[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 .
This extension of (STT) and (SH) is greatly facilitated if one takes the time t*
*o show
first that there is a natural construction of a 2-manifold with boundary M suc*
*h that
IntM = M - and M maps naturally onto M sending @M onto , _ indeed im-
mersively wherever is a 1-manifold. For pl pairs (M, ), this M is well know*
*n and
called the splitting of M at ; its construction in terms of a pl triangulation*
* of (M, )
is obvious. Given M , the relevant (generalized) collarings of in M are in*
* natural
one-to-one correspondence with the usual sort of collarings for @M in M .
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