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Homotopy string links over surfaces Yurasovskaya, Ekaterina 2008

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Homotopy string links over surfaces by Ekaterina Yurasovskaya  B.Sc., The University of Science and Arts of Oklahoma, 1998 M.A., The University of Oklahoma, 2001  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Doctor of Philosophy in The Faculty of Graduate Studies (Mathematics)  The University Of British Columbia August 1 2008 c Ekaterina Yurasovskaya 2008  Abstract In his 1947 work ”Theory of Braids” Emil Artin asked whether the braid group remained unchanged when one considered classes of braids under linkhomotopy, allowing each strand of a braid to pass through itself but not through other strands. We generalize Artin’s question to string links over orientable surface M and show that under link-homotopy surface string links form a group PBn (M ), which is isomorphic to a quotient of the surface pure braid group PBn (M ). Surface braid groups and their properties are an area of active research by Gonz´ alez-Meneses, Paris and Rolfsen, Gon¸calves and Guaschi, and our work explores the geometric and visual beauty of this subject. We compute a presentation of PBn (M ) in terms of the generators and relations and discuss the orderability of the group in the case when the surface in question is a unit disk D. Ekaterina Yurasovskaya. yurasoe@math.ubc.ca  ii  Table of Contents Abstract  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ii  Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . .  iii  List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  v  Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . viii Dedication  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  ix  1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  1  2 Braid group . . . . . . . . . . . . . . . . . . . . 2.1 Geometric and algebraic definitions . . . . . 2.2 Braid groups and configuration spaces . . . . 2.3 Artin Combing . . . . . . . . . . . . . . . . . 2.4 Surface braids: the question of presentations  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  3 Effect of link-homotopy on the complement of string over surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Definitions and notation . . . . . . . . . . . . . . . . . 3.2 Link-homotopy and the reduced group of a string link . 3.3 Homotopy string links form a group . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  links . . . . . . . . . . . .  6 6 8 9 10  12 12 14 17  4 Homotopy string links over a punctured orientable surface 4.1 Geometric view of surface braids. . . . . . . . . . . . . . . . 4.2 PBn (S): split exact sequence and semi-direct product. . . . . 4.3 PBn (S): structure of the group. . . . . . . . . . . . . . . . .  21 21 23 26  5 PBn (S): surface with a puncture and group 5.1 Presentations of groups . . . . . . . . . . . 5.2 Auxilary presentation . . . . . . . . . . . . 5.3 PBn (S): more pictures of braids . . . . . .  33 34 35 36  presentation . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . .  iii  Table of Contents 5.4  Presentation of PBn (S)  . . . . . . . . . . . . . . . . . . . . .  6 Homotopy string links over closed orientable 6.1 PBn (M ) and PBn (S): the group presentations 6.2 Isotopy braids . . . . . . . . . . . . . . . . . 6.3 Link-homotopy: finger-loop moves . . . . . . 6.4 PBn (M ): group presentation . . . . . . . . .  surface . . . . . . . . . . . . . . . . . . .  38  . . . .  . . . . .  . . . . .  . . . . .  48 48 51 52 53  7 Invariants of homotopy string links over a disk . . . . 7.1 Two definitions of reduced free group . . . . . . . . . . 7.2 Basic commutators . . . . . . . . . . . . . . . . . . . . 7.3 Magnus expansion and basic commutators with repeats 7.4 Injective expansion of RF(k) . . . . . . . . . . . . . . . 7.5 Integer invariants of homotopy string links . . . . . . .  . . . . . .  . . . . . .  . . . . . .  55 55 56 57 59 61  8 PBn (D) is bi-orderable. . . . . . . . . . . . . . . . 8.1 Orderability: definitions and examples . . . . . 8.2 Ordering semi-direct products . . . . . . . . . 8.3 Ordering RF (k) . . . . . . . . . . . . . . . . . 8.4 Automorphisms of RF (k) and order on PBn (D)  . . . . .  . . . . .  . . . . .  64 64 66 68 71  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  74  . . . .  . . . . .  . . . . .  . . . . .  . . . . .  iv  List of Figures 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11  Non-trivial.. . . . . . . . . . . . . . . . . ..braid.. . . . . . . . . . . . . . . . . . . .... . . . . . . . . . . . . . . . . . . . . . which is... . . . . . . . . . . . . . . . . . ..link-homotopically . . . . . . . . . . . ..trivial . . . . . . . . . . . . . . . . . . . The Whitehead link is link-homotopic... ...to the trivial link . . . . . . . . . . . . Hopf link with a knotted component... . ...is link-homotopic to the Hopf link . . Borromean rings . . . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  . . . . . . . . . . .  1 1 1 2 2 2 2 2 3 3 3  2.1 2.2 2.3 2.4  Generator σi . . . . . . σi σk = σk σi . . . . . . . σi+1 σi σi+1 = σi σi+1 σi . Artin Combing of a pure  . . . . . . . . . . . . braid  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  . . . .  7 7 7 10  3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9  Example of a string link . . . . Crossing change . . . . . . . . . Complement of σ . . . . . . . . Complement of σ . . . . . . . . Loop representing [ti , ti ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... is link-homotopically trivial  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  . . . . . . . . .  13 14 15 15 18 18 18 18 18  4.1 4.2 4.3 4.4 4.5  Fundamental polygon L for surface Side view of a braid . . . . . . . . View from above . . . . . . . . . . Generator ai,2k+1 . . . . . . . . . . Generator ai,2k . . . . . . . . . . .  S . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  22 22 23 25 25 v  List of Figures 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17  Generator ti,j . . . . . . . . . . P1 -based strand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P2 -based strand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . crosses itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . crosses itself . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  . . . . . . . . . . . .  25 30 30 30 30 31 31 31 31 32 32 32  5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21  Generator ai,2k+1 . . . . . . . . . . Generator ai,2k . . . . . . . . . . . Generator ti,j . . . . . . . . . . . . Braid Ti,j . . . . . . . . . . . . . . Braid Aj,r in polygon L . . . . . . Transforming polygon L... . . . . . ...into the Pi -polygon . . . . . . . . Braid Aj,r in Pi -polygon . . . . . . PR3 . . . . . . . . . . . . . . . . . PR3 . . . . . . . . . . . . . . . . . PR3 . . . . . . . . . . . . . . . . . PR4 . . . . . . . . . . . . . . . . . PR4 . . . . . . . . . . . . . . . . . PR5: LHS . . . . . . . . . . . . . . PR5: RHS . . . . . . . . . . . . . . PR5: LSH and RHS are equivalent PR6: 1 . . . . . . . . . . . . . . . . PR6: 2 . . . . . . . . . . . . . . . . PR7: braid β over Pj -polygon . . . PR7 simplified . . . . . . . . . . . PR7 in Pi -polygon . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . .  37 37 37 38 38 39 39 39 45 45 45 45 45 46 46 46 47 47 47 47 47  6.1 6.2 6.3 6.4 6.5  −1 braid Ti,n−1 Ti,n . . . . . . . . . −1 braid equivalent to Ti,n−1 Ti,n in relation (PR1) . . . . . . . . . Braid γ in Pi -polygon... . . . . ..becomes.. . . . . . . . . . . .  . . . . . . . Pn -polygon . . . . . . . . . . . . . . . . . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  . . . . .  50 50 50 50 50 vi  List of Figures 6.6 6.7 6.8 6.9 6.10 6.11  equivalent braid in Pj -polygon . . . . . . All components of (PR8)... . . . . . . . . ... in Pj -polygon . . . . . . . . . . . . . . Arc s in a ball B . . . . . . . . . . . . . . Motion of B and s under ambient isotopy Finger-loop move along s . . . . