q) in Tn+1 . Then induction on n completes Step (1). * * |_| Step (1+) For each (unoriented) edge 1(p, q) of T define a homeomorp* *hism __ h| : 1(p, q) ! ffi(p,tq)hat maps p to c(p), q to c(q), and is pl on 1(p, q) -* * {p, q}. |_| Here and elsewhere, h| informally denotes a map that is going be a restrictio* *n of h. Observe that, after Step (1+), the map h is well defined and continuous on the * *boundary @S of every 2-simplex S = 2(p, q, r) of T , and pl except (possibly) at the 3 * *ver- tices; it maps onto the frontier of the sealed Jordan region B(p, q, r) in B wi* *th boundary ffi(p, q) [ ffi(q, r) [ ffi(r, p). Step (2) For each (unoriented) 2-simplex S = 2(p, q, r) of T define a homeomo* *rphism h| : 2(p, q, r) ! B(p,,q,sr)o that: (i) On each 1-simplex face of S , this h|S is the 1-simplex mapping defined in * *Step (1+). (ii)h|S is pl except at the vertices p, q, r _ which are the points of (h|S)-1(* *C). Construction for (2) The required extension h|S is provided by the Almost pl * *__ Schoenflies Theorem (APLST) with its complement (APLST+), see x5. * *|_| At this point h : B2 ! B is well defined as a map of sets and it is clearly p* *l on IntB2. 8 6.3 Proof that h : B2 ! B is a homeomorphism Assertion (A) The map h : B2 ! B is injective. Proof of (A). By Steps (0), (1), and (1+) of its construction, this h is inject* *ive on the union of the simplices of T of dimensions 0 and 1. By Step (2) it is inje* *ctive on each individual 2-simplex. Then (A) follows from the Jordan Domain Disjunction_* *Lemma_ of x3. |* *_| Assertion (B) h|IntB2 is continuous. Proof of (B). For this, it suffices to verify continuity of h|U on a collection* * of open sets U of B2 that cover IntB2. Now h|Tk is continuous since it is continuous on each* * simplex, and Tk is a finite complex. A fortiori, h|IntTk is continuous. But the open set* *s IntTk form an open cover of IntB2 because (2n) contains the scaled 2-disk cos(2ss=2n* *+1)B2._ |_| Assertion (C) There exists a sequence j1, j2, j3, . .c.onverging O (n) such tha* *t, for every chordal sector X of B2 that is one of the 2n sealed components of B2 - (* *n), the diameter Diam h(X) of h(X) is jn . Proof of (C). By definition, X is the convex hull in R2 of a unique circle a* *rc A = A(jffn, (j + 1)ffn) where ffn = 2ssi=2n and 0 j < 2n . Also X \ (n) is = 1(jffn, (j + 1)ffn). By Step (1) of the construction of h, this linear ch* *ord of B2, which is the frontier of X in B2, is mapped by h to a nice chord lying i* *n the "n -neighborhood in B of h(A). Claim Diam h(X) = Diam h(@X) Diam h(A) + 2"n (*) Proof of claim. In (*), the relation follows from the metric triangle inequal* *ity, but the equality = is not obvious. However, in place of = is obvious from inclusi* *on. So the remaining task is to establish in place of =, which we do as follows. The Euclidean Metric Property of x6.1 tells us that Diam h(@X) = Diam (Y ), w* *here Y denotes the sealed Jordan domain with frontier h(@X). It is not (yet) clear * *that Y is h(X). But Y contains h(A) by its definition, and it contains h(Si) for every t* *riangular tile Si of T lying in the chordal sector X ; this is a consequence of the Jord* *an Subdomain Lemma of x3. Hence Y contains all of h(X) and so Diam h(X) Diam Y , as requir* *ed_to complete the proof of the claim. * * |_| Continuing the proof of (C), note that, since the restriction h|S1 is continu* *ous and S1 is compact, h|S1 is uniformly continuous, in the sense that, for any i > 0, the* *re exists a , > 0 such that if A0 S1 is of diameter < , then the diameter Diam h(A0) is < * *j . It follows that, if in = Max Diam (h(A)), where A ranges over the 2n arcs of S1 in* *to which the 2n -th roots of 1 cut S1, then in converges O(n). From (*) we conclude, setting jn = in+2"n , that Diam h(X) jn , where jn co* *nverges_ O(n). This proves Assertion (C). * * |_| Assertion (D) h : B2 ! B R2 is continuous. Proof of (D). In view of (B), it suffices to prove continuity of h at an arbitr* *ary point p in S1. We distinguish two cases: Case (I) The angle of p is not a dyadic rational multiple of 2ss . Proof of Case (I) _ see Figure 6-d. For all n 1, the point p of S1 lies in the interior in B2 of exactly one ch* *ordal sec- tor X = Xp(n) cut out of B2 by (2n) as described in Assertion (C). By Assertio* *n (C), Diam h(Xp(n)) converges O(n), which proves continuity of h at p, thus completin* *g_the_ proof of Case (I). * * |_| Case (II) The angle of p is a dyadic rational multiple of 2ss . Proof of Case (II) _ see Figure 6-e. Suppose p has angle 2ss`=2m where ` is odd and m 1. Then p is a vertex of * *the regular polygon (2n) precisely if n m. The new difficulty here is