Groupoids, the Phragmen-Brouwer Property,
and the Jordan Curve Theorem
Ronald Brown
University of Wales, Bangor
January 3, 2006
University of Wales Math Preprint 06.01
Abstract
We publicise a proof of the Jordan Curve Theorem which relates it to th*
*e Phragmen-Brouwer
Property, and whose proof uses the van Kampen theorem for the fundamental *
*groupoid on a set of
base points.
1 Introduction
This article extracts from [Bro88] a proof of the Jordan Curve Theorem based on*
* the use of groupoids,
the van Kampen Theorem for the fundamental groupoid on a set of base points, an*
*d the use of the
Phragmen-Brouwer Property. In the process, we give two results on the Phragmen-*
*Brouwer Property
(Propositions 4.1, 4.3), which may not have been published other than in [Bro88*
*]. There is a renewed
interest in such classical results1, as shown in the article [Sie] which revisi*
*ts proofs of the Schoenflies
Theorem.
There are many books containing a further discussion of this area. For more*
* on the Phragmen-
Brouwer property, see [Why42 ] and [Wil49]. Wilder lists five other properties *
*which he shows for
a connected and locally connected metric space are each equivalent to the PBP. *
*The proof we give
of the Jordan Curve Theorem is adapted from [Mun75 ]. Because he does not have *
*our van Kampen
theorem for non-connected spaces, he is forced into rather special covering spa*
*ce arguments to prove
his replacements for Corollary 3.3 and for Proposition 4.1.
The intention is to make these methods more widely available, since the abo*
*ve book has been out
of print for at least ten years. A new edition is in preparation under the titl*
*e `Topology and groupoids'
which better reflects the character of the book.
_____________________________________
www.bangor.ac.uk/r.brown. This work was partially supported by a Leverhulme *
*Emeritus Fellowship (2002-2004).
1See also the web site on the Jordan Curve Theorem: http://www.maths.ed.ac.uk*
*/,aar/jordan/
1
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 2
I mention in the same spirit that the results from [Bro88] on orbit spaces *
*have been made available
in [BH02 ]. A further example of the use of groupoid methods, this time in comb*
*inatorial group theory,
is in [Bra04], which gives a new result combining the Kurosch theorem and a the*
*orem of Higgins
which generalises Grusko's theorem.
Note that we use groupoids not to give a nice proof of theorems on the fund*
*amental group of a
space with base point, but because we maintain that theorems in this area are a*
*bout the fundamental
groupoid on a set of base points, where that set is chosen in a way appropriate*
* to the geometry of the
situation at hand. The set of objects of a groupoid gives a spatial component t*
*o group theory which
allows for more powerful and more easily understood modelling of geometry, and *
*hence for more
computational power.
2 The groupoid van Kampen theorem
We assume as known the notion of the fundamental groupoid ss1XJ of a topologica*
*l space X on a set
J: it consists of homotopy classes rel end points of paths in X joining points *
*of J " X. We say the pair
#X, J# is connected if J meets each path component of X. The following theorem *
*was proved in [Bro67,
6.7.2].
Theorem 2.1 (van Kampen Theorem) Let the space X be the union of open subsets *
*U, V with intersec-
tion W, let J be a set and suppose the pairs #U, J#, #V, J#, #W, J# are connect*
*ed. Then the pair #X, J# is
connected and the following diagram of morphisms induced by inclusion is a push*
*out in the category of
groupoids:
ss1WJ____//_ss1VJ
| |
| |
fflffl| fflffl|
ss1UJ____//_ss1XJ
This has been generalised to arbitrary unions in [BR84 ].
3 Pushouts of groupoids
In order to apply Theorem 2.1, we need some combinatorial groupoid theory. Thi*
*s was set up in
[Hig05], [Bro88], and here we just quote some first main facts, for example how*
* to compute an object
group H#x# of a groupoid H # G=R given as the quotient of a groupoid G by a tot*
*ally disconnected
graph R # fR#x# j x 2 Ob#G#g of relations: of course G=R is defined by the obvi*
*ous universal property,
and has the same object set as G.
Recall from [Bro88, 8.3.3]that:
Proposition 3.1(a) If G is a connected groupoid, and x 2 Ob#G#, then there is a*
* retraction r # G ! G#x#
obtained by choosing for each y 2 Ob#G# an element oy 2 G#x, y#, with ox # 1x.
(b) If further R # fR#y# j y 2 Ob #G#g is a family of subsets of the object gro*
*ups G#y# of G, then
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 3
the object group #G=R##x# is isomorphic to the object group G#x# factored by th*
*e relations r#ae# for all
ae 2 R#y#, y 2 Ob#G#.
