"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Wilmarth, Constance"@en . "2009-07-20T20:25:06Z"@en . "2000"@en . "Master of Science - MSc"@en . "University of British Columbia"@en . "This thesis examines some connections between topology and group theory, in particular\r\nthe theory of orderable groups. It investigates in close detail some landmark results on\r\nthis mathematical interface, beginning with Holder's Theorem, and touches upon some\r\nrecent results in this expanding field of research.\r\nSimply stated, Holder's Theorem asserts that Archimedean orderable groups are none\r\nother than subgroups of the group of real numbers under addition. Since Holder proved\r\nthis in 1902, only one significant refinement, due to Paul Conrad, has been made, so these\r\npowerful theorems provide the foundation for our understanding of orderable groups.\r\nIn particular this understanding has served topologists well. This thesis is mostly a\r\ndistillation of work done in connection with topological applications of the theory, which\r\nare surprisingly varied and diverse. Burns and Hale's work on local indicability and right\r\norderability is considered, as well as Bergman's study of the universal covering group of\r\nSL(2,R). In addition N. Smythe's extension of a classical result of Alexander's via the\r\nleft orderability of the fundamental groups of certain surfaces is investigated."@en . "https://circle.library.ubc.ca/rest/handle/2429/11015?expand=metadata"@en . "1835460 bytes"@en . "application/pdf"@en . "ORDERABLE GROUPS AND TOPOLOGY by CONSTANCE WILMARTH B.Sc. (Mathematics) University of Oregon, 1997 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics WevaccepA this thesis as conforming to thfe teiJ*rfrefl standard THE UNIVERSITY OF BRITISH COLUMBIA November 2000 \u00C2\u00A9 Constance Wilmarth, 2000 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University- of British Columbia, I agree that the Library shall make it freely available for refer-ence and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Date Ze=>ab>a=$- aba-1 > e b'1 > e, a contradiction as b > e => 6 _ 1 < e. It is natural to inquire, in the wake of such definitions and distinctions, why orderable groups might be worth studying. To give an idea of the value of orderability as a math-ematical tool, we note that if G is right orderable, then it is torsion free. For take g ^ e, say g > e. Then g2 > g => g3 > g2 ... =4> gn > e for every n \u00C2\u00A3 N, implying that gn zfz e Vn. It is known that if G is a right-orderable group and R is a domain, then the group ring RG has no zero divisors and only trivial units. It is also known that if G is an orderable group, then ZG embeds in a division algebra. So knowing that a group is right-orderable or orderable gives important information, not just about the group itself, but about some related mathematical structures as well. The purpose of this thesis is to investigate the theory of orderable groups in relation to topology, and is organized as follows. The second chapter is devoted to Holder's Theorem, which states that a group G is and Archimedean if and only if it is isomorphic to a subgroup of (R, +). Paul Conrad's refinement of this result, which allows for a weakening of the hypotheses of the theorem to right orderability and Archimedean, is noted. The next chapter positions right orderability in the larger group theoretic context with respect to the class of locally indicable groups. In this chapter major contributions by Burns and Hale and Bergman are investigated. In the final chapter, orderable groups are featured in their interactions with knot theory, and Neville Smythe's generalization 2 Chapter 1. Introduction of a result in classical knot theory is studied. 3 Chapter 2 Holder's Theorem As the study of orderable groups properly begins with Holder's characterization of Archimedean ordered groups, and as the proof of the result is obtainable in English only in the form of a brief sketch [22], it seems appropriate to begin this study of orderable groups and topology by working out the details of Holder's argument. But first we will need some preliminary definitions. Definition 2 Define the positive cone P C G of an orderable group G to be the set {x G G : x > e}. Definition 3 Let < be an order on an orderable group G. Forx G G define the absolute value of x as For x,y G G write x \y\ and yt > \x\, so xst = (xsY > > y* > \z\; it follows easily that x ~ z. So ~ is an equivalence relation; the equivalence classes so defined are called Archimedean classes. Definition 4 If all the non-identity elements of G are equivalent, then (G, <) is an Archimedean group, that is, an Archimedean order is one in which G has only two Archimedean classes. Note: in an Archimedean group, for all x, y such that x < y, x ^ e there exists n \u00C2\u00A3 Z with \xn\ > y. Now it is not the case that every ordered group is Archimedean. For instance, consider (Z x Z, +) with the dictionary order (ai, bi) < (a2, b2) if a\ < a2, or if ai = a 2 and b\ 0. Theorem 1 (Holder) Let < be an ordering on an orderable group G. Then < is an Archimedean order if and only if G is order-isomorphic to a subgroup of the additive group of the real numbers under the natural order. Proof. J (<=) (K, <) is Archimedean, and so any subgroup of it is. (=>) The strategy is to show first that G is an abelian group, and then to use this fact to define an isomorphism. Assume that < is an archimedean order on G. The claim is that G is abelian. For consider any t G G such that t > e. 5 Chapter 2. Holder's Theorem Case 1. Suppose that e < t < x for every x G P C G. Then G Archimedean implies the existence of an integer nx for each x such that tn* < x < tn*+l => e < xt~nx \u00E2\u0080\u00A2 x = tUx =>G=(t) So G is cyclic and hence abelian. Case 2. Assume now that there exists u G G such that e < u < t. If t < u2, that is, if e < u < t < u2, then t < u2 =>\u00E2\u0080\u00A2 < u , u~Hu~l < e => (wH)2 < t. So relabelling we have found u (choose u~lt above) such that e < u < u2 < t; thus in all cases we may assume that there exists u such that e < u < u2 < t. Now if G is not abelian, then there exist x,y G G such that commutator [x, y] ^ e. Without loss of generality take [x, y] > e. Then with [x, y] playing the role of t in the inequalities above we obtain, as G Archimedean implies that there exist m , n G Z such that um < x < um+l and un < y < un+1, that [x,y] < u~mu~num+lun+l = u2, which is a contradiction as e < u < u2 < [x, y]. Therefore we conclude that G is abelian. 