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  . . . . . .  50 51 51 52 52 52  vii  Acknowledgements I thank my parents for their love and support, and I thank my Grandmother for setting an amazing example of unending intellectual curiosity, desire for learning and dedication to her profession. I would like to thank my advisor Dr Dale Rolfsen for his support, advice, patience, and amazing ability of geometric vision. My gratitude also goes to the graduate secretary at the Department of Mathematics, Ms Marija Zimonja, for answering innumerable questions on paperwork matters which invariably surround the pure art of mathematics.  viii  Dedication To my parents and my Grandmother.  ix  Chapter 1  Introduction Milnor’s relation of link-homotopy first appeared in [Mil54] and has enjoyed a long history ever since. By imposing link-homotopy on an object made of strands, such as a knot or a link, we allow each strand to pass through itself, but not through other strands. As a result knots in each individual strand vanish, leaving us with a potentially simpler object. In his paper ”Theory of Braids” [Art47] Emil Artin proposed to apply the relation of link-homotopy to the braid group over a disk, allowing each individual strand of a braid to pass through itself, but not through other strands. Artin then asked whether the braid group under link-homotopy acquired additional relations or remained the same as the classical braid group PBn (D). The question remained unanswered for a long time until in her 1974 paper ”Homotopy of Braids - in Answer to a Question of E. Artin” [Gol74], Deborah Goldsmith described a subgroup of isotopically non-trivial braids that became trivial under the relation of link-homotopy. The group of braids under link-homotopy, which we denote PBn (D) following Goldsmith, is a quotient group of the pure braid group PBn (D). Figures 1.1 through 1.6 demonstrate original Goldsmith’s construction of a non-trivial braid that is link-homotopically trivial.  Figure 1.1: trivial..  Non-  Figure 1.2: ..braid..  Figure 1.3: ....  1  Chapter 1. Introduction  Figure 1.4: which is...  Figure 1.5: homotopically  ..linkFigure 1.6: ..trivial  Besides a purely aesthetical attraction of the construction, the new group PBn (D) was destined to play a fundamental role in the solution of another long-standing question which we describe below. A topological link is a finite collection of disjoint circles embedded in a 3-manifold. Possible arrangements of circles are endless, giving rise to a problem of classification: determine when one link can be deformed into another by moving strands without breaking them. In order to simplify the question, we apply the relation of link-homotopy to the link. Link-homotopy removes knots from each individual link component, allowing us to study linking in its pure form. Thus, for example, the Whitehead link in the Figure 1.7 and the trivial link in the Figure 1.8 are link-homotopic. The link in the Figure 1.9 is link-homotopic to the Hopf link in the Figure 1.10.  Figure 1.7: The Whitehead link is link-homotopic...  Figure 1.8: ...to the trivial link  2  Chapter 1. Introduction  Figure 1.9: Hopf link with a knotted component...  Figure 1.10: ...is link-homotopic to the Hopf link  The foundational paper by John Milnor [Mil54] appeared in 1954, where Milnor described the result of link homotopy on the link and the structure of the surrounding space. Milnor succeeded in classifying links with 3 components, as well as links for which every sub-link is trivial, for example the Borromean rings in Figure 1.11 are not link-homotopic to the trivial 3-component link. In addition Milnor obtained a set of numerical invariants  Figure 1.11: Borromean rings that classify links up to a certain indeterminacy. The indeterminacy proved to be a significant impediment. Despite the efforts of many mathematicians no further progress was made until 1988, when Jerome Levine classified links of 4 components [Lev88]. In 1990 Nathan Habegger and Xiao Song Lin solved the question and obtained a full classification of links up to link-homotopy embedded in Euclidean space or in the 3-sphere [HL90]. A fundamental tool in the solution was a group of link-homotopy classes of string links based in a disk. A string link is a pure braid with the monotonicity requirement dropped: its strands are possibly knotted upon themselves and can make any possible 3  Chapter 1. Introduction tangles with other strands. The link-homotopy relation turns each string link into a pure braid, and shows the group of link-homotopy string links to be precisely Goldsmith’s quotient group PBn (D) described above. The action of PBn (D) was proved to be algorithmically computable, determining equivalence of two links in a finite number of steps. In the present work we consider the group of link-homotopy classes of string links based on an orientable surface. The delicate geometry of the setup and close connection with the surface braid group indicate that the group of homotopy string links possesses a refined and interesting structure. The new results include proofs that the set of homotopy string links in fact forms a group, a detailed description of the group structure, and group presentations in terms of the generators and relations for the surfaces with and without boundary. In Chapter 2 we present the necessary background and definitions of classical and surface braid groups. In Chapter 3 we define string links over a surface S and discuss the effect of link-homotopy on the string link. In addition we consider the complement of the string link in the product space S × I of the surface S with the unit interval I. To this complement we associate an invariant: a certain ”reduced group” that remains invariant under link-homotopy. The reduced group serves as a main tool in subsequent computations where we decide whether a particular string link is link-homotopically trivial. In Chapter 4 we consider a surface S obtained by deleting a single point from a compact, connected, orientable surface without boundary, different from the sphere S2 . We describe the structure of the group of link-homotopy classes of string links over S, which we shall denote by PBn (S). As a main result of Chapter 4 we obtain an exact sequence of groups of string links (Corollary 4.8), that we use in Chapter 5 to build a presentation of PBn (S) in terms of the generators and relations (Theorem 5.1). In Chapter 6 we turn our attention to the case of a closed orientable surface M . We combine the presentation of PBn (S) with presentations of braid groups over M and over S and apply certain geometric arguments on the structure of link-homotopy. The main result of both Chapter 6 and of this work as a whole is the presentation in terms of generators and relations of the group of link-homotopy classes of string links over a closed orientable surface (Theorem 6.3). Chapters 7 and 8 focus on the ”classical” group PBn (D) of homotopy 4  Chapter 1. Introduction string links over a disk D as described in [Gol74] and [HL90]. Chapter 7 describes a complete set of link-homotopy invariants of PBn (D) by giving a new proof of a previously known result by John Milnor [Mil54] on the expansion of a certain quotient of a free group into a polynomial ring (Theorem 7.11). Chapter 8 builds on the results of Chapter 7 and presents a construction of a strict total ordering of PBn (D) that is invariant under left and right multiplication (Theorem 8.17). Ordered groups connected to topology have been a subject of extensive study in recent years ([DDRW02], [KR02]), and our work adds PBn (D) as an example to the collection of such groups.  5  Chapter 2  Braid group 2.1  Geometric and algebraic definitions  Let M be a compact connected surface without boundary, orientable or nonorientable, possibly with a finite number of punctures. Let P = {P1 , ..., Pn } be a set of distinct points in the interior of M . Let I1 , ..., In be n copies of the unit interval I. Let nj=1 Ij denote the disjoint union of these intervals. Definition 2.1. A braid β on n strands over a surface M is a smooth or piecewise linear proper imbedding k  Ij → M × I  β: j=1  such that 1. β|(Ij (0)) = (Pj , 0) and β|(Ij (1)) ∈ P × 1. 