that the 2* *n chordal 9 (j + 1)ffn (j + 1)ffn Ap(n) A00p(n) c(Ap(n)) p X00p(n) jffn = p h 1 jffn 0 X h 1 Ap(n) p(n) Yp(n) (of length f* *fn) hXp(n) X0p(n) (j - 1)ffn Figures 6-d and 6-e sectors X of B2 cut out by (2n) do not include any neighborhood of p in B2, wh* *ilst we want a basis of neighborhoods as n varies to be able to test continuity of h* * at p. However, for all n m, the point p in S1 lies at the intersection of exactly* * two adjacent chordal sectors X0p(n) and X00p(n) of B2, having respective extremities (j - 1)* *ffn , jffn and jffn , (j + 1)ffn , where ffn = 2ssi=2n , and 0 j < 2n . Their union is n* *ot quite, but almost, a neighborhood of p in B2. We define the neighborhood Xp(n) of p in B2 * *to be (see Figure 6-e illustrating Xp(n) as a glider in flight): Xp(n) = X0p(n) [ X00p(n) [ Yp(n) where Yp(n) is a small neighborhood of p in the convex full regular 2n -gon (2* *n), so chosen that: Diam h(Yp(n)) < jn (**) This is possible because h|Tn is continuous by construction. We conclude that the image under h of the neighborhood Xp(n) of p in B2 has d* *iameter bounded by the sum of the diameters of its three parts: Diam h(Xp(n)) Diam h(X0p(n)) + Diam h(X00p(n)) + Diam Yp(n) 3jn where the last uses (*) of Assertion (C) and (**) above. It follows that Diam* * h(Xp(n)) converges O(n), which proves continuity of h at any dyadic point p. Assertion (D) is now proved in both possible cases. * * __|_| Assertion (E) The restriction h| : IntB2 ! D = B - C is an open map. Proof of (E). Apply the invariance of domain theorem (IOD) in the easy case of_* *pl maps. |_| Lemma Let f : X ! Y be a closed and continuous map, and let U be open in Y . T* *hen __ the restriction f | : f-1 (U) ! U is a closed map. * * |_| Assertion (F) h : B2 ! B is bijective. Proof of Assertion (F). By Assertion (A), h is injective. Since h maps S1 bijec* *tively to C , it suffices to show that h maps IntB2 = B2 - S1 surjectively to D = B - * *C . Since h is continuous, and B2 is compact, h is closed. It follows, from the a* *bove lemma, that h| : IntB2 ! D = B - C is a closed map. As it is also open by Assertion (E* *), and both IntB2 and D are connected, we conclude that the open and closed subset h(I* *ntB2) __ of D is all of D ; thus h is surjective as well as injective. * * |_| Since h is, by Assertion (F), a bijective continuous map of compact spaces, i* *t is a home- omorphism (see x2). This completes the proof of the Schoenflies Theorem (ST)_a* *nd its complement (ST+). |_| 10 Comment. The pace of the above proof is extremely leisurely, indeed prudish _ l* *ike that in [Keld] _ in comparison with many, say those in [Bing5][Newm][Ker'e]. This is* * perhaps advisable given the history in x4; and it is possible without straining the rea* *der's stamina, thanks to our simplified outline. 7 Proof of some PL Schoenflies Theorems _ see x5 This is the first of two sections establishing tools for the proof of (ST). W* *e begin with Classical PL Schoenflies Theorem (PLST) _ see [Scho4, x13] 1906. Every pl Jordan curve C in R2 bounds a pl 2-disk. We will need some easy lemmas that are left as exercises, cf. [RourS]. Lemma (1) A compact and convex or star-shaped sealed Jordan domain in R2 with pl frontier is necessarily pl homeomorphic to B2. In particular, B2 is pl homeomor* *phic_to the pl cone Cone (B1) on B1 and also to the pl cone Cone (S1) on S1. * * |_| Lemma (2) Any pl self-homeomorphism of S1 extends (by coning) to a pl * *self- homeomorphism of B2 ~= Cone(S1). Likewise any pl self-homeomorphism of a close* *d __ interval in S1 extends (by coning) to a pl self-homeomorphism of B2 ~=Cone (B1)* *. |_| Lemma (3) Any naturally cyclicly or anti-cyclicly ordered finite sequence of N * *points_in S1 is equivalent to any other by a pl self homeomorphism of S1. * * |_| Lemma (4) If X is a finite simplicial 2-complex expressed as a union X = X1 [ X* *2 of two subsomplexes, each ~=B2, so that X1 \ X2 ~=B1, then X is pl homeomorphic_to_ B2. |_| Topological versions Four lemmas parallel to the above four, but without the ep* *i- thets pl, and with ordinary homeomorphism ( ) in place of pl homeomorphism (~=)* *, are_ also true and have similar proofs. * * |_| Proof of (PLST). We proceed by induction on the number N(C) of 2-simplices, lin- ear in R2, needed to triangulate the (compact) sealed Jordan domain B with fron* *tier C . Since, by Lemma (1), (PLST) is true for N(C) = 1, we can assume N(C) > 1. There is always a 2-simplex S in the triangulation of B with at least one edg* *e in C . By a short case analysis, one or two edges of the boundary of this S must alway* *s split the triangulation X of B as envisioned in Lemma (4). Since X1 and X2 are ~=B2 b* *y_ inductive assumption, Lemma (4) shows that B ~=B2. * *|_| Remarks on proofs of the pl case of (JCT) The above easy proof of (PLST) requires the Jordan curve theorem for pl Jordan curves _ call this (PL JCT). In* *deed, (PL JCT) provides us with the sealed Jordan domain B on which the above inducti* *on turns. (i) A geometrical proof of (PL JCT) One proceeds by induction on the number of corners of C R2. We merely give some hints: Since the case when C is conve* *x is trivial to prove directly, one can assume there is a linear segment 1 in the f* *rontier of the convex hull of C that intersects C in its end points @ 1 only. This provides tw* *o Jordan curves C0 and C00with intersection 1 and union C [ 1. Since each has fewer co* *rners_ than C , one can argue inductively . . . * * |_| (ii) A combinatorial homology proof of (PL JCT). The homology in question has coefficients in Z2 = Z=2Z, and one calculates it for a finite simplicial 2-comp* *lex X using the chain complex C*(X; Z2) : 0 C0 C1 C2 0 in which Ci= Ci(X; Z2) can be identified to the set of finite subsets of the set of (unoriented) i-simplices * *of X . Now let X be a linear triangulation of the convex hull of C in R2, making the given pl * *Jordan curve C a subcomplex; then note that the set of 1-simplexes of X in C is natura* *lly a cycle in C1(X; Z2). Since X is contractible, H*(X; Z2) = 0; in particular, the * *1-cycle C is the boundary of a 2-chain B ; this B is a set of 2-simplices whose union is * *a compact connected 2-manifold with boundary C . Using these facts as a lemma, one shows * *that any pl Jordan curve C in S2 is the common boundary and frontier of compact conn* *ected submanifolds B1 and B2, so that C =B1 \ B2. Then B1 [ B2 is a closed pl sumanif* *old_ of S2 and hence all of S2. * *|_| 11 The proof of (APLST) uses three more lemmas. Lemma (5) _ see Figure 7-a. If a pl 2-manifold Y is expressed as an infinite u* *nion Y = Y1 [ Y2 [ Y3 [ . . . where Yj ~=B2, and Yj \ Yj+1 ~=B1 for all j 1, while Yj \ Yj+k = ; for k > 1,* * thenp_ Y ~=[0, 1) x B1 ~=B1 x [0, 1) ~=(B1 * i) - {i} C, where * denotes join and_i_* *= -1 . |_| Y1 Y2 Y3 Y4 Figure 7-a B1 0 1 2 3 Sublemma (6) Let 2 in R2 be a linear 2-simplex. Consider a self-homeomorphism ' of a 1-simplex 1 that is a face of 2. Suppose that ' fixes the end points @ 1* * and is pl on Int 1 = 1 - @ 1. Then ' can be extended to a self-homeomorphism of 2 that is pl on 2 - @ 1 and fixes pointwise @ 2 - Int 1. Proof of (6). Form an infinite but locally finite linear triangulation of 2 - * *@ 1 such that ' is linear on simplices _ see Figure 7-b. Then, on 2 - @ 1, the extensio* *n is the unique symplexwise linear map that fixes all vertices outside of 1; then o* *ne further_ extends by the identity on the two points @ 1 to define on all of 2. * * |_| Figures 7-b and 7-c 8 < (3) 2: 1 Lemma (7) Consider a self-homeomorphism ' of @B2 that respects a finite set Y * *of N points in @B2 and is pl on @B2-Y . Such a ' can be extended to an self-homeomor* *phism of B2 that is pl on B2 - Y . Proof of (7). By Lemmas (2) and (3), it suffices to prove this when OE fixes X * *pointwise. We initially assume that N 3. Identify B2 by a pl homeomorphism to the standa* *rd solid regular N -gon (N), sending Y to the vertices. Triangulate (N) by coni* *ng the natural triangulation of the N -gon frontier to the origin of R2; see Figure 7-* *c. Applying Lemma (6) to the N resulting 2-simplexes simultaneously, but independently, we * *get the wanted extension . If N = 0, then Lemma (2) establishes Lemma (7). If N = 1, (respectively N = 2), then the proof for N = 4 applied with data sy* *mmetric under rotation of angle ss=2 (respectively angle ss ) establishes Lemma (7) by * *passage_to the pl orbit space, which is a pl 2-disk. * * |_| In x5, we mentioned that (APLST) follows from the known pl classification of * *surfaces. Here is the relevant part of the classification; it is formulated to additional* *ly help to prove the Side Approximation Lemma (SAL) in the next section. Definition A pl surface M is irreducible if every pl circle embedded in M is * *the boundary of a pl 2-disk in M . Noncompact Irreducible PL Surface Classification Theorem (PLCT) Let M be a connected noncompact irreducible pl surface. (i) If @M = ;, then M ~=R2. (ii)If M has N 1 boundary components, then M ~=B2 - bXwhere Xb is any set of N points of @B2 = S1. 