6 Chapter 2. Holder's Theorem The next step is to show that G is order-isomorphic to a subgroup of (R, +) by defining explicitly an isomorphism G -\u00C2\u00BB (R, +) taking x G G to the limit of a certain sequence to be defined for each x. First fix some t > e, t G G. Given x G G and n G Z + , there exists m G Z (m dependent on x,n,t) such that tm < x2\" < tm+l. So for each x, for every n, define (n(x) = | e S so that tm < x2\" < tm+1. Claim: l im^oo (n(x) exists (for each x). To prove the claim, it suffices to show that {(n} is a Cauchy sequence, that is, it suffices to show that 1 Cn < Cn+1 < Cn + 7^ \u00E2\u0080\u00A2 Lemma: For any four elements a, b, c, d of a bi-orderable group with a < b and c < d, we have ac < 6d (and so, in particular, an < bn for every n e R Proof of Lemma: ac < be < bd. Now, if tm < x2n < tm+\ then it follows from the lemma that t2m < x2n+1 < t2^m+l). This implies that _m_2m 2(m+1) _ m + 1 J_ C\" \u00E2\u0080\u0094 2 \" \u00E2\u0080\u0094 2 n T T - < 2^+1 _ 2 \" ~ + 2\"' as desired. Thus the sequence is Cauchy, hence convergent. We are therefore in a position to define a map 4>>\u00E2\u0080\u00A2: G \u00E2\u0080\u0094>\u00E2\u0080\u00A2 (R, +), (a;) = lim^^oo Cn(^)-Claim: (j) is order-preserving (that is, x < y =>\u00E2\u0080\u00A2 (xy) = \"+2 = (n(x) + (n(y) + 2^n\ Then taking n oo we have Chapter 2. Holder's Theorem 4>(xy) = {x) + 4>{y). So 2\" Vn (as t < x). It follows that mn > 2\" - l = > f f > l - ^ > 0 V n > l = ^ l im n _ > 0 0 Cn(x) 7^ 0 as (n(x) is monotonically increasing. Thus (j>{x) ^0. Case 2: x < t. Here there exists N G N such that t < x2\". Again mn satisfies tmn < x2n < tm\"+1 Vn G N, implying that f,m\u00C2\u00BB < x^N+in-N) < t\u00E2\u0084\u00A2n+i_ B u t ^ > t ^ m n + x > 2n-N =>mn + l > 2n2~N =>mn> 2n2~N - 1 ^ ^ > ^ - J r = > l i m ^ Cn(x) > 0. Finally, x \u00C2\u00A3 P, x ^ e a ; - 1 G P (z-1) = -{x) ^ 0 (as x'1 ker by the above) =*> ^(x) ^ 0. \u00E2\u0080\u00A2 This, then, is the proof of Holder's Theorem. Paul Conrad [9] added a remarkable footnote in 1959: Theorem 2 (Conrad) / / G is right orderable and archimedean, then G is orderable. Thus by Holder's Theorem, G is order-isomorphic to a subgroup o/(R,+). Conrad's proof is short and simple, and depends upon the fact, easily verified, that G is orderable if and only if P is normal in G. 9 Chapter 3 Some Group Theory and Satan's Parkade In attempting to understand right orderable groups it becomes important to situate them in relation to other classes of groups. Within this context the class of locally indicable groups assumes particular prominence, as it has been shown that locally indicable implies right orderable, but right orderable does not imply locally indicable. We now turn to these results with a view to understanding a remarkable topological application, the universal covering group of SL(2,R), which we will denote by SL. Definition 5 A group G is said to be locally indicable if each of its non-trivial finitely generated subgroups can be mapped homomorphically onto a non-trivial subgroup o/Z. More generally, if X is a class of groups closed under forming isomorphic images then G is locally X-indicable if every non-trivial finitely generated subgroup admits a non-trivial homomorphism onto a group in X. The following results are stated without proof ([9], [8]): Theorem 3 (Conrad) A group G is right orderable if and only if for every finite subset {xi, ...,xn} C G that does not contain e, there exist Ci = \u00C2\u00B1 1 , 1 < i < n, such that e does not belong to the subsemigroup of G that is generated by {xf, ...,xe\u00E2\u0084\u00A2}. 10 Chapter 3. Some Group Theory and Satan's Parka.de Theorem 4 (Burns and Hale) If G is locally RO-indicable, then it is right orderable. Coro l la ry : If G is locally indicable, then G is right orderable, as Z is right orderable. An example of a right orderable group that is not locally indicable will be found in the universal covering group of SL(2, E). But this is not a simple construct and will require some additional work to understand. We begin with some definitions. Def in i t ion 6 A topological group G is a group that is also a topological space, sat-isfying the requirements that the multiplication map m : G x G \u00E2\u0080\u0094> G sending (x, y) to x \u00E2\u0080\u00A2 y and the inversion map sending x to x~l are continuous. An important example: the set M\u00E2\u0080\u009E(E) of n by n matrices is a euclidean space of dimension n2. As the determinant function det : M\u00E2\u0080\u009E(E) \u00E2\u0080\u0094> E is continuous, dei _ 1(0) is a closed set, and its complement, the group GL (n , E), is thus an open subset of E \" 2 . Note that matrix multiplication, which is given by polynomials in the coefficients, is continuous, and the inversion map, which by Cramer's rule is a rational function of the coefficients, is also continuous. Def in i t ion '7 Let G be a topological group with operation \u00E2\u0080\u00A2, and assume thatp : G \u00E2\u0080\u0094> G is a simply connected covering space of G. Then if G is locally path-connected, there exists a multiplication map rh on G relative to which G is a topological group and p is a homomorphism. The group G is called the universal covering group of G. Let V : G x G ->\u00E2\u0080\u00A2 G x G be the map p x p, and let e e p _ 1(e) be selected, where e = l o For the existence of rh required by the definition, we note that by the lifting theorem mV lifts to rh with rh(e,e) = e if and only if (mV)*(iTi(G x G, (e, e))) = m*V*{iTi(G x G, (e, e))) C p*(iri(G, e)). Since G x G is simply connected the containment 11 Chapter 3. Some Group Theory and Satan's Parkade is automatic and we are guaranteed the lift m, which as the lift of a continuous map is continuous. For the condition that p is a homomorphism,. the multiplication map fn is defined as follows. Let x,y G G. Choose paths a from e to x and ft from e to y. Let a(t) = pa(t), and (3(t) = pJ3(t). Then take the path j(t) = a(t) \u00E2\u0080\u00A2 /3(t) (where \u00E2\u0080\u00A2 is the group operation in G, not a path product) and lift it to a path 7 in G that begins at e. Then x \u00E2\u0080\u00A2 y is defined to be the endpoint of 7 . Note that we have p(x \u00E2\u0080\u00A2 y) = p(7(l)) = 7(1) = a( l ) \u00E2\u0080\u00A2 0(1) = P&(1) \u00E2\u0080\u00A2 PP(1) = p{x)p(y). Now SL(2,K) is not its own universal covering group, as we establish in the following claim. Claim: 7r 1(SL(2, K), I) = Z (where J is the identity matrix). Proof of