2. β|(Ij (t)) ∈ M × t for all t ∈ Ij . Two braids are equivalent if one can be deformed to the other through the family of braids, with endpoints fixed throughout the deformation. The set of all equivalence classes of n-strand braids over M forms a group Bn (M ), with concatenation serving as group operation. The inverse of a braid β is given by mirror reflection. When the surface in question is the unit disk D, the braid group Bn (D) can be defined purely algebraically via generators and relations that were shown by Emil Artin to be a complete set. Definition 2.2. The classical braid group on n strands Bn (D) has generators σ1 , σ2 , ..., σn−1 and relations: σi σk = σk σi ,  where |i − k| ≥ 2,  σi σi+1 σi = σi+1 σi σi+1 6  Chapter 2. Braid group  Figure 2.1: Generator σi  Figure 2.2: σi σk = σk σi  Figure 2.3: σi+1 σi σi+1 = σi σi+1 σi  The generating braid σi appears in Figure 2.1. Figures 2.2 and 2.3 illustrates the defining relations of Bn . A braid β defines a permutation of its n endpoints, resulting in a welldefined surjective homomorphism and an exact sequence 1 → PBn (D) → Bn (D)  Σn → 1  where Σn is the symmetric group on n letters. The kernel of this exact sequence is a subgroup of index n!, called the group of pure braids PBn (D). A Pi -based strand of a pure braid shall end at the point (Pi , 1).  7  Chapter 2. Braid group  2.2  Braid groups and configuration spaces  A particular beauty of the braid groups lies in the multitude of ways by which a braid group can be defined. Here we give an alternative definition that is far less intuitive than the geometric one. This definition, due to Fadell and Neuwirth [FN62], views the braid group as a fundamental group of a configuration space, which we describe below. Let Fn M denote the space of n-tuples of distinct points in M . In fact, Fn M is a subspace of the n-fold product of M with itself, minus the big diagonal ∆n Fn M = M n − ∆n , where ∆n = {(x1 , ...xn ) ∈ M n |xi = xj for some i = j}. Fadell and Neuwirth define the surface pure braid group PBn (M ) as the fundamental group π1 Fn M . To obtain the full braid group Bn (M ) we consider the action of the symmetric group on n letters Σn on the space Fn M . Σn acts on Fn M by permuting coordinates, and when we consider the space of orbits Fn M/Σn , we see that Bn (M ) is the fundamental group π1 (Fn M/Σn ). Note that if M is a connected surface, then both PBn (M ) and Bn (M ) are independent (up to isomorphism) of the choice of basepoint (P1 , ..., Pn ). The view of PBn (M ) as the fundamental group of a space allows us to use tools of homotopy theory. It is shown in [FN62] that Fn M has the structure of a fiber bundle. Given 1 ≤ m < n p : Fn M → Fm M (x1 , ..., xn ) → (xn−m+1 , ..., xn ) is a fibration with a fiber Fn−m (M \ {Pn−m+1 , ..., Pn }) - notice that the surface M has m points removed. We can now work with a long exact sequence of homotopy groups of these spaces. Setting m = n − 1 and leting Pn−1 denote the set of points {P2 , ..., Pn }, we obtain the fiber F1 (M \ Pn−1 ), which by definition is just the punctured surface M \ Pn−1 . We then have an exact sequence ... → π2 Fn M → π2 Fn−1 M → π1 (M \ Pn−1 ) → PBn (M ) → PBn−1 (M ) → 1. Note that we can build the same exact sequence for the fibrations Fn−1 M → Fn−2 M with fiber M \ {P3 , ..., Pn }, etc.  8  Chapter 2. Braid group With exception of the sphere S2 and the projective plane RP 2 , the punctured surface M \ Pn−1 has the homotopy type of a wedge of circles, hence higher homotopy groups of M \Pn−1 are trivial: πk (M \Pn−1 ) = 1 for k ≥ 2. The above long exact sequence implies that πk Fn M  πk Fn−1 M  πk Fn−2 M  ...  πk F1 M = πk (M )  for k ≥ 3  and π2 Fn M ⊂ π2 Fn−1 M ⊂ π2 Fn−2 M ⊂ ... ⊂ π2 F1 M = π2 (M ). It is well-known that πk (M ) = 1 for k ≥ 2, and immediately we see that Fn M is an Eilenberg-MacLane space, or K(π, 1), i.e. πk Fn M = 1 for k ≥ 2. Since Fn M is a covering space of Fn M/Σn , it follows that Fn M/Σn is an Eilenberg-MacLane space as well. From the long exact sequence of homotopy groups with fiber M \ Pn−1 we have now obtained the classical exact sequence of braid groups (PBS): 1 → π1 (M \ Pn−1 ; P1 ) → PBn (M )  p  PBn−1 (M ) → 1  In terms of braid groups defined geometrically the map p corresponds to removing the P1 -based strand of a pure braid. The kernel π1 (M \ Pn−1 ; P1 ) corresponds to the subgroup of braids whose P1 -based strand winds around the straight strands based at points P2 , ..., Pn .  2.3  Artin Combing  In case the surface M is a unit disk D, we obtain an exact sequence that splits: p 1 → F(n − 1) → PBn (D) PBn−1 (D) → 1 The splitting is obtained by adding a trivial strand at the point P1 to the braid on n − 1 strands. The kernel of this exact sequence is the free group F(n − 1) on n − 1 generators, which is the fundamental group of a punctured disk D\Pn−1 . We see that PBn (D) has the structure of a semi-direct product PBn (D)  F (n − 1)  PBn−1 (D).  The semi-direct product decomposition can be repeated on PBn−1 (D) and present us with what is commonly known as Artin Combing or Artin normal form([Bir74]): PBn (D)  F (n − 1)  (F (n − 2)  ...  (F (2)  F (1))...). 9  Chapter 2. Braid group The Artin normal form solves the word problem in PBn (D): each pure braid β over D can be written uniquely as a product β = β1 ...βn−1 , with βi ∈ F(n − i). Figure 2.4 shows a braid in the Artin normal form.  Figure 2.4: Artin Combing of a pure braid In fact, the word problem is solved in Bn (D) as well. Given a word w in Bn (D), consider the permutation that w induces in the symmetric group Σn . If the permutation is non-trivial, then so is the word w. If the permutation is trivial, then the braid is pure, and we use Artin Combing to decide if w is trivial or not. To conclude the section on the normal form, we cite Artin’s own opinion from [Art47]: ”Although it has been proved that every braid can be deformed into a similar normal form the writer is convinced that any attempt to carry this out on a living person would only lead to violent protests and discrimination against mathematics. He would therefore discourage such an experiment”  2.4  Surface braids: the question of presentations  Artin’s original interest concerned the classical groups of braids over a disk, but the definition that connected braid groups and configuration spaces of points on a surface gave rise to particular interest in surface braid groups and their properties. Many mathematicians computed presentations of surface braid groups at various times. A presentation for the braid groups over the sphere S2 was given by W. Magnus in 1934 [Mag34] and by O. Zariski in 1936 [Zar36]. In 1937 Zariski gave a presentation of surface braid groups [Zar37]. Van Buskirk 10  Chapter 2. Braid group studied groups of braids over the sphere S2 and projective plane RP 2 and showed that these braid groups possess elements of finite order, while braid groups over any other surfaces are torsion-free. A presentation of Bn (S2 ) was computed [FVB62]. G. P. Scott computed presentations of braid groups over all closed surfaces [Sco70]. Of more recent works on presentations one can name the ones by D.L. Gon¸calves and J. Guaschi [GG04] and P. Bellingeri [Bel04]. Especially significant for our current work is an article of Juan Gonz´alezMeneses [GM01],”New presentations of surface braid groups” that appeared in 2001. Our own calculations rely on his methods of computation and geometric visualization. Particular details of his methods shall be discussed in Chapters 4-6 of the present work.  11  Chapter 3  Effect of link-homotopy on the complement of string links over surfaces. 