12 For the proof, we fix any pl triangulation T of M . Proof of (i). Arbitrarily large connected compact subcomplexes K of T are easil* *y con- structed. For " > 0 small, the closed "-neighborhood L = N"(K) of K in the simp* *licial metric of T is a compact pl submanifold of M . By irreducibility of M , any b* *ound- ary component C of L, is the boundary of a pl 2-disk BC in M . The connected * *set IntL K lies entirely in one of (a) BC or (b) M - BC . Case (b) cannot hold * *for all the finitely many components C of @L, for then M would be compact. Hence case (* *a) occurs. We now know that M is an ascending union of open sets IntBC ~=IntB2 ~=R2. But* * it is well known that any pl manifold that is an ascending union of open sets ~=Rn* * is itself ~=Rn . The proof is an interesting exercise. See [RourS]. * * __|_| Exercise Combining (i) above with (PL JCT) and (PLST) prove the following: If X* * is__ a compact connected subset of S2 and U is any component of S2 - X , then U ~=R2* *. |_| Proof of (ii) for N = 1. Build L an arbitrarily large compact connected pl subm* *an- ifold as for case (i), but assure that that meets @M . Consider its frontier ff* *iL in M . It is a compact pl 1-manifold and each component of ffiL is a circle C in IntM or * *an arc J with @J = J \ @M . A disk BC with boundary a circle component C of ffiL cann* *ot contain L since it lies in IntM . For each arc component J of ffiL, there is a * *unique arc J0 @M with the same boundary @J = @J0. Together they form a circle CJ , which* *, by irreducibility of M , bounds a pl disk BJ in M . Arguing as for (i), we see tha* *t exactly one such BJ contains L. We now know that M is an ascending union of open sets IntBJ ~=R2+~=R1x [0, 1)* * ~= B2 - {i}. But any pl manifold that is an ascending union of open sets ~= Rn+ i* *s itself ~=Rn+. Alternatively, one can apply Lemma (5). * * __|_| Proof of (ii) for N > 1. We induct on N . Since M is connected, there exists a pl path g : [0, 1] ! M joining two distinct components of @M and meeting @M only at its endpoints. By a canonical shortcutting procedure, one can make g an embe* *dding onto a pl arc fl g([0, 1]); for more on this shortcutting, see the proof of (* *PLCL) in the next section; it was already used by Jordan [Jord2] in 1887. Splitting M at* * fl yields two components M1 and M2 with intersection fl . Indeed, if the splitting yielde* *d just one component, M would not be irreducible. Each Mi is a connected noncompact irredu* *cible pl surface with Ni 1 boundary components. Further, N = N1 + N2; whence N1 and N2 are < N . Applying induction on N and Lemma (7), part (ii) easily follows. I* *n more detail: choose a plJordan arc J in B2-Xb so that J \S1 = @J , and J splits B2 i* *nto two pl disks Bi containing respectively subsets Xbiof Ni points of Xb. Then, induct* *ive hy- pothesis offers plhomeomorphisms hi: Bi-Xbi! Mi, and Lemma (7) allows us to adj* *ust them so that the restrictions h1|J and h2|J induce one and the same pl homeomor* *phism onto fl . Now h1 and h2 together define the required pl homeomorphism B2 - bX~=* *M_. |_| The following was used directly (but for N = 3 only) in proving (ST) in x6. Almost pl Schoenflies Theorem (APLST) Let C be a Jordan curve in R2 that is pl (= piecewise linear) except at a (possibly empty) finite set X of N = N(X) o* *f points in C . A homeomorphism H can be built from the standard pldisk B2 onto the sealed * *Jordan domain B with frontier C ; furthermore, H can be pl except possibly at h-1X @* *B2. Proof of (APLST) for N = 0. This is exactly (PLST). __|_| Proof of (APLST) for N > 0. By (PLCT) there is a pl homeomorphism H0 : B2 - bX-! B - X where Xb and X are sets of exactly N points in S1 and C respec- tively. The following Compactification Lemma (8) from general topology assures * *us that there is an extension of H0 to a continuous surjection H : B2 ! B with H(Xb) = * *X . Since a surjection of sets of N elements is bijective, H is bijective. Then, by* *_compactness, H is a homeomorphism (see x2). |* *_| 13 Compactification Lemma (8) Let X and Y be compact Hausdorff spaces; let A and B be finite subsets of X and Y respectively such that X - A is dense in X and * *Y - B is dense in Y . Consider a continuous map g : X - A -! Y - B that is proper in the* * sense that, for every compact set K in Y - B , the preimage g-1(K) is compact in X - * *A . Suppose that each point a 2 A has arbitrarily small compact neighborhoods Xa in* * X such that Xa - {a} is connected. Then g extends uniquely to a continuous map G : X !