Claim. Recall the following definitions. Def in i t ion 8 An or thogonal mat r i x is a matrix A for which AA1 = I. The set O(n) of orthogonal (real) matrices forms a subgroup of G L ( n , R ) ; and SO(n) = {A G O(n) | det(A) = 1}, the special orthogonal group, is a subgroup o /SL(n , R). It is a standard result that O(n) is compact. Def in i t ion 9 A subspace Y of a space X is called a deformat ion retract if there exists a continuous retract r : X \u00E2\u0080\u0094>\u00E2\u0080\u00A2 Y such that the identity map on X is homotopic to the map i o r, where i is the inclusion of Y in X. A subspace Y C X is a strong deformat ion retract of X if there exists a continuous map H : X x I \u00E2\u0080\u0094> X such that H{x, 0) = x for every x e X, H(x, 1) e Y Va; G X, and H(s, t) = s Vs G Y and V i G I. 12 Chapter 3. Some Group Theory and Satan's Parkade Recall that if Y is a strong deformation retract of X, with y0 6 Y, then the inclusion map i : (Y, y0) \u00E2\u0080\u0094> (X, y0) induces an isomorphism of fundamental groups. With this artillery we are ready to prove that SL (2 ,E) is not simply connected, by showing first that TTI(SL(2,R)) = TTI(S0.(2)), and then that TTI(S0(2)) = TTI(51). Subclaim #1: SO(2) SL (2 ,E) is a strong deformation retract. Proof of Subclaim. We construct the homotopy as follows. Let A \u00C2\u00A3 SL (2 ,E ) , let ex = (J),e2 = (\u00C2\u00B0). Let Vi = Aei,V2 = Ae2. Since det(A) = 1, A is invertible, v\ and v2 are linearly independent. We use the Gram Schmidt process on vx and v2 to orthnormalize via three homotopies: 1). Set hi(t) = tfy + (1 -t)vi, sending vx \u00E2\u0080\u0094> = v'{. 2. Replace v2 by v'2 = v2 \u00E2\u0080\u0094 ^ p j p W i via the map h2(t) = v2 \u00E2\u0080\u0094 t(v2 \u00E2\u0080\u00A2 v\")vi. 3. Replace v'2 by v'2' = ^ via h3{t) = t ^ + (l- t)v'2. After reparametrizing these three homotopies we obtain the single homotopy desired. Thus to prove that 7Ti(SL (2 ,E)) = Z, it suffices to show that S0(2) is homeomorphic to S 1 . Subclaim #2: SO(2) ^ 5 1 . Proof of Subclaim. We will use the fact that for X compact and Y Hausdorff, if / : X \u00E2\u0080\u0094> Y is continuous, one-to-one, and onto, then / is a homeomorphism. Since det: 0(2) \u00E2\u0080\u0094> E is continuous, d e \u00C2\u00A3 - 1 ( l ) = S0(2) is closed, hence compact. Moreover S1 is Hausdorff. We define / : S O ( 2 ) \u00E2\u0080\u0094> S1, f(A) = A(l). Then / takes a matrix to where it moves the point e\u00C2\u00B1, and since SO(2) C 0(2) preserves lengths of vectors / maps Sl into itself. Note that / is 13 Chapter 3. Some Group Theory and Satan's Parkade clearly onto, as ^ cos 9 \u00E2\u0080\u0094 sin 9 ^ sin 9 cos 6 G S0(2) for every 9 G [0, 27r). Moreover the invertibility of elements in SO(2) gives A(l) = B(l) 1(1) = A^BQ) B = A. Continuity is also immediate from the continuity of the sine and cosine functions and the fact that elements of SO(2) acts as rotations on S1: for if Rg G SO(2) is rotation by angle 9 with image Rgei * = (cos 9, sin 6), we can always choose rotation by angle (j), R < 9 + e to obtain images that are arbitrarily close. Summarizing: we have seen that SL(2,R) is not simply connected; and in the ensuing discussion we will investigate more closely the geometry associated with its covering group. Definition 10 A (right) group action is a map X x G\u00E2\u0080\u0094> X, (x,g) H-> x-g satisfying x \u00E2\u0080\u00A2 e = x and x \u00E2\u0080\u00A2 (hg) = (x \u00E2\u0080\u00A2 h) \u00E2\u0080\u00A2 g for all g,h G G,x G X. The kernel of the action is the set {g G G | x \u00E2\u0080\u00A2 g = x Vx G X}. An action is faithful provided that the kernel is trivial. In equivalent terminology that is often used: an action is effective if (x \u00E2\u0080\u00A2 g = x Vx G X) g = e. Of course, a left action is defined in the obvious way. If a set X admits a right (respec-tively, left) group action, X is said to be a right (respectively, left) C7-space. It is easy to check that if X is a left C7-space and if we define x \u00E2\u0080\u00A2 g = (#-1) \u00E2\u0080\u00A2 x for all x G X and g G G, then X is a right G-space. To show that G is right orderable, we will need the following characterization of right orderable groups: 14 Chapter 3. Some Group Theory and Satan's Parkade Proposition: G is right orderable if and only if G acts effectively on an ordered set by order-preserving bijections. Proof: (=>\u00E2\u0080\u00A2) Let G act on itself by right multiplication. Then right invariance gives precisely the order-preserving property desired. (<=) Assume now that G acts effectively on an ordered set X by order-preserving bi-jections. Assuming the Well Ordering Principle, let X be well-ordered by some or-der (X,-<). The idea is to define an order (G,<) in the following way. Since G acts effectively on X, if g, h e G are such that g ^ h, then there exists x E X such that x-g ^ x \u00E2\u0080\u00A2 h. Under the well-ordering, there exists a minimal such x at which g and h differ: denote this Say g < h provided \u00E2\u0080\u00A2 g < x{g,h) \u00E2\u0080\u00A2 h. For right invariance assume that g < h, that is, \u00E2\u0080\u00A2 g < x{g,h) \u00E2\u0080\u00A2 h; then for any / \u00E2\u0082\u00AC G we require gf < hf, but in fact X(gf,hf)'' 9f < x(gf,hf) ' hf follows at once from the assumption that G acts in an order-preserving manner. Finally we show that g < h,h < k =>\u00E2\u0080\u00A2 g < k. There are a couple of cases to consider. If x^h) = x{h,k), then easily X'9th) \u00E2\u0080\u00A2 g < X(9th) \u00E2\u0080\u00A2 h = X'h,k) \u00E2\u0080\u00A2 h < x^h%k) \u00E2\u0080\u00A2 k = x^h) \u00E2\u0080\u00A2 k, so g < k. Otherwise assume that X(9th) ^ \u00C2\u00A3 ( / i , / c ) ; let us first suppose that \u00E2\u0080\u00A2< X(h,k)- Now g and h differ at but h and k must agree at x^^ -< x^k), so necessarily g and k disagree there, that is, x{g,h) \u00E2\u0080\u00A2 9 < x(g,h)\u00E2\u0080\u00A2 h = \u00E2\u0080\u00A2 k. At no point less than x^g^) do g and k differ, since k = h and h = g prior to X(9th)- Thus we obtain g < k. If on the other hand x^k) < x{g,h), then g = hat X(htk) and therefore x>hjk) \u00E2\u0080\u00A2 g = x^k) \u00E2\u0080\u00A2 h < x^k) \u00E2\u0080\u00A2 k. Since is the first point at which g, h, and k differ, we again have g < k. \u00E2\u0080\u00A2 1 Now it is evident that SL(2, R) acts by matrix multiplication on the set of rays through 0 in the plane: if A e SL(2, R), and x = (x, y) is a point on a ray p through the origin, A(tx) = tA(x) for every t e R by linearity, so rays are indeed