3.1  Definitions and notation  To be consistent with existing literature we will follow the notation set before in [Mil54],[Lev88] and [HL90]. Let S be a compact orientable surface of genus g ≥ 1, perhaps with a single puncture. Let I be the unit interval [0, 1]. Choose k points P = {P1 , ...Pk } to lie in the interior of S. Let I1 ...Ik be k copies of the unit interval I. Let ki=1 Ii denote the disjoint union of these intervals. A string link σ on k strands over a surface S is a smooth or piecewise linear proper imbedding k  Ii → S × I  σ: i=1  such that σ|(Ii (0)) = (Pi , 0) and σ|(Ii (1)) = (Pi , 1). When the surface S is understood, σ shall be called simply a ”string link”. Informally, we can say that a string link is a pure braid with the monotonicity requirement relaxed, whose endpoints are still fixed and whose strands may knot upon themselves and on other strands. We orient the strands downwards from S × 0 to S × 1. Every pure braid is in fact a string link in itself. Figure 3.1 shows an example of a string link on 2 strands in the cylinder D × I, where D stands for the unit disk. An ambient isotopy between string links σ and σ is an orientation-preserving diffeomorphism of S × I which maps σ onto σ while keeping the boundary S × {0, 1} point-wise fixed and is isotopic to the identity, relative S × {0, 1}. Let S × I − σ denote the complement of the image of σ in S × I. The group of a string link σ, denoted G(σ), is the fundamental group S × I − σ.  12  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces.  Figure 3.1: Example of a string link We define a set of meridians {t1 , ...tk } explicitly as follows. Choose an orientation in S × I − σ and let mi be an oriented circle {∗} × S 1 around a tubular neighbourhood I × D of the Pi -based strand. Let x∗ be a chosen basepoint in the space S × I − σ and connect x∗ to mi by a path γi in S × I − σ. The meridian ti is an element of G(σ) represented by γi mi γi−1 . Choosing a different path γi from x∗ to mi takes ti to its conjugate ti by some ζ ∈ G(σ), where ti = ζti ζ −1 . Definition 3.1. [HL90] We say that two string links σ and σ are linkhomotopic, if there is a homotopy of the strings in S × I, fixing S × {0, 1} and deforming σ to σ , such that the images of different strings remain disjoint during the deformation. During the course of deformation, each individual strand is allowed to pass through itself but not through other strands. As an example, we see that the string link of Figure 3.1 is not ambient isotopic with a braid, but is link-homotopic to the braid σ1−2 . An alternative definition of link-homotopy is more convenient for our purposes (see [Mil54], [Lev88], and [HL90]) Link-homotopy is an equivalence relation on string links, that is generated by a sequence of ambient isotopies of S × I fixing S × {0, 1}, and local crossing changes of arcs from the same strand of a string link. See Figure 3.2. We now define the main tool of this chapter. Let J(σ) be the normal subgroup of G(σ) generated by elements of the form [ti , ti ], where ti and ti are conjugates of the meridian ti for any i = 1, ..., k. Note that J(σ) remains invariant under ambient isotopy, since an ambient isotopy of S × I − σ takes conjugates of meridians to conjugates of meridians. We allow conjugates of 13  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces.  Figure 3.2: Crossing change the meridian ti to commute, for any i = 1, ..., k, and obtain a quotient group G(σ), given by G(σ)/J(σ). We call G(σ) the reduced group of σ. Remark 3.2. We obtain the same presentation of the group G(σ) if instead of saying ”all conjugates of ti commute”, we say ”each meridian ti commutes with all of its conjugates”: [ti , ti ] = 1 if and only if [ti , ti ] = 1 Proof. ⇒ Obvious. ⇐ [ti , ti ] = ti ti (ti )−1 (ti )−1 −1 = ζti ζ −1 ti ζt−1 (ti )−1 ζζ −1 i ζ −1 −1 = ζ(ti ti t−1 i (ti ) )ζ  = ζ[ti , ti ]ζ −1 = ζζ −1 = 1  3.2  Link-homotopy and the reduced group of a string link  The reduced group G(σ) is an invariant of the string link σ under linkhomotopy. Theorem 3.3. If string links σ and σ over surface S are link-homotopic, then their reduced groups G(σ) and G(σ ) are isomorphic. Proof. Here we view link-homotopy in the complement of a string link σ as a sequence of ambient isotopies and a finite number of crossing changes confined to small disjoint ball neighbourhoods. Each such neighbourhood 14  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces. contains a crossing of an individual strand with itself. Ambient isotopy leaves the fundamental group G(σ) invariant, and hence it leaves invariant the reduced group G(σ). In what follows we explore the effect of a single crossing change on G(σ). Let us say that string links σ and σ are related by a single crossing change of the jth strand. Consider the Wirtinger presentation argument in the Figures 3.3 and 3.4, that refer to string links σ and σ respectively. In  Figure 3.3: Complement of σ  Figure 3.4: Complement of σ  both figures, the arrows A, B, C and D represent loops which are conjugates of the meridian tj . From Figure 3.3 we obtain the Wirtinger relations AC = DB and A = B. Figure 3.4 gives us the Wirtinger relations AC = DB and C = D. Under link-homotopy Figures 3.3 and 3.4 are equivalent, hence link-homotopy introduces relations AC = DB A=B C=D Solving this system we see that under link-homotopy we obtain a set of equalities: AC = CA, AD = DA, etc - i.e. a set of relations saying that certain conjugates of tj now commute with each other. We see that these relations have no effect on the reduced groups and hence G(σ) and G(σ ) are isomorphic. We now define an important quotient of the free group which shall serve as a major building block in many subsequent constructions. Definition 3.4. Let F be the free group on generators {x1 , ..., xn }∪{t1 , ..., tk }, for non-zero n and k. We add relations saying that the conjugates of ti commute for all i = 1, ..., k and we denote the resulting partially reduced free group by F(n, k). 15  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces. The reduced group of a link-homotopically trivial string link has a particularly nice structure. Let Ik denote the trivial braid on k strands based at points P1 , ..., Pk . Corollary 3.5. Let σ be a link-homotopically trivial string link over a surface S. Then the reduced group G(σ) is isomorphic to the partially reduced free group. Proof. If the surface S of genus g ≥ 1 is closed, let n = 2g − 1, otherwise let n = 2g. The fundamental group G(Ik ) is isomorphic to a free group on n + k generators. This fact is easy to see, since G(Ik ) is isomorphic to the fundamental group π1 (S \ P) of the surface S with the point set P deleted. Note that the projections of the meridians {t1 , ..., tk } onto the surface S are part of the generator set of π1 (S \ P). Thus G(Ik ) is isomorphic to the partially reduced free group F(n, k). The string link σ is link-homotopic to Ik . Therefore by Theorem 3.3, G(σ) is isomorphic to F(n, k) as well. Let γ be a loop in the complement S × I − σ of a string link σ . We shall say that γ is H-trivial if there is a link-homotopy H of σ ∪ γ such that 1. H(γ) represents the identity in G(H(σ)). 2. During the link-homotopy H images of the strands of σ and the image of the loop γ remain disjoint. Corollary 3.6. Let σ be a string link, and let γ be a loop in S × I − σ. Then γ is H-trivial if and only if γ represents the identity in the reduced group G(σ). Proof. =⇒ Let γ in S ×I −σ be H-trivial. Then H is the link-homotopy that takes σ to a string link σ , such that H(γ) represents the identity element in the fundamental group G(σ ). Consequently the loop class [H(γ)] goes to the identity in the corresponding reduced group G(σ ) under the quotient map. J(σ ) → G(σ )  G(σ )  [H(γ)] → [H(γ)] By Theorem 3.3, the reduced groups G(σ) and G(σ ) are isomorphic, consequently γ represents the identity in G(σ) under the isomorphism H∗ : G(σ) → G(σ ) [γ] → [H(γ)] 16  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces. ⇐= In [Mil54], John Milnor presents an example of a homotopically nontrivial loop γ in the complement of a link, where γ becomes trivial if the link is affected by link-homotopy. The sequence of Figures 3.5 through 3.9 shows Milnor’s original argument. For our proof, the ith strand represents a strand of a string link σ. The picture argument shows the sequence of steps leading the ith strand to pass through itself as the loop representing [ti , ti ] becomes trivial. A loop class that represents identity in G(σ) is given by a product of conjugates of commutators [ti , ti ] for 1 ≤ i ≤ k. Given a loop γ representing identity in G(σ) we repeat the picture argument successively for each commutator of the form [ti , ti ]. As a result we obtain a link-homotopy H of σ ∪ γ and demonstrate loop γ to be H-trivial.  