* * Y . Moreover, G is surjective if g is surjective. Remarks. (i) The connectivity assumption is essential, as the homeomorphism (S1-(1, 0)) * * IntB1 reveals. (ii)This lemma can be regarded as a fragment of the `end compactification' theo* *ry of K'er'ekjart'o and Freudenthal. (iii)The assumption that the Xa in (8) are compact is inessential because the c* *losure of a connected set is always connected. Proof of (8) It suffices to establish the existence of a continuous extension G* * of g . Indeed, the asserted uniqueness of G follows from the density of X - A in X . * *The implication "g surjective implies G surjective", follows from the density of Y * *- B in Y . As for existence, we begin by disposing of the case when B consists of a sing* *le point. Then Y is the well known Alexandroff or one-point compactification of the space* * (Y - B) by the `infinity' point B . The extension G is obtained by sending all of A to * *B . Its conti- nuity is an easily proved and standard fact about Alexandroff compactification,* * cf. [Dugu]. Note that, in this case, the assumptions mentioning connectivity are superfluou* *s; also A need only be compact, rather than finite. It remains to prove existence of G in the case when B is 2 points. Assertion (*) With the data of (8), for any point a in A, the following exist:* * a point b in B ; a compact neighborhood Yb of b in Y with Yb \ B = {b}; and a compact neighborhood Xa of a in X such that g(Xa - {a}) Yb- {b}. Proof that G exists if (*) is true. With the data provided by (*), consider, for each a in A the (proper!) restriction ga : Xa - {a} -! Yb- {b} of g . The prove* *d case of (8) where B is a single point assures us a continuous extension of this ga to a* * continuous map Ga : Xa ! Yb ,! Y . These Ga together yield an extension G of g . It is con* *tinuous since g and these Ga are continuous and agree on the open cover of X formed by * *X_-_A and the interiors in X of these Xa. * * |_| Proof of (*) _ using connectivity. Choose, for the points b in B , pairwise d* *is- joint compact neighborhoods Yb in Y ; we denote by YB their union. Since YB -* * B is a neighborhood of (Alexandroff) infinity of Y - B , (i.e. Y - YB has compact c* *losure in Y - B ), while g is proper, it follows that g-1(YB - B) is a neighborhood of in* *finity in X - A, whence A [ g-1(YB - B) is a neighborhood of A in X . For any given a in * *A, choose a neighborhood Xa of a in X , with (Xa - {a}) connected, and so small th* *at (Xa - {a}) g-1(YB - B). As the preimages g-1(Yb - {b}) form a closed partiti* *on of g-1(YB - B), this connectedness implies that (Xa - {a}) lies entirely in som* *e_one g-1(Yb- {b}). |_| It remains to prove (APLST+), which supported (APLST) in x6. Complement (APLST+) to (APLST) _ see x5. If f: @B2 ! C is a homeomorphism that is pl except at f-1 (X), then H : B2 ! B offered by (APLST) can be (re)chosen to extend f . Proof of (APLST+). This follows immediately from (APLST) and Lemma (7). __|_| 8 Proof of the Side Approximation Lemma (SAL) _ see x5 This is the second and last section devoted to tools for the proof of the Sch* *oenflies Theorem (ST). We begin with a couple of lemmas valid in all dimensions. 14 Linear Access Lemma (LAL), cf. [Vebl][Brou1]. Consider any open subset D of Rn with its closure B and its frontier C . In C consider any point P and an " > 0.* * There exists a compact linear arc L in B such that L \ C is a single point "-near to * *P . If C is pl near P then L \ C can be P itself. U 0 Q Figure 8-a P C P Proof of (LAL) _ see Figure 8-a. The case when C is pl near P is immediate from the locally cone-like behavior of a pl object [RourS]. Otherwise, since C is th* *e frontier of D = B - C , there is a point Q of D that is "-near to P (for Euclidean distance* *). This assures that the oriented linear segment = 1(Q, P ) in Rn running from Q to * *P is also "-near to P . Letting P 0be the first point on that lies on C , the line* *ar_segment L = 1(Q, P 0) establishes the lemma. * * |_| PL Chord Lemma (PLCL) Given again the data of (LAL), suppose that D is con- nected, and that P 06= P is a second given point in C . Then there exists a co* *mpact pl arc L in B such that L \ C = @L and L joins a point "-near P to a point "-near * *P 0. Furthermore, if C is pl near P and near P 0, then one can have @L = {P, P 0}. Proof of (PLCL). Applying (LAL) twice, and then the connectivity of B - C assur* *ed by (JCT), we get a pl path ~ : [0, 1] ! B so that = ~([0, 1]) is as required * *for L, except for one fault: ~ may not be injective. Taking, in order, all possible sh* *ortcuts to ~ offered by the self-intersections of this path, we (canonically!) derive an inj* *ective_pl path whose image L is a pl Jordan arc with the required properties. * * |_| Data We now restrict to dimension 2 and consider a Jordan curve C R2 that by * *(JCT) is the frontier of a unique Jordan domain D . In the sealed Jordan domain B = C* * [ D we consider what has been called an nice chord J of B , which is, by definition, a* * Jordan