taken to rays. The 15 Chapter 3. Some Group Theory and Satan's Parkade kernel of this action is trivial, since if Cl denotes the set of rays through the origin and A G SL(2,R) fixes every ray through the origin, then certainly AQ) = (*\u00E2\u0080\u00A2) for some A > 0, and = (\u00C2\u00B0), some (5 > 0; but then det(,4) = 1 (3 = A\" 1 . Now A fixes every ray, so the point x = (}) is mapped to (^_i) = t(l) for some t G R + =>- A = 1 and A = I. In the action just described, the angle through which a ray is moved is specified only modulo 27T for A G SL(2,'R), but an element of the covering group SL which projects down into SL(2,]R) may be thought of as a linear transformation of the plane in which the angle a ray is moved is specified continuously in the real number s: if A moves p by angle s, there exist elements of SL that move p by angles s 4- 27r/c for every k G Z. Thus the action of SL on Cl is certainly not faithful (there are infinitely many elements that project to / G SL(2,'R)), but if we \"unwind\" the.circle of rays through the origin and identify them with R, then SL indeed acts faithfully on this set. If one is mathematically inclined to lurid geometrical visions, one can visualize this set-which is the set of rays through 0 in the infinite-sheeted branched covering of the plane with branchpoint 0-as a sort of infinite parkade spiralling above the plane (in which parking space 2nk is directly overhead parking space 2ir(k \u00E2\u0080\u0094 1)). Hence the name Satan's Parkade. The claim is that SL is right orderable, preserving the usual ordering of the line R. We will use the proposition above, showing that SL acts effectively on R by order-preserving bijections. Note that SL(2, R), in acting on the rays through the origin of the plane, can also be considered as acting on 5 1 by orientation-preserving homeomorphisms, where x G S1 is taken by A G SL(2,R) to pffy- So we will prove the more general result that if A : R \u00E2\u0080\u0094>\u00E2\u0080\u00A2 R is a lift of any homeomorphism A : S1 \u00E2\u0080\u0094> S1, then A is also a homeomorphism; and that if A preserves orientation on the circle, then A preserves the ordering on the line. 16 Chapter 3. Some Group Theory and Satan's Parkade For given A a homeomorphism from Sl to itself, A will be a lift of the map Ap and is thus continuous: R,x~0 R,y0 S1,x0 >-S1,y0 A To see that A has a (continuous) inverse, let B = A~l. That is, B is the lift of the map A_1p, and we obtain pBA = A~xAp = Ip. Since Ipx~o \u00E2\u0080\u0094 Ix0 = x0, pBAfo \u00E2\u0080\u0094 x0; but also plx0 \u00E2\u0080\u0094 px0 = XQ, so BA = I as maps. Analogous reasoning yields AB \u00E2\u0080\u0094 I, and we can conclude that the inverse of A i s . A - 1 ; note that A'1 is continuous. Thus homeomorphisms of S1 lift to homeomorphisms of R. We now assume that A : S1 \u00E2\u0080\u0094> S1 preserves orientation, and show that A lifts to an order-preserving homeomorphism on R. Consider the points (*) and A^) = (c\u00C2\u00B0^e9) (we will write (cos9,sin9) for convenience). Let Rg be rotation on S1 by angle \u00E2\u0080\u00949. Note that Re is an orientation-preserving homeomorphism and that (Re o A ) Q = (J). Let g : Sl \u00E2\u0080\u0094>\u00E2\u0080\u00A2 S1 be this map i?e \u00C2\u00B0 A. Then it suffices to show that g : R \u00E2\u0080\u0094> R preserves order, where (R,p) is a covering map (that is, pg = gp; we take p(\u00C2\u00A3) = (cost, sint)). We already know that homeomorphisms of the circle lift to homeomorphisms of the line; so as g is either an increasing or a decreasing function it suffices to show that g maps R + to R + . Since p is a covering map there exists an open neighbourhood U C S1 of (J) such that each component of p~l(U) is mapped topologically onto U. So there exists an interval V = (\u00E2\u0080\u0094e, e) C R such that p restricted to V is a homeomorphism. Since g is continuous there exists 5i > 0 (take Si < e) such that d(0,y) < 51 =4> d(g(0),g(y)) < e. Then putting Ui = p(\u00E2\u0080\u00948i,5i) we have Ui C J7. Moreover since ^ is continuous we can choose a neighbourhood U2 C C/x small enough that flf(f/2) C C/x. Let V - p~l(U2) = (-5,5) for some 5 > 0. To show that g takes R + to itself we need only show that 0. Recall 17 Chapter 3. Some Group Theory and Satan's Parkade that g fixes (J) on S1, so that on V we have g(0) = 0. To see why #(\u00C2\u00A7) > 0, consider P 0 on V C R. Thus orientation-preserving homeomorphisms on S~l lift to order-preserving homeomor-phisms on R; so finally we can deduce that SL is a a right-orderable group. We obtain additional information about SL in the following Lemma: For any A G SL(2,R), with t0 and tf G R and t0 < tf, if |t0 - tf \ < 2n, then \A(t0) - A(tf)\ < 2ir for every A G SL projecting to A. Proof of Lemma: Suppose for a contradiction that \A(tf) \u00E2\u0080\u0094 A(to)\ > 27r; let's first assume that A(t0) < A(tf). Then since A is continuous, there exists ti G (to,i/) s u c n that A(ti) = A (t0) + 27r. Letting p0 \u00E2\u0082\u00AC ^ denote the ray of angle t0, p/ the ray of angle tf, and so on, it follows that A(pi) = A(p0) + 27r = A(p 0). So we have t0 < h but A(t0) = A(ti), contradicting that det(A) = 1. (If we assume A (to). > -4(t/), then there exists ti \u00E2\u0082\u00AC (to,t/) with A(ti) = A(tf) + 27T and we have ti < t/ but A(ti) = -A(t/), contradiction.) \u00E2\u0080\u00A2 Now suppose A takes the point t0 G R to r 0. Then from the lemma it follows that |A(t0+27r) \u00E2\u0080\u0094 .A(to) | < 27r, and since A is.order-preserving, as well as a bijection, A(to+2ir) must be A(t0) + 2TT = r 0 + 27r. In fact this observation generalizes to the following proposition, of which it is the base case: Proposition: If A : R \u00E2\u0080\u0094> R is an order-preserving homeomorphism such that A(Z) = Z, then if A(0) - 0, we also have A(n) = (n) Vn \u00E2\u0082\u00AC Z. Proof: Since A(\u00E2\u0080\u0094n) = \u00E2\u0080\u0094 A(n) for every n, it suffices to show that the Proposition is true for n \u00E2\u0082\u00AC N . Assume that the assertion is true for all k < n \u00E2\u0080\u0094 1, but suppose that A(n) = m,m > n. Then since A is onto, there exists s G N, s > n, such that A(s) = n, 18 Chapter 3. Some Group Theory and Satan's Parkade contradicting that A is order-preserving. \u00E2\u0080\u00A2 Thus we now know that S L acts on 27r-multiples of vectors in fl in predictable fashion, rather than shuffling them around. Our goal is to define a nontrivial subgroup of the covering group of SL(2 ,R) that will map