3.3  Homotopy string links form a group  For the remainder of the section let M denote an orientable surface, with a finite number of punctures or without punctures at all. The next important theorem shows that any string link is link-homotopic to a pure braid. The proof is due to Roger Fenn and Dale Rolfsen, and we repeat it here almost unaltered. Theorem 3.7. Every n-strand string link over a surface M is link-homotopic to a pure braid. Proof. Our proof is geometric, and will use induction on n. For the basis of the induction, first consider a string link L of one component, which runs from (q, 0) to (q, 1) in M × I. By a preliminary homotopy, we may assume that, under the natural projection p : M × I → M , the composite p ◦ L has only double points (including the endpoints) and that they project transversely in M . Orienting L in a direction starting at (q, 1) and proceeding along L in that direction, we say that L is descending if each time one encounters a double point in the projection it is on the part of the strand which is “over” , i.e., with greater I coordinate. By a well-known method of knot theory (see, for example, [Ada94]), if L has such a descending projection, we may construct an ambient isotopy of M × I which moves points only in the I direction, is fixed on M × {0, 1}, and moves L so that L(t) lies in M × {t}. But if the projection of L is not descending, that is the first encounter of a double point is “under” instead of “over” one can 17  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces.  Figure 3.5: Loop representing [ti , ti ]  Figure 3.6:  Figure 3.7:  Figure 3.8:  Figure 3.9: ... is link-homotopically trivial  18  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces. easily modify L by a link homotopy which moves the under strand upward, crossing through the other strand, moving points only vertically (that is, without changing the M coordinate), and thus changing L to another string link with the same projection, but which is now descending, if one performs this modification in a neighbourhood of each crossing of the projection which violates the descending condition. This finishes the basis for the induction. Now suppose n > 1 and consider an n-strand string link L over M . Let Li denote the ith strand of L, for 1 ≤ i ≤ n. For induction, we may assume the theorem is true for all (n − 1)-strand links over all surfaces. Again by a preliminary homotopy, we may also assume the projection of L on M has only double points. Moreover, by considering the operation of the previous paragraph, and the fact that each double point of L1 in its projection on M has a neighbourhood disjoint from projections of the other components of L, we may move L1 by a homotopy not passing through the other components, and ambient isotopy (which may move the other components) so that L1 (t) lies in M × {t} for all t in I. Thinking of p ◦ L1 (t) as a point moving continuously in the interior of the surface M , this motion may be extended to a continuous family of homeomorphisms ht : M → M ; that is, ht (q1 ) = p ◦ L1 (t) for all t ∈ I. It may be further assumed that ht (qi ) = qi for 2 ≤ i ≤ n. Then we may construct a homeomorphism H : M × I → M × I by the formula H(x, t) = (ht (x), t). Note that H −1 ◦L1 (t) = (q1 , t) ∈ M ×{t}. Therefore the string link H −1 ◦L in M ×I has its first component “vertical” – the image of the first component is simply {q1 } × I. Because of this, we may think of the remaining components {H −1 ◦ L2 , . . . , H −1 ◦ Ln } as comprising a string link of n − 1 components over the surface M \ {q1 }. By inductive hypothesis, this string link is linkhomotopic to a pure (n−1)-strand braid, say β, in (M \{q1 })×I. Composing such a homotopy with the homeomorphism H, produces a homotopy of L to L1 ∪ (H ◦ β). Since H preserves t-levels, the resulting string link L1 ∪ (H ◦ β) is a braid. Lemma 3.8. Link-homotopically trivial surface braids Hn (M ) are a normal subgroup of PBn (M ) Proof. Concatenation of two link-homotopically trivial braids produces a link-homotopically trivial braid. If a braid β is link-homotopically trivial, then β −1 is also link-homotopically trivial. To see this, move β −1 by isotopy to be in the mirror-reflection position with respect to β and use the reflection 19  Chapter 3. Effect of link-homotopy on the complement of string links over surfaces. of the link-homotopy. Note that intermediate stages of link-homotopy may be string links, rather than braids. If β is link homotopically trivial, then clearly for any x ∈ PBn (M ), xβx−1 is link-homotopically trivial, hence Hn (M ) is normal in PBn (M ). Let PBn (M ) denote the set of link-homotopy classes of string links over surface M , which we shall call simply homotopy string links. Proposition 3.9. Under concatenation PBn (M ) is a group isomorphic to the quotient of the pure braid group PBn (M ) by the subgroup of linkhomotopically trivial braids Hn (M ) PBn (M )  PBn (M )/Hn (M ).  Proof. By Theorem 3.7 each string link is link-homotopic to a pure braid. Lemma 3.8 lets us express PBn (M ) as a quotient of the pure braid group PBn (M ): PBn (M ) PBn (M )/Hn (M ). A quotient of a group by its normal subgroup is a group. Thus PBn (M ) inherits from PBn (M ): operation - concatenation of homotopy string links. inverse - mirror reflection, up to link-homotopy equivalence.  20  Chapter 4  Homotopy string links over a punctured orientable surface The first two sections of this chapter discuss the structure of the surface pure braid group PBn (S) for the particularly convenient case of an orientable surface with a single puncture. We are then able to demonstrate by geometric arguments that the group of homotopy string links PBn (S) has the structure of an iterated semi-direct product. The goal of this chapter is to establish a certain short exact sequence for PBn (S). The new sequence is similar to the pure braid group exact sequence from Chapter 2 and serves as a fundamental tool for finding a presentation of PBn (S) in Chapter 5.  4.1  Geometric view of surface braids.  The following geometric method of easily depicting braids over a surface appears in [GM01] for the case of closed surfaces. We adapt the construction to surfaces with a single puncture. Surface. For the remainder of this chapter let S be the surface obtained by deleting a single point from a closed orientable surface of genus g ≥ 1. Let us represent S by its fundamental polygon L with 4g sides, with pairs labeled α1 , ..., α2g . Let the sides of L be identified as shown in Figure 4.1. Neighbourhoods of the vertices of L are removed and represent the puncture on S - there is thus a ”hole” in place of each vertex. Choose n basepoints P = {P1 , P2 , ...Pn } across the diameter of L. L with the above identification does indeed give us a punctured orientable surface of genus g: observe that the boundary is connected, the identification space is connected and orientable with Euler characteristic 1 − 2g. One may picture L as a disk with 2g one-handles attached. Drawing braids over S. Let I = [0, 1] be the unit interval. We represent S × I by the cylinder L × I with opposite sides identified, following the original identifications of L. Let us assume that S × 0 is the top of the  21  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.1: Fundamental polygon L for surface S cylinder. A surface braid β over S appears in Figure 4.2, with strands oriented downwards. Note that a string of a braid may ”go through the wall” of the cylinder L × I and re-appear from the opposite ”wall”.  