arc J in B such that @J = J \ C and J \ D is pl. A nice sector X of B is define* *d to be a sealed Jordan domain in R2 with frontier J [ J0 where J is a nice chord of* * B and J0 is a Jordan arc in C with @J = @J0. Side Compression Lemma (SCL) With the above data, let J be an nice chord of B and let X be the nice sector of B whose frontier in R2 is the Jordan curve J [ * *J0. Then for any open neighborhood U of J0 in B there exists another nice sector X0 of B* * such that (a) X0 U \ X , (b) X0\ C = J0, and (c) the frontier J0 of X0 in B is an * *nice chord of B that coincides with J near C . Proof of (SCL). The pl manifold X - J0 with boundary J is irreducible by (PLST) of the last section; hence, by the pl classification theorem of the last sectio* *n, there is a pl homeomorphism H : R1 x [0, 1) ! X - J0. Since X - U is a compact set in X - J0, it is contained in the image by H of any square [-r, r] x [0, 2r] with r > 0 su* *fficiently_ large. Then the required nice sector of B can be X0 = X - H( (-r, r) x [0, 2r) * *). |_| Remark Nothing has been done to assure nice convergence of H in B ; it is quite* * pos- sible, for example, that the image by H of some infinite linear ray in R2+= R1 * *x [0, 1) (but not in R1 x {0}), has all of J0 as limit set. Peripheral Niceness Proposition (PNP) With the same data, let p be any point of the Jordan curve C . There exist arbitrarily small compact neighborhoods of p i* *n B that are nice sectors of B . Proof of (PNP). Let V be any open neighborhood of p in B . By the pl Chord Lemma (PLCL), there exists a nice sector X of B such that X \ C is a Jordan arc* * J0 and p 2 (J0 - @J0). Now (SCL) provides a nice sector X0 V of B with X0\ C = * *J0.__ It is the required neighborhood of p in B . * * |_| Piecewise Linear Access Theorem (PLAT) Let C R2 be a Jordan curve and B its sealed Jordan domain. 15 (i) For any point p in C , there exists a Jordan arc J in the sealed Jordan reg* *ion B with frontier C such that J \ C = p and J - p is pl. (ii)For any two distinct points p and q in C , there exists a Jordan arc J in B* * such that J \ C = {p, q} and J - {p, q} is pl. Proof of (i). By the Peripheral Niceness Proposition (PNP) there exists an infi* *nite de- scending sequenceTX1, X2, X3, . .o.f compact neighborhoods of p in B such that * *their intersection iXi is p and their respective frontiers in B are pairwise disjoi* *nt and nice chords Ji of B . Choose a point pi in Ji-@Ji. Let Yi be the compact closure of * *Xi-Xi+1. Its frontier ffiYi in R2 is a Jordan curve containing the disjoint Jordan arcs * *Ji and Ji+1 By (PLCL)Sthere is a pl Jordan arc Ki in Yi such that @Ki = {pi, pi+1}. The un* *ion_ {p} [ iKi is the wanted Jordan arc J . * * |_| Proof of (ii). One proof is similar. Alternatively, one can piece together two * *arcs_offered by (i) using the method of the proof of (PLCL). * * |_| The result needed in the proof of (ST) is now within easy reach. Side Approximation Lemma (SAL) Let J be any Jordan arc in the Jordan curve C R2 and let B R2 be the sealed Jordan domain with frontier C . There exis* *ts a Jordan arc J0 B with the same end points {P0, P1} as J , such that J0\ C = {P* *0, P1} and J0- C is pl. Further, one can choose J0 in any prescribed neighborhood of J* * . Proof of (SAL). This is an immediate consequence of part (ii) of (PLAT) and the* *_Side Compression Lemma (SCL). |_| 9 Some consequences of (ST) and (PLST) Many proofs will be left partly or wholly as exercises _ hopefully all pleasa* *nt! Unknotting Theorem [Scho4, x13]. If C and C0 are Jordan curves in R2 there exis* *ts a self-homeomorphism H : R2 ! R2 such that H(C) = C0. Furthermore, if a homeomo* *r- phism h : C ! C0 is given, then H can coincide with h on C . Hints for proof. Given h, (ST) of x4 provides a homeomorphism H0 : B ! B0 ex- tending h where B and B0 are the sealed Jordan domains with frontiers C and C0 respectively. The same argument extends H0 to a self-homeomorphism H1 of the 1-* *point compactification R2[1 = S2. One can adjust H1 on R2-B to fix 1. Then the requir* *ed_ H is the restriction of H1 to R2. * * |_| Graph Data A graph is a topological space homeomorphic to a locally finite simp* *licial 1-complex. Consider a graph embedded as a closed subset of a topological 2-ma* *nifold M without boundary. Local Graph Taming Theorem (LGTT) With above data, let P be any point in . There exists an open neighborhood U o* *f P in M and an embedding f : U ! R2 such that the image h(U \ G) is a pl subset of R2. Here are three exercises leading to a proof of (LGTT) from (ST). (1) Without loss of generality, is homeomorphic to the cone on N 1 points a* *nd M is S2. (This is assumed