We assume as understood the notion of free groupoid on a (directed) graph. *
*If G, H are groupoids
then their free product G H is given by the pushout of groupoids
Ob#G# " Ob#H#___i__//H
j|| ||
fflffl| fflffl|
G __________//G H
where Ob#G# " Ob#H# is regarded as the subgroupoid of identities of both G, H o*
*n this object set, and
i, j are the inclusions. We assume, as may be proved from the results of [Bro88*
*, Chapter 8]:
Proposition 3.2If G, H are free groupoids, then so also is G H.
If J is a set, then by the category of groupoids over J we mean the categor*
*y whose objects are
groupoids with object set J and whose morphisms are morphisms of groupoids whic*
*h are the identity
on J.
Proposition 3.3Suppose given a pushout of connected groupoids over J
i
C _____//_A
| | *
* (1)
j || |u|
fflffl|fflffl|v
B _____//_G
Let p be a chosen element of J. Let r #A! A#p#, s # B ! B#p# be retractions obt*
*ained by choosing elements
ffx 2 A#p, x#, fix 2 B#p, x#, for all x 2 J, with ffp # 1, fip 2 1. Let fx # #u*
*ffx# 1#vfix# in G#p#, and
let F be the free group on the elements fx, x 2 J, with the relation fp # 1. Th*
*en the object group G#p# is
isomorphic to the quotient of the free product group
A#p# B#p# F
by the relations
#rifl#fx#sjfl# 1fy1 # 1 *
* (2)
for all x, y 2 J and all fl 2 C#x, y#.
Proof We first remark that the pushout (1) implies that the groupoid G may be p*
*resented as the
quotient of the free product groupoid A B by the relations #ifl##jfl# 1 for a*
*ll fl 2 C. The problem is
to interpret this fact in terms of the object group at p of G.
To this end, let T, S be the tree subgroupoids of A, B respectively generat*
*ed by the elements ffx, fix,
x 2 J. The elements ffx, fix, x 2 J, define isomorphisms
' # A ! A#p# T, _ # B ! B#p# S
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 4
where if g 2 G#x, y# then
'g # ffy#rg#ffx1, _g # fiy#sg#fix1.
So G is isomorphic to the quotient of the groupoid
H # A#p# T B#p# S
by the relations
#'ifl##_jfl# 1 # 1
for all fl 2 C. By Proposition 3.1, the object group G#p# is isomorphic to the *
*quotient of the group
H#p# by the relations
#r'ifl##r_jfl# 1 # 1
for all fl 2 C.
Now if J0 # J n fpg, then T, S are free groupoids on the elements ffx, fix,*
* x 2 J0, respectively. By
Proposition 3.2, and as the reader may readily prove, T S is the free groupoi*
*d on all the elements
ffx, fix, x 2 J0. It follows from [Bro88, 8.2.3] (and from Proposition 3.1), th*
*at #T S##p# is the free
group on the elements rfix # ffx1fix # fx, x 2 J0. Let fp # 1 2 F. Since
r'ifl # rifl, r_jfl # fy#sjfl#fx1,
the result follows. *
* 2
Remark 3.4 The above formula is given in essence in van Kampen's paper [Kam33 ]*
*, since he needed
the case of non connected intersection for applications in algebraic geometry. *
*However his proof is
difficult to follow, and a modern proof of the case of connected intersection w*
*as given by Crowell in
[Cro59]. *
* 2
There is a consequence of the above computation which we shall use in the n*
*ext section in proving
the Jordan Curve Theorem.
First, if F and H are groups, recall that we say that F is a retract of H i*
*f there are morphisms
- # F ! H, ae # H ! F such that ae- # 1. This implies that F is isomorphic to a*
* subgroup of H.
Corollary 3.5Under the situation of Proposition 3.3, the free group F is a retr*
*act of G#p#. Hence if
J # Ob#C# has more than one element, then the group G#p# is not trivial, and if*
* J has more than two
elements, then G#p# is not abelian.
Proof Let M # A#p# B#p# F, and let -0 # F ! M be the inclusion. Let ae0 # M*
* ! F be the retraction
which is trivial on A#p# and B#p# and is the identity on F. Let q # M ! G#p# be*
* the quotient morphism.
Then it is clear that ae0preserves the relations (2), and so ae0defines uniquel*
*y a morphism ae # G#p# ! F
such that aeq # ae0. Let - # qi0. Then ae- # ae0i0# 1. So F is a retract of G#p*
*#.
