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 References [BH02] Brown, R. and Higgins, P. J. `The fundamental groupoid of the quotient* * of a Hausdorff space by a discontinuous action of a discrete group is the orbit groupoid of* * the induced action'. arXiv math.AT/0212271 (2002) 18 pages. [BR84] Brown, R. and Razak Salleh, A. `A van Kampen theorem for unions of non* *-connected spaces'. Arch. Math. 42 (1984) 85-88. [Bra04] Braun, G. `A proof of Higgins' conjecture'. Bull. Austral. 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[Wil49] Wilder, R. L. Topology of manifolds, AMS Colloquium Publications, Vol* *ume 32. American Mathematical Society, New York (1949).