for the rest of the proof.) * * __|_| (2) The case N 2. Hints for (2). Suppose then that N is 1 or 2. The graph is then a Jordan arc * *in R2. Identify S2 R2 [ 1 by homeomorphism, and form the 2-fold branched cyclic cove* *ring of S2 branched at the two end points @G. In it apply (ST) to the Jordan curve c* *overing_ and examine the quotient map to S2 R2. * *|_| (3) The case N 3. Hints for (3). Proceed by induction on N . Choose a Jordan arc J in joining a* *ny pair among the N distinct extremal (i.e. non-separating) points. Then form the* * 2-fold branched cyclic covering of S2 branched at the two end points @J , and complete* * the_ induction by proceeding much as for (2). * * |_| 16 Next come basic uniqueness theorems for pl structures on 2-manifolds. See [Mo* *is2] for alternative proofs also accessible to students. Surface Hauptvermutung (SH) Consider any topological homeomorphism g : M ! W of pl 2-manifolds. Let A M be closed and B M be compact. Suppose that g is pl near A. Then there exists another homeomorphism h : M ! W that is pl ne* *ar A [ B , and coincides with g near A and outside a compact subset of M . Surface Hauptvermutung with Approximation (SH+) Consider any topological homeomorphism g : M ! W of metric pl 2-manifolds. Let A be a closed subset of M near which g is pl, and let " : M ! R be any strictly positive continuous fun* *ction. Then there exists a self-homeomorphism fl : M ! M fixing all points near A, suc* *h that gfl : M ! W is a pl homeomorphism, and such that, for all points x in M , the d* *istance in M from x to fl(x) is < "(x). These statements make sense in all dimensions (i.e. with any integer n > 0 in* * place of 2) and are collectively known as the "Hauptvermutung for manifolds" (first f* *ormulated in print by Steinitz in 1907). However, they are only true for n 3 (see [Mois* *1-2]), or with extra hypotheses (see [KirS1] and [FreeQ]). See [Sieb] for more historical* * notes and references. The proofs will be rather formal consequences of the following three `handle * *lemmas': H(2, 0), H(2, 1), H(2, 2): H(2, k) Handle lemma for dimension 2 and index k 2, _ cf. [KirS1]. The Surface Hauptvermutung (SH) holds true under the following set of extra con* *ditions: (i) M = R2 and W is an open subset of R2. (ii)The closed set A is empty if k = 0 and is (Rk - IntBk) x R2-k in general. (iii)The compact set B (which is is called the k-handle core) is the origin if * *k = 0 and is in general the k -disk Bk x 0. Condition (i) permits us to use in M and W our results on Jordan domains. The following exercises outline proofs of these lemmas needing little more th* *an (ST) and (PLST). (A) Hints for H(2, 0). Apply (ST+) in W to the Jordan curve g(@ 2B2). * *__|_| (B) Hints for H(2, 2). Choose ~ < 1 so near to 1 that h is pl on (R2- @~B2). Th* *en__ apply (PLST) just once to the pl circle h(R2 - @~B2) in W R2. * * |_| (C) Hints for H(2, 1) _ the hardest case. Figure 9-a R2 anInd W(,STapply) (PLST)twonce,i* *ce, as indicated in Figure 9-a, to three adjacent se* *aled (ST) Use (PLST) Jordan domains. For convenience Jordan arcs here. we have identified W to Mo where pl for o B1 x {0} o is the unique pl manifold stru* *c- ture on M making h : Mo ! W (ST) a pl homeomorphism. The two (2nd factor) shared pl Jordan arcs in the frontier of the pl sealed Jordan Here o is domain are provided by the easy standard. (1st factor) pl Chord Lemma (PLCL) of x8. The coning lemmas of x7, particularly the pl and topological versions of Lemma * *(2),_can_ serve to fit together the embeddings that (PLST) and (ST) provide. * * |_| Remarks (1) The proof for index 0, in case (A) above, could imitate case (C) using, in * *place of (ST+), the simpler statements (ST) and (PLST). (2) The use of (ST) in the above proof is very mild since the Jordan curves met* * obviously can be (bi-)collared, cf. [Brow2][KirS2]. Thus, for example, the proof of (ST) * *by M. Brown [Brow1] under this stronger hypothesis (it is valid for all dimensions) could h* *ere replace our proof of (ST). 