homomorphically to the identity, thus establishing that our group is not locally indicable. To do this we begin by specifying some matrices of interest in SL (2 ,R) . Let T a 1 -r \ 0 1 , r = i 1 0 U= 0 where r \u00E2\u0082\u00AC R, s G R + . In fact ar is a horizontal shear, while br is a vertical shear. One easily computes that ar has repeated eigenvalue 1 and eigenspace {(*) | t G R}, while br also has repeated eigenvalue 1 though with eigenspace {(\u00C2\u00B0) | t G R}, and cs has eigenvalues i/s and both positive by the condition on s with eigenspaces {(*) : t G R} and {(\u00C2\u00B0) : s G R}. One also obtains the following identities: T Tl a a 19 Chapter 3. Some Group Theory and Satan's Parkade The following easy calculations are also very much to our purpose: V ^ 0 W l - r \ / 4 - 0 \ 1 -sr caarc-i =. I ^ 1 = 1 I = a 0 ^ / \ 0 1 / \ 0 I \ 0 1 , 4= 0 U l 0 W v's 0 \ [ 1 0 , c:lbrcs = I ^ v = 1 = 6 0 V~* I \ f 1 J \ 0 ) \ rs 1 Finally, letting d = aba 0 -1 1 0 we note that sr - 1 0 / V 0 Ts ) \ 1 0 / V 0 c:1 . 0 -1 \ / 1 - r \ / 0 -1 \ / 1 0 , d~1ard= | = | =br -1 0 / \ 0 1 / \ 1 0 / \ r 1 . 0 1 \ / 1 0 \ / 0 -1 \ 1 -r . - 1 0 / \ r l / \ l 0 / \ 0 \u00E2\u0080\u00A2 1 Now a,b,c,d e SL(2,R) satisfy these relations, but the immediate goal is to show that certain elements in their fibres satisfy the same relations in S L ; for once this is established it will not be too difficult to define a finitely generated subgroup of S L which maps 20 Chapter 3. Some Group Theory and Satan's Parkade homomorphically to the identity in Z, thus showing that S L is not locally indicable. Suppose h G SL(2,R) has a positive eigenvalue A > 0 (relevant aside: a,b,c above all do). Then there exists v G R 2 with hv \u00E2\u0080\u0094 Xv, so in terms of the action of SL(2, R) on Cl, h fixes a ray v, and in terms of the action of S L on Satan's Parkade, each element of p~l(h) must move v by 2nk, some k G Z (p here is the projection map in the definition of the universal covering group above). We define Ar,Br, Cs to be the liftings of of, br, and cs that actually fix the rays in the Parkade (rather than moving them by 27rA;). SO in order for this definition to even make sense we need to check that only one of h's infinitely many liftings to S L fixes the rays that map into the eigenspace of A. Moreover, since cs has two distinct positive eigenvalues, it is necessary to check that the lift that fixes the rays that map into one eigenspace is the same lift that fixes the rays that map into the other. To see why this latter claim is so, we claim that if / G SL (2 ,R) has at least one positive eigenvalue, then / moves every ray through an angle strictly less than it. For let A > 0 be an eigenvalue1, with v its eigenvector. Suppose w is a ray moved by angle > it. Note that w ^ \u00E2\u0080\u0094v, as then w is also fixed by / . Thus v and w are linearly independent and form an ordered basis for R 2 , but if / moves w by an angle greater than or equal to it, then / is in fact orientation-reversing, contradicting that det(f) = 1. But since / moves every ray through an angle of magnitude less than it', it cannot be lifted to a map / that fixes a ray v[ of angle vy and moves some ray v2 of angle u2 by some 2itk, where k > 1. For (assuming without loss of generality that v\ < u2) f is continuous, and by the Intermediate Value Theorem there exists c G R, with associated vector w of angle c with ui < c < v2 such that f(w) = it, so that / moves w by it, a contradiction. Now to see that only one of h's infinitely many liftings to S L fixes the rays that map into the eigenspace of A, suppose that there exist hi,h2 G p~l(h), and hi and h2 both fix the eigenvector v of h in the Parkade. We must show that hi = h2. Let h \u00E2\u0080\u0094 hih^1. Then ph = p(hih2l) \u00E2\u0080\u0094 p{hi)p(h2)-1 = hh~l = / . Consider w G Q,w ^ v and chosen to be 21 Chapter 3. Some Group Theory and Satan's Parkade linearly independent from v. Then of course ph(w) = w. But then h also fixes w in the Parkade, since we saw above that I cannot lift to a map h that moves v by 0 degrees and w by 2nk, k ^ 0 (else there exists u between v and w that is moved by 7r. We conclude that h is the identity, so hi = h^. Thus it makes sense to speak of the unique liftings Ar,Br, and Cs as defined above. Claim: Ar,Br, and Cs satisfy the corresponding identities in SL, namely: A r A r , = Ar+r^ W Q T , = gr+r^ = CsArC~x = Ars, C^B'Cs = Brs. Proof of Claim. We already know that in each case, both sides the equation project to the same element in SL(2,R). So by the preceding arguments we simply need to show that both sides of each equation have fixed rays, rather than moving them by some 27rfc. But by definition the right hand side of each equation has fixed rays. Now consider ArArl. Recall that ar has eigenspace {spanQ} for every r \u00C2\u00A3 R. Since matrix multiplication is composition of transformations the eigenvectors stay fixed. So the same holds true for the chosen lift. This argument applies to BrBrf and CSCS, as well. For the last two equations the left hand sides are conjugates of elements having fixed rays, hence have fixed rays. If we define D = ABA we obtain in similar fashion the identities: D~lCsD = c;1 22 Chapter 3. Some Group Theory and Satan's Parkade D-lArD = Br D~lBrD = Ar. Theorem: SL is not locally indicable. Proof. Let n e N > 1 and let H = (A,B,Cn) < SL. Let $ : H \u00E2\u0080\u0094> (Z,+) be any homomorphism. The identities above give that CnAC~x = An,C~1BCn = Bn, and D~xCnD = C~l. So A,B, and Cn are conjugate in this group to powers of themselves. Then ^(CnAC-1) = $(A n ) $(C*n) + $(A) - $ (C n ) = $(An) = n(A) for some n > 1, implying that $(A) = 0, with similar arguments holding for the other generators. \u00E2\u0080\u00A2 23 Chapter 4 Knot Theory and Orderable Groups Knot (and braid) theory crop up notoriously in all sorts of different mathematical con-texts, so it is hardly surprising that orderable groups should intersect knot theory in nontrivial ways. This chapter undertakes an investigation of a result of N. Smythe which exploits order considerations to generalize a classical theorem of Alexander's pertaining to knots to all surfaces. Once again we begin with some definitions. Def in i t ion 11 A surface is a connected 2-manifold (that is, a connected Hausdorff space such that each point P has an open neighbourhood homeomorphic to the open disk D\u00C2\u00B0 = {(x, y) G R 2 \ x2 + y2 < I}). A surface w i th boundary is a Hausdorff space such that every