The concluding statements are clear. *
* 2
We use the last two statements of the Corollary in sections 4 and 5 respect*
*ively.
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 5
4 The Phragmen-Brouwer Property
A topological space X is said to have the Phragmen-Brouwer Property (here abbre*
*viated to PBP) if X is
connected and the following holds: if D and E are disjoint, closed subsets of X*
*, and if a and b are points
in X n #D [ E# which lie in the same component of X n D and in the same compone*
*nt of X n E, then a and b
lie in the same component of X n #D [ E#. To express this more succinctly, we s*
*ay a subset D of a space
X separates the points a and b if a and b lie in distinct components of X n D. *
*Thus the PBP is that: if
D and E are disjoint closed subsets of X and a, b are points of X not in D [ E *
*such that neither D nor
E separate a and b, then D [ E does not separate a and b.
A standard example of a space not having the PBP is the circle S1, since we*
* can take D # f#1g,
E # f 1g, a # i, b # i. This example is typical, as the next result shows. But*
* first we remark that
our criterion for the PBP will involve fundamental groups, that is will involve*
* paths, and so we need
to work with path-components rather than components. However, if X is locally p*
*ath-connected, then
components and path-components of open sets of X coincide, and so for these spa*
*ces we can replace
in the PBP `component' by `path-component'. This explains the assumption of loc*
*ally path-connected
in the results that follow.
Proposition 4.1Let X be a path-connected and locally path-connected space whose*
* fundamental group
(at any point) does not have the integers Z as a retract. Then X has the PBP.
Proof Suppose X does not have the PBP. Then there are disjoint, closed subsets *
*D and E of X and
points a and b of X n #D [ E# such that D [ E separates a and b but neither D n*
*or E separates a and b.
Let U # XnD, V # XnE, W # Xn#D[E# # U"V. Let J be a subset of W such that a, b *
*2 J and J meets
each path-component of W in exactly one point. Since D and E do not separate a *
*and b, there are
elements ff 2 ss1U#a, b# and fi 2 ss1V#a, b#. Since X is path-connected, the pa*
*irs #U, J#, #V, J#, #W, J#
are connected. By the van Kampen Theorem 2.1 the following diagram of morphism*
*s induced by
inclusions is a pushout of groupoids:
i1
ss1WJ________//ss1UJ
| |
| |
i2|| |u1|
| |
fflffl| fflffl|
ss1VJ___u2__//_ss1XJ.
Since U and V are path-connected and J has more than one element, it follow*
*s from Corollary 3.5
that ss1XJ has the integers Z as a retract. *
* 2
As an immediate application we obtain:
Proposition 4.2The following spaces have the PBP: the sphere Sn for n > 1; S2n *
*fag for a 2 S2; Sn n
if is a finite set in Sn and n > 2. *
* 2
In each of these cases the fundamental group is trivial.
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 6
An important step in our proof of the Jordan Curve Theorem is to show that *
*if A is an arc in S2, that
is a subspace of S2 homeomorphic to the unit interval I, then the complement of*
* A is path-connected.
This follows from the following more general result.
Proposition 4.3Let X be a path-connected and locally path-connected Hausdorff s*
*pace such that for each
x in X the space X n fxg has the PBP. Then any arc in X has path-connected comp*
*lement.
Proof Suppose A is an arc in X and X n A is not path-connected. Let a and b lie*
* in distinct path-
components of X n A.
By choosing a homeomorphism I ! A we can speak unambiguously of the mid-poi*
*nt of A or of
any subarc of A. Let x be the mid-point of A, so that A is the union of sub-ar*
*cs A0 and A00with
intersection fxg. Since X is Hausdorff, the compact sets A0and A00are closed in*
* X. Hence A0n fxg and
A00n fxg are disjoint and closed in X n fxg. Also A n fxg separates a and b in *
*X n fxg and so one at least
of A0, A00separates a and b in X n fxg. Write A1 for one of A0, A00which does s*
*eparate a and b. Then
A1 is also an arc in X.
In this way we can find by repeated bisection a sequence Ai, i > 1, of sub-*
*arcs of A such that for
all i the points a and b lie in distinct path-components of X n Ai and such tha*
*t the intersection of the
Aifor i > 1 is a single point, say y, of X.
Now X n fyg is path-connected, by definition of the PBP. Hence there is a p*
*ath ~ joining a to b in
X n fyg. But ~ has compact image and hence lies in some X n Ai. This is a contr*
*adiction. 2
Corollary 4.4The complement of any arc in Sn is path-connected. *
* 2
In this theorem the case n # 0 is trivial, while the case n # 1 needs a spe*
*cial argument that the
complement of any arc in S1 is an open arc. The case n > 2 follows from the abo*
*ve results.