17 (3) Some classical proofs of (SH) use no more than (PLST), but they seem to sha* *re some of the complexities of our proof of (ST). (4) The Alexander isotopy theorem of [Alex4] shows that the homeomorphism h pro* *mised by H(2, k) is (topologically) isotopic to g fixing points in A and all points o* *utside a com- pact set in R2. We delay the indications for the proof of (SH) because the proof of (SH+) see* *ms more interesting. Definition For any (closed) simplex oe of a simplicial complex T , the star St(* *oe) of oe in T is the union of all (closed) simplices of T that contain oe . Proof of (SH+). Let U be an open neighborhood of A in M on which g is pl. Choose a pl triangulation T of M so fine that: (a) if a simplex oe of T intersects A* * then it lies in U , (b) for any vertex v of T the diameter of its star neighborhood St(v, T* * ) in T is less than the minimum of " on the compact set St(v), and (c) some open neighbor* *hood of g(St(v)) in W is a pl chart of W in the sense that it admits an open pl embe* *dding into R2. For each vertex v of T , pl identify to R2 a small open neighborhood N(v) S* *t(v) while assuring that the N(v) are disjoint for distinct vertices v . Apply H(2, * *0) in N(v) to obtain a self-homeomorphism gv of N(v) with compact support in N(v) such tha* *t the composition gflv is pl near v . Doing this independently for all such N(v), we * *obtain a self-homeomorphism g0 of M so that the composition h0 = gfl0 : M ! W is pl near* * all vertices of T . Next, for each edge e of T disjoint from A, pl identify to R2 = R1 x R1 a sm* *all open neighborhood N(e) St(e) of e - @e in M , in such a way that: (a) R1 x 0 corre* *sponds to e-@e, (b) the homeomorphism h0 is pl near (R1-IntB1)xR1, and (c) the sets N(* *e) are disjoint for distinct edges. Apply H(2, 1) in N(e) to obtain a self-homeom* *orphism ge of N(e) with compact support in N(e) such that the composition he = h0fle is* * pl near e. Doing this independently for all such N(e), we obtain fl1 : M ! M so * *that h1 = h0fl1 = hfl0fl1 is pl near all vertices and edges of T . Finally, for each face (2-simplex) f of T disjoint from A, let N(f) be f - @* *f pl iden- tify it to R2 in such a way that h1 is pl near R2- IntB2. The sets N(f) are nec* *essarily disjoint for distinct faces f . Apply H(2, 2) in N(f) to obtain a self-homeomor* *phism gf of N(f) with compact support in N(f) such that gflf is pl near f . Doing this i* *ndepen- dently for all such N(f), we obtain fl2 : M ! M such that that h2 = h1fl2 is pl* * on all simplices of T , hence on all of M . Observe that, by construction, each of fl1, fl2, fl3 respects the vertex star* * St(v) for each vertex v of T . Hence the composition fl = fl0fl1fl2 does likewise and it follo* *ws, for_the metric d on M , that d(fl(x), x) < "(x) for all x. * * |_| Hints for proof of (SH). Carry through the proof of (SH+) above with the follow* *ing modifications: (a) Eliminate the approximation conditions _ say by choosing " identically 1 an* *d re- placing metric d by d=1 + d. (b) Replace T by the finite subcomplex of all (closed) simplices that meet the* * compact set B . The resulting self-homeomorphism fl of M then has compact support since it is a* * finite composition of homeomorphisms with compact support. The required h is the compo* *si-_ tion h = gfl . * *|_| Exercise Deduce (SH) from (SH+). __|* *_| Exercise Using Remark (4) following H(2, k), establish versions with isotopy_of* *_(SH) and (SH+). |_| Surface Triangulation Theorem (STT) _ see [Rad'o] 1925. Consider any topological 2-manifold M without boundary. There always exists a p* *l struc- ture o on M . Furthermore, if oe is a given pl structure on an open subset U of* * M , then o can coincide with oe on U . 18 Proof of (STT). This proof is valid for non-metrizable manifolds. Zorn's Lemma * *clearly applies to the partially ordered set of all pairs (V, ff) consisting of an open* * subset V of M and a pl structure ff on V _ the ordering being by pl inclusion. Thus, there* * is a maximal pair (V, o) containing (U, oe). Seeking a contradiction, suppose V is not all of M ; select a point x in M -* * V and an open neighborhood W of x in M that is homeomorphic to R2; use the homeomorphism to endow W with a pl structure fi . Let " : W ! R be a continuous function th* *at is positive on W \ V and zero elsewhere on W . (SH+) then provides a pl homeomorp* *hism f : (W \ V )fi! (W \ V )o that is "-near the identity map of W \ V , and hence * *extends by the identity to a pl isomorphism g : Wfi! Wg(fi). Now g(fi) and o agree on (* *W \ V ) and hence define a pl structure on (W [ V ) showing that (V, o) is not maximal,* *_the_ contradiction proving (SH+). * *|_| Remarks on boundaries and embedded graphs (a) In this section we have paid little or no attention to 2-manifolds with non* *-empty boundary. However, similar results can easily be deduced for them with the hel* *p of M. Brown's collaring theorems [Brow1], [KirS2, App A of Essay I,]. (b) In a similar manner, one can deduce versions of (STT) and (SH) for pairs (M* *, ) with a graph as introduced for (LGTT). Indeed, using (LGTT), one can adapt M. Brow* *n's collaring theorems to deal with a slightly generalized notion of collaring for * * in M . 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