Figure 4.2: Side view of a braid We obtain a less ambiguous view of the braid if we look on the cylinder L × I from the top - see Figure 4.3. This way we see the strings of the braid as paths in the surface S. At a crossing of two strands, the top strand is the one that reaches the crossing first.  22  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.3: View from above  4.2  PBn (S): split exact sequence and semi-direct product.  Let M be a compact surface without boundary, different from a sphere S2 or a projective plane RP 2 . Let Pn−1 denote the set of points {P2 , ..., Pn } in M . Recall that by removing the P1 -based strand of a braid we obtain a homomorphism PBn (M ) PBn−1 (M ) which fits in the classical exact sequence of pure braid groups (PBS): 1 → π1 (M \ Pn−1 ; P1 ) → PBn (M ) → PBn−1 (M ) → 1  (PBS)  The question ”When does PBS split?” has long been of interest. Here we restrict our attention to orientable surfaces. In [Art47] Artin showed that in the case of M = D - a disk - the PBS splits for all n. In [FVB62] Fadell and Van Buskirk established a geometric section for M = S2 - a sphere. In [Bir69] Joan Birman established a splitting for the case of M = T2 - a torus, and posed the question of splitting in the case of surfaces of genus g ≥ 2. Recent work by Gon¸calves and Guaschi [GG04] completely answered Birman’s question. Given a compact, connected, orientable surface M without boundary of genus g ≥ 2, the PBS splits if and only if the number of strands n equals 2. The case of a surface with punctures is much simpler, and the PBS splits for all n ≥ 2. The following proposition is proved in [GG04].  23  Chapter 4. Homotopy string links over a punctured orientable surface Proposition 4.1. Let M be the surface obtained by deleting a single point x0 from a compact, connected surface without boundary different from S2 and RP 2 . For each integer n > 1, the pure braid short exact sequence (PBS) splits. Proof. For a complete proof see [GG04]. The following points are important for our subsequent constructions: (1.) The splitting is geometric, and is equivalent to adding a strand that runs ”parallel” to the puncture x0 × I. (2.) The proof does not depend on the presentation of the fundamental group π1 (M \ Pn−1 ; P1 ).  We now return to the surface S obtained by deleting a single point from a closed orientable surface of genus g ≥ 1. Let Pn−i denote the set of points {Pi+1 , ..., Pn }. P0 is understood to be an empty set. As a consequence of Proposition 4.1 we have the following corollary: Corollary 4.2. Let S be the surface obtained by deleting a single point from a closed orientable surface of genus g ≥ 1. PBn (S) is isomorphic to an iterated semi-direct product of fundamental groups π1 (S \ Pn−1 ; P1 )  (π1 (S \ Pn−2 ; P2 )  (...  (π1 (S \ P1 ; Pn−1 )  π1 (S; Pn ))...))  Statement (2.) in the proof of Proposition 4.1 implies that the iterated semi-direct product above is independent of the presentation of π1 (S \ Pn−i ; Pi ) for each i = 1, ..., n. Let us therefore choose a set of generators of π1 (S \ Pn−i ; Pi ) that shall be the most convenient for further proofs and drawings. It is a well-known fact that the fundamental group π1 (S \Pn−i ; Pi ) is free for each i = 1, ..., n − 1 on two types of generators ai,r and ti,j : • Let 1 ≤ r ≤ 2g. Figures 4.4 and 4.5 show the generator ai,r . Note that ai,r shall go up in L if r is odd and downwards if r is even. • Let i + 1 ≤ j ≤ n. Figure 4.6 shows the generator ti,j . Note that the fundamental group π1 (S; Pn ) has {an,k }2g k=1 as a free system of generators. Let us now describe π1 (S \ Pn−i ; Pi ) in terms of braids. We can view π1 (S \ Pn−i ; Pi ) as a free subgroup of PBn (S), which we shall denote by 24  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.4: Generator ai,2k+1  Figure 4.5: Generator ai,2k  Figure 4.6: Generator ti,j F(2g + n − i). The strands based at {P1 , ..., Pi−1 , Pi+1 , ..., Pn } are trivial and go vertically down without winding. The Pi -based strand winds around the straight strands based at {Pi+1 , ..., Pn }, and through the walls of L × I. Generators of the free subgroup F(2g + n − i) correspond precisely to those of π1 (S \ Pn−i ; Pi ). We label generators of both groups in the same way, to avoid excessive notation. We re-use Figures 4.4, 4.5, and 4.6 to picture the two types of generators of F(2g + n − i) and imagine that we are now looking at a braid from the top of cylinder L × I. The strand based 25  Chapter 4. Homotopy string links over a punctured orientable surface at (Pi , 0) is the time-history in L × I of a point following the generator of π1 (S \ Pn−i ; Pi ). We can now repeat our previous corollary in terms of braids Corollary 4.3. Let S be the surface obtained by deleting a single point from a closed orientable surface of genus g ≥ 1. PBn (S) is isomorphic to the iterated semi-direct product of braid subgroups F(2g + n − 1)  (F(2g + n − 2)  (F(2g + n − 3)...(F(2g + 1)  F(2g))...))  We obtain a presentation for PBn (S) that is very similar to the classical Artin Combing of the pure braid group PBn (D) over a disk .  4.3  PBn (S): structure of the group.  Recall from Proposition 3.9, that the group of link-homotopy string links PBn (S) is the quotient group of the pure braid group PBn (S) by the normal subgroup Hn (S) of braids that are link-homotopic to the identity braid. In this section we determine the necessary and sufficient condition for a pure braid over S to be link-homotopically trivial. Our argument is an extension of a similar construction used by Deborah Goldsmith [Gol74] for the case of the classical pure braid group PBn (D). Let S be the surface obtained by deleting a single point x0 from a closed orientable surface of genus g ≥ 1. Let k be a braid in PBn (S). By Corollary 4.3 k can be written as the product k = k1 ...kn where ki belongs to the braid subgroup F(2g + n − i) as described above, for 1 ≤ i ≤ n. If we were to draw a picture of the braid k, it would look very similar to a picture of the normal form of a classical pure braid over a disk D, such as in Figure 2.4, to which we refer the reader. Proposition 4.4. k is link-homotopically trivial if and only if each ki is link-homotopically trivial for i = 1, ..., n Proof. ⇐= Obvious. =⇒ Suppose k = k1 ...kn is link-homotopically trivial. Remove strands based at P1 through Pi . Clearly, the braid k on the remaining strands is still linkhomotopically trivial. Let L denote the link-homotopy that takes k to a 26  Chapter 4. Homotopy string links over a punctured orientable surface trivial braid on n − i strands. Add i trivial strands to k to obtain the braid ki+1 ...kn in PBn (S). It can be easily seen that the braid ki+1 ...kn is link-homotopically trivial: Consider a small annulus -neighbourhood D that contains the puncture x0 on S and avoids the pointset P. The link-homotopy L can be approximated by a link-homotopy L that avoids D × I, as well as the -neighbourhoods U0 of the ”ceiling” S × 0 and U1 of the ”floor” S × 1 in cylinder S × I. By an isotopy we deform the i trivial strands based at P1 through Pi to lie entirely inside the neighbourhood U0 ∪ (D × I) ∪ U1 . We then apply the link-homotopy L that takes the strands based at Pi+1 , ..., Pn to a trivial braid on n − i strands, and then by an isotopy move the strands based at P1 , ..., Pi out of the neighbourhood U0 ∪ (D × I) ∪ U1 into their original straight position. Since Hn (S) is a subgroup of PBn (S) we know that (ki+1 ...kn )−1 is also link-homotopically trivial. Thus k · (ki+1 ...kn )−1 = k1 ...ki ∈ PBn (S) is linkhomotopically trivial as well. Now remove the strands based at P1 through Pi−1 , and replace them with trivial strands. By the same argument as above we see that the braid ki is link-homotopically trivial. Let us now adopt a new notation. Given