point P has a neighbourhood homeomorphic either to the open disk or to the half-disk {{x,y) \u00C2\u00A3 D\u00C2\u00B0 \ y > 0}. If the boundary is empty and the surface is compact, then it is said to be closed. Def in i t ion 12 A closed non-self-intersecting polygonal line in R 3 is called a polygonal knot. A smooth knot is the image of an infinitely differentiable embedding f : Sl \u00E2\u0080\u0094 r R3J(t) = (x(t),y(t),z(t)), with ( f , f , | ) \u00C2\u00B1 (0,0,0). Thus we can envisage a knot as an entwined polygon in R 3 (with finitely many edges), or we can envisage it as an entwined circle in that space. Two polygonal knots K0 and Ki are said to be equivalent provided Ki can be obtained from K0 via a finite sequence 24 Chapter 4. Knot Theory and Orderable Groups of \"elementary moves\" (or their inverses), where these are as follows: let E{ and Ej be adjacent edges of the polygonal knot K0, and assume that the triangle spanned by E{ and Ej does not intersect K0 in any other points. Then an elementary move is simply the replacement of Et and Ej by the third leg of the triangle spanned. On the other hand, two smooth knots K0 and Kx are equivalent if there exists a one-parameter family ft : R 3 \u00E2\u0080\u0094> R3,t \u00C2\u00A3 [0,1] of diffeomorphisms smoothly depending on the parameter t such that fo(K0) = KQ and fi(K0) = K\. Here, \"smoothly depending\" means that the map F : R 3 x [0,1] \u00E2\u0080\u0094 \u00E2\u0080\u00A2 R3,F(x,t) = ft(x) is difTerentiable. The family of diffeomorphisms ft is called an isotopy, and equivalent knots are sometimes referred to as isotopic, or ambient isotopic. Note that if K0 and Ky are equivalent, there is a homeomorphism, namely, / i , between (R3,K0) and (R3,Kx). One might well ask whether the double-barreled definition of knots given above-polygonal versus smooth-reflects two different irreconcilable approaches to knots, or whether the definitions are interchangeable. It is the latter that is true: there is a process called \"smoothing\" taking equivalence classes of polygonal knots to equivalence classes of smooth knots (for details see [7]). The upshot is that we can choose either definition we prefer to work with. Definition 13 A knot which bounds a disc in R 3 is trivial. A further technicality that must be broached in any discussion of knots is the issue of projection and knot diagrams, which we treat briefly and informally here. In essence what is required in choosing a \"regular\" projection for a (smooth) knot is a plane such that: \u00E2\u0080\u00A2 no more than two distinct points of the knot are projected onto the same point on the plane \u00E2\u0080\u00A2 the set of such crossing points is finite, and at any such crossing point the projections of two tangent lines to the knot do not coincide 25 Chapter 4. Knot Theory and Orderable Groups \u00E2\u0080\u00A2 the tangent lines to the knot project onto lines on the plane, not to points. With polygonal knots some adjustment in vocabulary is necessary: we require that a vertex of the knot is never mapped onto a double point, as well as the conditions above that no more than two points are in the pre-image of any projection point (with at most finitely many such double points). Now there is an algorithm known as \"laying down the rope\", known at least as far back as Alexander and sometimes attributed to him, demonstrating that for any regular projection p(k) of a polygonal knot k, there is a trivial knot with the same projection (see, for instance, [26]). The procedure is simply to orient p(k), choose a starting point x on p(k), and at each double point P of p(k) with itself, in which arcs a(P) and /3(P) cross, to designate a{P) as an undercrossing if in the chosen orientation a(P) lies between x and arc 0(P). The knot that is obtained will be trivial. The question is whether this result generalizes to all surfaces. One generalization, to 5 2 x / , is immediate: that is, if k is embedded in S2 x I and p(k) is its projection onto S2 x {0}, Alexander's Algorithm applies since p(k) is contractible in the complement of a point Q of S2. Since S2 \u00E2\u0080\u0094 {Q} = R 2 , the algorithm proceeds exactly as above, and the post-algorithmic p(k) will still bound a disc. Here is the theorem in full generality: Theorem 5 (N . Smythe) Let S be a surface, orientable or not, compact or not. Let k be a polygonal knot contained in, and contractible in, the interior of S x I, with regular projection p(k) in S x {0}. Then there exists a knot k' C S x [|, |] which has the same regular projection, and which bounds a disc in S x [|, | ] . What is extraordinary is that the theorem hinges upon the left-invariant orderability of the fundamental group TTI(S), SO that a necessary preliminary involves the (non-trivial) verification that such orderings exist for every surface other than the projective plane RP2 (which has fundamental group Z / 2 Z , clearly non-orderable, and will require sep-26 Chapter 4. Knot Theory and Orderable Groups arate treatment). First we observe that the free group of finite or countable rank has a left-invariant ordering. Various proofs exists: see for instance [12] (another slick ap-proach uses the Magnus map). Note that this result will take care of finite surfaces with boundary (whose canonical polygons.have perforations or \"holes\" bounded by the boundary curves), since any such is homeomorphic to a disc with strips attached (double strips corresponding to handles, Mobius strips to crosscaps, and single strips for any ex-tra perforations); and such surfaces deformation retract onto bouquets of circles passing through the strips, hence have fundamental groups free of rank equal to the number of strips. In addition this will take care of the case in which the surface is infinite (that is to say, non-compact), since the fundamental group of an infinite surface is free. For the details of this result, see, for instance [32]. The basic idea is to represent an infinite surface T as a nested union ( J n Tn of finite surfaces with boundary; with some care one inductively establishes that the free generators of iti(J-n) remain free in ir^Tn+i); then, since any closed path based at x \u00C2\u00A3 J- is compact, it must lie in some Tn. We have as remaining cases the closed orientable surfaces which have fundamental group Gh = (fli, b i , b h I Ylaibiai~lbi~l = l),h> 0, and the closed non-orientable Hk = {ai,...,a,k I ai2a22 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 = 1), for k > 1. Note that (ai | a2 = 1) is the fundamental group of the projective plane, to be discussed separately, while if h = 0 the group is trivial, corresponding to the surface S2, and is thus orderable. Turning to the orientable case first we note that if h > 1 we can define a homomorphism ^ : Gh \u00E2\u0080\u0094* Z given by */j(ai) = 1,^(61) = 0,^h(ai) = ^ h(k) = OVi > 1. Note that the kernel Kh contains bi and aj, bi for all i > 2. In the non-orientable case we define the homomorphism : Hk Z given by * f c(ai) = 1, ^ (^2) = - 1 , ^(a*) = 0 Vi > 2. Note that if k > 2, the kernel contains all powers of at least one a;. In case k = 2 we observe that the kernel contains {a^a^- \u00E2\u0080\u00A2\u00E2\u0080\u00A2a1i\"a2*\" | \u00C2\u00A3 ' = 1 is = EZiJs}-27 Chapter 4. Knot Theory and Orderable Groups We are interested in the covering spaces corresponding to these (normal) subgroups of Gh and Hk. Recall that it is possible to define an action of the group ni{S, X) on the fibre set p_1(a:) for any rc: for chosen x \u00E2\u0082\u00AC p _ 1(x) and any a \u00E2\u0082\u00AC 7Ti(S,x), define x \u00E2\u0080\u00A2 a G p_1(a;) to be the terminal point of the unique path class a in S such that p*(&) = a and the initial point of a is the point x. Of course the actions by Gh and Hk, restricted to their respective kernels, are still group actions. The claim is that in each instance p~l(x) has infinitely many sheets. To see that this is so, consider first the case k = 2, that is, Hk = (a, b | a2b2 = 1). Then the kernel of the homomorphism to Z contains such elements of the fundamental group as a3b3, a3b3a3b3, and (a3b3Y for any j > 1; for chosen x0 G p_ 1(:r) we will denote by Xj = x0-(a3b3y the terminal point of the lift. If Xj = xk for some j, k, where j ^ k, then the respective lifts are identical, so p* {(a3b3)3) =p*{(a3b3y), implying, since is a monomorphism, that (a3b3Y = (a3b3)h in H2. Since j ^ k this implies that we have (a3b3)m = 1 for some m \u00C2\u00A3 Z, But H2 is not a commutative group, hence (a3b3)m ^ (a2b2)n for any n; so this is a contradiction. If k > 2, then there exists ai, i > 2, and all powers of in the kernel, and similar reasoning leads us to conclude that p~l(x) has infinitely many sheets, and if h > 1, the fact that all powers of bi are in the kernel gives the same result. In all three cases, therefore, the covering space corresponding to the kernel is an open'infinite surface, so the kernel is a free group, hence left-orderable. Since G/K is isomorphic to a subgroup of Z and left-orderable, it follows that Hk and Gh are left-orderable, since as mentioned earlier in this thesis, if K < G, and both K and G/K have left-invariant orderings, then so does G. With the left-orderability of the fundamental group of these surfaces now established, we turn to an investigation of the theorem in the case wherein 5 is any surface with the exception of S2 or RP2. By assumption, k, a polygonal knot, is contained in and contractible within M xl, where M denotes the interior of S. Let (M,e) be the universal covering space of M x I. Now it is a standard result of covering space theory that if 28 Chapter 4. Knot Theory and Orderable Groups e : X -\u00C2\u00BB Y is a covering map with X simply connected, then Aut(X) = TVi(Y,yo). We shall also make use of the fact that the universal covering surfaces are the sphere (for the sphere itself, and the projective plane), and the plane, which follows from every non-orientable surface having an orientable surface as a 2-sheeted cover [32]. Thus M = R2 xl, and Aut(M) =.7Ti(M X I) = ITI(S), which we know is left orderable. Let x be chosen as a basepoint in M and let k\ be the lift of p(k) through x. If we let Tu denote the covering translation corresponding to the element u G TT1(M X J), then p(k) is covered by Tu(ki), which we denote by ku. A chosen orientation for p(k) G M x {0} will induce an orientation in ku G R2 x 0. Let p : R 2 x / \u00E2\u0080\u0094> R 2 x {0} be the projection over p, that is, ep = p. Alexander's algorithm of \"laying down the rope\" can be generalized as follows. Choose neighbourhoods around the double points of the collection {ku} sufficiently small to be disjoint and contain only the two simple arcs of the singularity. Then focussing attention on k\, we notice that kx might cross itself, or ki might cross some other ku, where u ^ 1. If ki crosses itself at the point P, let a(P) denote whichever of the two arcs precedes with respect to x in the orientation of ki, let /3(P) denote the arc that succeeds. If ki crosses kv, v ^ 1, let 7(P) denote the arc at the crossing that belongs to ki, and v(P) be the arc belonging to kv. Then a simple closed curve kx G R 2 x / is constructed in the following manner. For any point (r, s, 0) G ki, if (r, s , 0) does not belong to any of the singularity neighbourhoods, or if (r, s, 0) G OL{P), or if (r, s, 0) G 7(P) where P is a crossing of ki and ku such that u > 1, then let (r, s, |) be a point of ki. If on the other hand (r, s, 0) G (3{P), or if (r-, s, 0) G 7 ( f ) , where P is a crossing of kx and kv such that v < 1, then let (r,s,\ + t),0 < t < | , be a point of A4, where i = \ at P itself (giving (r, s, | ) as the point above P on fci), and t varies continuously to approach 0 as (r, s, 0) approaches the boundary of the neighbourhood at P. Thus we have fashioned a closed curve ki G R 2 x I such that p(ki) = k\u00C2\u00B1. Note that a(P) is sent to an undercrossing on 29 Chapter 4. Knot Theory and Orderable Groups ku and ft(P) becomes an overcrossing. Finally we define ku = Tu(ki). The algorithm is rather dauntingly technical, but roughly speaking (and we return to make this more precise below), we are to think of the curves {kw} as suspended in E 2 x I with their \"altitudes\" on the interval dependent on the group element which indexes them: higher up for those whose group element is positive in the group ordering, lower down if the group element is negative. Claim: ku and kv are disjoint. Indeed, let Q G ku n kv. Consider Tu-i(Q) = P. Since ku = Tu(ki), we have P = Tu-i(Q) C Tu-iTu(ki) G h, but also P C Tu-iTv(ki) G 'ku-iv. Thus P G M ku-iv. Consider also P' = Tv-i(Q). Since kv = Tv(ki), we have P' = Tv-i(Q) C Tv-iTv(ki) G jfei and P' C Tv-iTu(ki) G kv-iu, so P' G ki fl kv-iu. Without loss of .generality, take u > v, i.e. u~lv < 1. Then we have p(P) a double point of ki and ku-iv by definition of p, so by construction P must be in E 2 x | . But turning to consider p(P') a double point of kx and we must have P' G E 2 x | , as > 1. But this