5 The Jordan Separation and Curve Theorems
We now prove one step along the way to the full Jordan Curve Theorem.
Theorem 5.1 (The Jordan Separation Theorem)The complement of a simple closed cu*
*rve in S2 is not
connected.
Proof Let C be a simple closed curve in S2. Since C is compact and S2 is Hausdo*
*rff, C is closed, S2n C
is open, and so path-connectedness of S2n C is equivalent to connectedness.
Write C # A [ B where A and B are arcs in C meeting only at a and b say. Le*
*t U # S2 n A,
V # S2 n B, W # U " V, X # U [ V. Then W # S2 n C and X # S2 n fa, bg. Also X i*
*s path-connected,
and, by Corollary 4.4, so also are U and V.
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 7
Let x 2 W. Suppose that W is path-connected. By the van Kampen Theorem ??, *
*the following
diagram of morphisms induced by inclusion is a pushout of groups:
ss1#W, x#________//ss1#U, x#
| |
| |
| |
| i|
| |
| |
fflffl| fflffl|
ss1#V, x#________//ss1#X, x#.
j
Now ss1#X, x# is isomorphic to the group Z of integers. We derive a contradicti*
*on by proving that the
morphisms i and j are trivial. We give the proof for i , as that for j is si*
*milar.
Let f # S1 ! U be a map and let g # if # S1 ! X. Choose a homeomorphism h #*
* S2 n fag ! R2
which takes b to 0 and such that hg maps S1 into R2 n f0g. Since hg#S1# is comp*
*act, there is only
one unbounded component of R2 n g#S1#, and we may assume this contains 0. Again*
* since hg#S1# is
compact, there is an r > 0 such that hg#S1# ` B#0, r#. Choose a point y in R2 s*
*uch that jjyjj > r. Then
there is a path ~, say, joining 0 to y in R2n hg#S1#, since 0 lies in the unbou*
*nded component of this set.
Define G # S1 I ! R2 by
(
hg#z# ~#2t# if0 6 t 6 1_2,
G#z, t# #
#2 2t#hg#z# yif1_26 t 6 1.
Then G is well-defined. Also G never takes the value 0 (this explains the choic*
*es of ~ and y). So G
gives a homotopy in R2 n f0g from hg to the constant map at y. So hg is inesse*
*ntial and hence g is
inessential. This completes the proof that i is trivial. *
* 2
As we shall see, the Jordan Separation Theorem is used in the proof of the *
*Jordan Curve Theorem.
Theorem 5.2 (Jordan Curve Theorem) If C is a simple closed curve in S2, then th*
*e complement of C
has exactly two components, each with C as boundary.
Proof As in the proof of Theorem 5.1, write C as the union of two arcs A and B *
*meeting only at a and
b say, and let U # S2n A, V # S2n B. Then U and V are path-connected and X # U *
*[ V # S2n fa, bg
has fundamental group isomorphic to Z. Also W # U " V # S2n C has at least two *
*path-components,
by the Jordan Separation Theorem 5.1.
If W has more than two path-components, then the fundamental group G of X c*
*ontains a copy of
the free group on two generators, by Corollary ??, and so G is non-abelian. Thi*
*s is a contradiction,
since G ,#Z. So W has exactly two path-components P and Q, say, and this proves*
* the first part of
Theorem 5.2.
Since C is_closed in S2 and S2 is locally_path-connected, the sets P and_Q *
*are open in S2. It follows
that if x 2 Pn P then x =2Q, and hence Pn P is contained in C. So also is Q n Q*
*, for similar reasons.
We prove these sets are equal to C.
__ *
* __
Let x 2 C and let N be a neighbourhood_of x in S2. We prove N meets Pn P. S*
*ince Pn P is closed
and N is arbitrary, this proves that x 2 Pn P.
Groupoids, Phragmen-Brouwer and the Jordan Curve Theorem *
* 8
Write C in a possibly new way as a union of two arcs D and E intersecting i*
*n precisely two points
and such that x is in the interior with respect to D of D. Choose points p in P*
* and q in Q. Since S2n E
is path-connected, there is a path ~ joining p to q in S2n E. Then ~ must meet *
*D, since p_and_q lie in
distinct path-components_of S2 n E. In fact if s # supft 2 I # ~#0, t# ` Pg, th*
*en ~#s# 2 Pn P. It follows
that N meets Pn P.
__ __
So Pn P # C and similarly Q n Q # C. *
* 2
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