elements t and g in a group G, let tg denote a conjugate of t by g: tg = gtg −1 Proposition 4.5. Let k ∈ F(2g + n − i) for some 1 ≤ i ≤ n − 1. Then k ∈ Hn (S) if and only if k is in the normal subgroup of the braid group PBn (S), generated by the commutators [ti,j , ti,j g ] for i + 1 ≤ j ≤ n, g ∈ F(2g + n − i). Proof. (⇐=) The proof is contained in the sequence of figures 4.7 through 4.17, which illustrates the case when the surface S is the 2-torus T2 , depicted without the puncture for convenience of drawing. We can, however, easily imagine the puncture obtained by removing small neighbourhoods of vertices of the fundamental polygon T2 without effect on the link-homotopy. The link-homotopy trivial braid τ is given by −1 −1 −1 a1,1 τ = t1,2 a1,1 t1,2 a−1 ] 1,1 t1,2 a1,1 t1,2 a1,1 = [t1,2 , t1,2  The points marked 1, 2, 3, 4 are located on the front face of the cylinder T2 × I and are identified with the points 1 , 2 , 3 , 4 on the back face of T2 × I. Note that in the process of link-homotopy the non-trivial strand of 27  Chapter 4. Homotopy string links over a punctured orientable surface the braid τ may cross itself. (=⇒) Let k ∈ F(2g + n − i). Then k is a braid with its i-th strand γ winding through the walls of L × I and among the straight strands based at Pi+1 through Pn . Remove the strands based at the points P1 , ..., Pi−1 . Let In−i denote the disjoint union of unit intervals nj=i+1 Pj × I. View γ as an element γ of the fundamental group G(In−i ) of S × I − In−i as follows. In the top L × 0 of the cylinder L × I connect point Pi × 0 to some vertex V × 0 by a straight line segment. On the ”floor” L × 1 draw a parallel segment from Pi × 1 to V × 1. Now join V × 0 and V × 1 by a straight line segment to obtain the loop γ. Recall that the fundamental group G(In−1 ) is isomorphic to the free group on 2g generators going through the ”walls” of L × I and on generators xj for i + 1 ≤ j ≤ n, where xj is a meridian that loops around the j th strand of In−1 . Since k is link-homotopically trivial, the winding strand γ is link-homotopic to a straight strand. By Corollary 3.6 the loop γ represents a trivial element in the reduced group G(In−i ), as defined in Chapter 3. By Corollary 3.5, G(In−i ) is isomorphic to a partially reduced free group F(2g, n − i). Consequently, γ belongs to the normal subgroup of the fundamental group G(In−i ), generated by commutators [xj , xfj ], for i + 1 ≤ j ≤ n, and f ∈ G(In−i ). Therefore the braid k must be in the normal subgroup of PBn (S), generated by [ti,j , ti,j g ] for i + 1 ≤ j ≤ n, g ∈ F(2g + n − i). Remark 4.6. The case of a link-homotopically trivial braid kn in the braid subgroup F(2g) is particularly simple. The braid kn consists of straight strands based at points P1 , ..., Pn−1 and a winding strand γ based at the point Pn . We remove strands based at P1 , ..., Pn−1 and transform γ into a loop γ in the space S × I as we have just described in the proof above. In S × I concepts of homotopy and link-homotopy coincide - since γ is the only strand left from the braid kn . After link-homotopy γ becomes straight, therefore the loop γ is homotopic to identity in the fundamental group π1 (S × I). Thus the braid kn is link-homotopically trivial if and only if kn is the identity in the braid subgroup F(2g). Corollary 4.7. PBn (S) is isomorphic to a semi-direct product of partially reduced free groups PBn (S)  F(2g, n − 1)  (F(2g, n − 2)  (F(2g, n − 3)...(F(2g, 1)  F(2g)...) 28  Chapter 4. Homotopy string links over a punctured orientable surface Proof. By Proposition 3.9, the group of homotopy string links PBn (S) is the quotient of the pure braid group PBn (S) by the normal subgroup of link-homotopically trivial braids Hn (S). We use the semi-direct product decomposition of the pure braid group PBn (S) of Corollary 4.3 and the results of Proposition 4.5 to obtain the iterated semi-direct product decomposition of PBn (S). Corollary 4.8. There is a short split exact sequence of groups i  1 → F(2g, n − 1) → PBn (S)  ρ  PBn−1 (S) → 1,  where ρ corresponds to removing the P1 -based strand of the string link. Proof. As a consequence of Corollary 4.8, we see that PBn (S) has a structure of a semi-direct product PBn (S)  F(2g, n − 1)  PBn−1 (S),  which implies existence of the above short split exact sequence.  29  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.7:  Figure 4.8: P1 -based strand crosses itself  Figure 4.9:  Figure 4.10: 30  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.11:  Figure 4.12:  Figure 4.13: P2 -based strand crosses itself  Figure 4.14: 31  Chapter 4. Homotopy string links over a punctured orientable surface  Figure 4.15:  Figure 4.16:  Figure 4.17: 32  Chapter 5  PBn(S): surface with a puncture and group presentation The goal of this chapter is to prove the following main theorem: Theorem 5.1. Let S be the surface obtained by deleting a single point from a closed orientable surface of genus g ≥ 1. The group of homotopy string links PBn (S) admits the following presentation: Generators: {ai,r ; 1 ≤ i ≤ n, 1 ≤ r ≤ 2g} ∪ {tj,k ; 1 ≤ j < k ≤ n} Relations: (LH1)  [tfi,j , tgi,j ] = 1 where f, g ∈ F(2g + n − i)  (PR2)  ai,r Aj,s = Aj,s ai,r for 1 ≤ i < j ≤ n;  (PR3)  r=s  −1 −1 −1 (ai,1 ...ai,r )Aj,r (a−1 i,j ...ai,1 )Aj,r = Ti,j Ti,j−1  for 1 ≤ i < j ≤ n; (PR4)  1 ≤ r, s ≤ 2g; 1 ≤ r ≤ 2g  Ti,j Tk,l = Tk,l Ti,j for 1 ≤ i < j < k < l ≤ n or 1 ≤ i < k < l ≤ j ≤ n  (PR5)  −1 −1 −1 −1 Tk,l Ti,j Tk,l = Ti,k−1 Ti,k Ti,j Ti,l Ti,k Ti,k−1 Ti,l  for 1 ≤ i < k ≤ j < l ≤ n (PR6)  ai,r Tj,k = Tj,k ai,r for 1 ≤ i < j < k ≤ n or 1 ≤ j < k < i ≤ n,  (PR7)  −1 ai,r (a−1 j,2g ...aj,1 Tj,k aj,2g ...aj,1 )  =  1 ≤ r ≤ 2g  −1 (a−1 j,2g ...aj,1 Tj,k aj,2g ...aj,1 )ai,r  for 1 ≤ j < i ≤ k ≤ n  33  Chapter 5. PBn (S): surface with a puncture and group presentation where F(2g + n − i) is a free group on generators {ai,r ; 1 ≤ r ≤ 2g} ∪ {ti,j ; i < j ≤ n}, −1 Aj,s = aj,1 ...aj,s−1 a−1 j,s+1 ...aj,2g ,  Ti,j = ti,j ...ti,i+1 We shall denote this presentation as Presentation 1. Note: In this chapter we use the method and construction used by Gonzal´ez-Meneses in [GM01] to compute the presentation of surface braids PBn (M ) over a closed surface M . We adopt the notation of [GM01] and follow his argument closely, to be able to draw clear connections between presentations of surface braid groups and groups of string links under linkhomotopy in Chapter 6.  5.1  Presentations of groups  Consider an exact sequence of groups A, G, G i  1→A →G  ρ  G→1  (∗)  Suppose that the groups A and G admit presentations A =< X|RA >, G =< Y |RG >, where X and Y are sets of generators, while RA and RG are sets of relators. The following well-known procedure outlines a method for putting together a presentation of G (see [Joh76]) Generators of G: Let X = {x = i(x)|x ∈ X} be the images of the generators X of A under i. Consider y ∈ Y . Let y denote a chosen pre-image of y under ρ in G, i.e. ρ(y) = y. Let Y = {y|y ∈ Y } be the set of all such pre-images. Then X ∪ Y constitute a set of generators for G Relations in G: There are three types of relations in G :  34  Chapter 5. PBn (S): surface with a puncture and group presentation Type 1 Relators of the form RA = {rA |rA ∈ RA }, where RA is the set of words in X obtained from RA by replacing each x by x. Thus each rA is an image under i of a relator rA in G. Type 2 Let rG be a word obtained from a relator rG in RG by replacing each y by its chosen pre-image y. We see that ρ maps rG in G to relator rG in G, therefore rG lies in the ker(ρ). Since the sequence (∗) is exact, we know that kerρ equals the image i(A) of A under map i. Thus rG = wr , where wr is a word in X. We thus have a second set of relators RG = {rG = wr |rG ∈ RG } Type 3 Choose any y from the set Y of chosen pre-images of the generator set Y under ρ. The image of A under i is a normal subgroup of G, therefore each conjugate of generator x again belongs to i(A). Thus yxy −1 can be written as a word wx over the generators X of the kernel. We put C = {yxy −1 = wx |x ∈ X, y ∈ Y } Proposition 5.2. With notation as above, the group G has a presentation < X, Y |RA , RG , C > Proof. See [Joh76], pp. 138-140.  