a contradiction, since TU(P) = TV[P') = Q, and covering translations do not affect the third coordinate. Thus ku and kv are disjoint for all u,v,u ^ v. Moreover since by the very definition of ku we have u > v implying that any crossing of ku lies above kv in E 2 x / , u > v implies that ku lies above kv; hence we can isotope the curves {kw} such that ku C R 2 x (0,,|) if u < 1, h remains in R 2 x [|, | ] , and kv CR2 x (|, 1) if v > 1; and in all cases p(kw) = kw remains a regular projection for all w. Note that is isolated in R 2 x [|, | ] , and that the construction of ki with specific reference to the double points of ki with itself, coincides exactly with the \"laying down the rope\" algorithm. Thus by Alexander, k\ is contractible in (U u ^ i ku) , and bounds a non-singular disc D in R 2 x [|, |] C M, the universal cover. Consider e(ki) C M x I: e(ki) bounds e(D). Since D e l 2 x [|, |] is isolated from the other curves {ku}, if e(D) has 30 Chapter 4. Knot Theory and Orderable Groups singularities, these will be points of self-intersection. We invoke now a celebrated result: Theorem 6 (Dehn's Lemma) Let M be a PL 3-manifold, compact or not, with bound-ary which may be empty, and in M let D be a two-cell with self-intersections (singulari-ties), having as boundary the simple closed polygonal curve C and such that there exists a closed neighbourhood of C in D which is an annulus (that is, no point of C is singular). Then there exists a two-cell D0 with boundary C semi-linearly embedded in M. By Dehn's Lemma, e(D) may be modified in a neighbourhood of these singularities, and thus within M x [|, | ] , so that e(ki) bounds a non-singular disc. There remains one final surface to attend to. As noted before, the fundamental group of the projective plane is not orderable, so this case requires an extra step that will reduce the argument to that of the Mobius band, which has fundamental group Z (the Mobius band deformation retracts to S1). If p(k) is the regular projection onto S x {0} of a knot in S x I, where S is now the projective plane, one can transform the closed curve p(k) into a finite number of simple closed curves C\, C 2 , C m , by choosing open neighbourhoods of each crossing point small enough to contain no other crossing point and \"splicing\" the crossing in the following way: if P is the crossing point of arcs a and /3, and N(P) is the selected small disc neighbourhood, one can choose a point A e a fl N(P) preceding P in the orientation of p(k), and a point B on /3 succeeding P in the orientation. Then one joins A and B by an arc 7 within N(P) that doesn't elsewhere intersect p(k), and by deleting the open arcs AP and PB one obtains disjoint closed curves. If we choose a basepoint x on p(k), p(k) is nomotopic to a product of conjugates of the Ci, namely WiCiW~l,i < i < m, where W{ is an approach path (the trivial path in the case i = 1) from the basepoint x that connects to Ci across the disc neighbourhood and coincides exactly with p(k) elsewhere. It will now be expedient to transcribe this set-up into the language of homology: note that as a closed path p(k) can be considered 31 Chapter 4. Knot Theory and Orderable Groups as a 1-cycle, that is, as a (formal) sum of oriented edges whose boundary is 0. So we have p(k) homotopic to a path product WiCiWi1w2C2w2l \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 wmCmw^-, but this latter is the same in homology as Y1T=\ since the formal sum of edges is unchanged under conjugation by the {wi}. Thus since p(k) is contractible, WiCiW^w^w^1 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 wmCmw^ is contractible, whence YHLI Ci is homologous to 0. It will follow that each d is null-homologous and thus contractible: for suppose that some Ci is not null-homologous. Since the fundamental group of R P 2 is already abelian, H^RP2) = TTI(RP 2) = Z / 2 Z implies that an even number of the C, must be non-bounding as Y^HLI Ci ~ 0- Assume that Ci and C2 are not null-homologous. Recall that R P 2 = N Ug D2, where iV is a Mobius band attached along the boundary of the disc JJ)2. If C\ does not bound a disc, it circles around the Mobius band. So C2 C S \u00E2\u0080\u0094 C\ is contained in the interior of D2, hence is contractible, contradicting the assumption that C2 is non-bounding. Thus each Ci bounds a disc. Since each d is bounding, each d separates RP2 into two connected regions, a disc A and the complement of the disc, which contains a Mobius band. Since no Cj intersects any Cj, i ^ j, their corresponding discs will either be disjoint or one will be contained in the other. If Di contains Cj, then Di contains Dj (as Di contains no Mobius band); so we can take Di, D2,Dn, n \u00C2\u00A3 { 1 , m } to be the outermost discs not contained in any other discs, and set S = (J\" = 1 Di U N(P), where N(P) is the disc neighbourhood for each crossing P of p(k); S is a proper connected subspace of R P 2 which contains p(k). Since within S we have p(k) homotopic to a product of conjugates of Ci, each of which is contractible in S, it follows that p(k) is contractible in S, hence contractible in the complement of a small disc in S, that is, within a Mobius band contained in S. With this, the argument is returned to the earlier case of the proof, and we are done. \u00E2\u0080\u00A2 32 Bibliography [1] J.W. Alexander. Topological invariants of knots and links. Trans. Am. Math. Soc, 1928. [2] George Bergman. Ordering co-products of groups. Journal of Algebra, 1990. [3] George Bergman. Right orderable groups that are not locally indicable. Pacific Journal of Math., 1991. 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Braids, orderings, and zero divisors. Journal of Knot Theory and its Ramifications, 1998. N . Smythe. Trivial knots with arbitrary projection. J. Austr. Math. Soc, 1967. A.B. Sossinsky and V.V. Prasolov. Knots, Links, Braids, and 3-Manifolds. AMS, 1991. E. H. Spanier. Algebraic Topology. McGraw-Hill Book Co., 1966. N . Steenrod. The Topology of Fibre Bundles. Princeton University Press, 1951. John Stillwell. Classical Topology and Combinatorial Group Theory. Springer-Verlag, 1980. V . M . Tararin. On convex subgroups of right ordered groups. Siberian Math. Journal, 1994. V . M . Tararin. On the theory of right orderable groups. Math. Notes, 1994. 34 "@en . "Thesis/Dissertation"@en . "2000-11"@en . "10.14288/1.0080041"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Orderable groups and topology"@en . "Text"@en . "http://hdl.handle.net/2429/11015"@en .