5.2  Auxilary presentation  To help us with further proofs we now define an alternative group presentation that is equivalent to the group presentation of the Theorem 1. First we state two auxilary lemmas: Lemma 5.3. [GM01] Let F (2g) be the free group freely generated by {x1 , ...x2g }. Set −1 Xr = x1 ...xr−1 x−1 r+1 ..x2g . Then {X1 , ..., X2g } is a free system of generators of F (2g).  35  Chapter 5. PBn (S): surface with a puncture and group presentation Proof. See [GM01]. The following are the formulae of the change of generators: −1 −1 −1 xk = (X1 X2−1 ...Xk−2 Xk−1 )(Xk+1 Xk+2 ...X2g−1 X2g )  x−1 k  =  if k is odd,  −1 −1 −1 (X1 X2−1 ...Xk−2 Xk−1 )(Xk+1 Xk+2 ...X2g−1 X2g )  if k is even.  Lemma 5.4. Let F (n−1) be a free group freely generated by a set {Y2 , ...Yn }. Let Y1 denote the identity element. Set −1 for 2 ≤ r ≤ n yr = Yr Yr−1  Then {y2 , ...yn } is a free system of generators of F (n − 1). Proof. The following is the formula for the change of generators: Yr = yr ...y2 To see that the set of generators is in fact free, notice that both sets {Y2 , ...Yn } and {y2 , ...yn } make two complete and free sets of generators of the fundamental group π1 (D \ Pn−1 ) of a disk D with n − 1 punctures. Use Figure 5.4 to visualize the generator Yi , and Figure 5.3 to visualize the generator yj . Definition 5.5. Define a group G by the following set of generators and relations: Generators: {Ai,r ; 1 ≤ i ≤ n; 1 ≤ r ≤ 2g} ∪ {Tj,k ; 1 ≤ j < k ≤ n} Relations: the same as Presentation 1, where −1 −1 −1 ai,k = (Ai,1 A−1 i,2 ...Ai,k−2 Ai,k−1 )(Ai,k+1 Ai,k+2 ...Ai,2g−1 Ai,2g )  if k is odd  −1 −1 −1 −1 a−1 i,k = (Ai,1 Ai,2 ...Ai,k−2 Ai,k−1 )(Ai,k+1 Ai,k+2 ...Ai,2g−1 Ai,2g )  if k is even  ti,j =  −1 Ti,j Ti,j−1  where Ti,i denotes the identity  We denote the above presentation as Presentation 2  5.3  PBn (S): more pictures of braids  Theorem 3.7 tells us that every string link is link-homotopic to a pure braid, therefore to obtain a picture of a homotopy string link σ we can draw a picture of a braid β that is link-homotopic to σ. We adopt the same geometric 36  Chapter 5. PBn (S): surface with a puncture and group presentation picture of braids over a surface as was described in the previous chapter and use the fundamental polygon L of surface S, with the boundary shifted to the vertex of L. Figures 5.1, 5.2, and 5.3 recall the pictures of braids ai,r and ti,j drawn in the fundamental polygon L. In addition let us define two more types of braids that help illustrate  Figure 5.1: Generator ai,2k+1  Figure 5.2: Generator ai,2k  Figure 5.3: Generator ti,j relations in Presentation 1 and serve as a generators for Presentation 2. Let 1 ≤ i < j ≤ n. Define Ti,j = ti,j ...ti,i+1 37  Chapter 5. PBn (S): surface with a puncture and group presentation Figure 5.4 shows braid Ti,j . Let 2 ≤ j ≤ n and 1 ≤ r ≤ 2g. Define −1 Aj,r = aj,1 ...aj,r−1 a−1 j,r+1 ...aj,2g  Figure 5.4: Braid Ti,j  Figure 5.5: Braid Aj,r in polygon L  We draw the braid Aj,r in the fundamental polygon L - however we immediately see that the picture is quite complicated. The following alternative representation of the fundamental polygon L for the surface S is convenient for depicting Aj,r , as well as various relations in PBn (S). Let i be fixed for some 1 ≤ i ≤ n. Denote by si,r the non-trivial string of ai,r and consider paths si,1 ...si,2g in the surface S, as in Figure 5.6 Cut the original polygon L along these paths and glue pieces along paths α1 , ...α2g as in Figure 5.7. We obtain a new polygon of 4g sides, with all vertices identified to Pi and sides labeled by si,1 , ...si,2g . The empty circle between the dots indicates the puncture. Figure 5.8 shows the braid Aj,r in the Pi -polygon for i < j. Note that Aj,r always points upward in the Pi -polygon for all 1 ≤ r ≤ 2g  5.4  Presentation of PBn (S)  We now proceed with the main section of this chapter. Let Γn be the abstract group defined by Presentation 1. Our goal is to show that the groups Γn  38  Chapter 5. PBn (S): surface with a puncture and group presentation  Figure 5.6: Transforming polygon L...  Figure 5.7: ...into the Pi -polygon  Figure 5.8: Braid Aj,r in Pi -polygon and PBn (S) are isomorphic. Define a map ϕ : Γn → PBn (S) ai,r → ai,r tj,k → tj,k Remark 5.6. Generators of Γn and corresponding braids in PBn (S) bear the same name, to avoid excessive notation. 39  Chapter 5. PBn (S): surface with a puncture and group presentation We shall demonstrate that the map ϕ is a homomorphism. To show that ϕ is an isomorphism we shall use induction on n and combine procedure of Section 5.1 with a short exact sequence from Chapter 4 that we shall recall later. Proposition 5.7. ϕ is a homomorphism Proof. We shall illustrate by pictures that the relations (LH1) and (PR2)(PR7) of the abstract group Γn do hold in PBn (S). For the sake of formatting all the pictures appear at the end of this section. (LH1) Relations of the type (LH1) describe string links that become trivial under the relation of link-homotopy. We re-use the sequence of pictures 4.7 through 4.17 that were used in the proof of Proposition 4.5 in Chapter 4. (PR2) Draw braids ai,r and Aj,s in the Pi -polygon as in Figure 5.8. Since r = s it is clear that the braids commute. (PR3) Draw braids ai,1 ...ai,r , Aj,r , and their inverses in Pi -polygon. Figures 5.9, 5.10, and 5.11 show transformation of the braid [ai,1 ...ai,r , Aj,r ] into the braid ti,j . (PR4) See Figures 5.12 and 5.13. Note, that both (PR4) and (PR5) can be regarded as braids over a disk. (PR5) Notice that by (PR4) the braids Ti,l and Tk,l commute. As a result the original relation (PR5) −1 −1 −1 −1 Tk,l Ti,j Tk,l = Ti,k−1 Ti,k Ti,j Ti,l Ti,k Ti,k−1 Ti,l  for 1 ≤ i < k ≤ j < l ≤ n −1 can be written as an equivalence of certain conjugates of the braid Ti,j Ti,l : −1 −1 −1 −1 −1 Tk,l Ti,j Tk,l Ti,l = Ti,k−1 Ti,k Ti,j Ti,l Ti,k Ti,k−1 −1 −1 −1 −1 −1 Tk,l (Ti,j Ti,l )Tk,l = Ti,k−1 Ti,k (Ti,j Ti,l )Ti,k Ti,k−1  The LHS and RHS of the above equation are illustrated in Figures 5.14 and 5.15; equivalence in Figure 5.16 follows. (PR6) Draw braids ai,r and Tj,k in Pi -polygon, as in Figures 5.17 and 5.18. (PR7) Figures 5.19 and 5.20 show the braid −1 β = a−1 j,2g ...aj,1 Tj,k aj,2g ...aj,1  drawn in the Pj -polygon. Now re-drawing β and ai,r in the Pi -polygon we see that these braids commute, see Figure 5.21. 40  Chapter 5. PBn (S): surface with a puncture and group presentation We now recall the exact sequence of groups of link-homotopy classes of string links from Corollary 4.8 and obtain the following diagram Γn ϕ 1  >  F(2g, n − 1)  i  >  ∨  PBn (S)  ρ  >  PBn−1 (S)  >  1,  where F(2g, n − 1) is a partially reduced free group whose definition we shall recall momentarily. We shall assume inductively the presentation of PBn−1 (S). The method of Section 5.1 gives us a presentation of PBn (S), knowing presentations of F(2g, n − 1) and PBn−1 (S). Case n=1: We have PB1 (S) = PB1 (S) = π1 (S, Pn ). Since all the braids in question have one single strand, the concepts of homotopy and link-homotopy coincide, thus PB1 (S) is a free group on 2g generators. All relations hold trivially, hence PB1 (S) is isomorphic to Γ1 . Case n-1: Assume inductively that PBn−1 (S) is isomorphic to Γn−1 as given by Presentation 2. The choice of Presentation 2 is a matter of convenience, as shall become obvious when working with Type 3 relations of PBn (S). Generators of PBn (S) Apply the procedure of Section 5.1 to the short exact sequence i  1 → F(2g, n − 1) → PBn (S)  ρ  PBn−1 (S) → 1  Recall that F(2g, n − 1) admits the presentation F(2g, n − 1) =< a1,1 , ..., a1,2g ;  t1,1 , ..., t1,n ; where [t1,i f , t1,i g ] = 1 >  where f and g belong to F(2g + n − 1), the free group on generators {a1,1 , ..., a1,2g } ∪ {t1,1 , ..., t1,n }. The images of generators F(2g, n − 1) in n PBn (S) under the map i are sets of braids {a1,i }2g i=1 and {t1,j }j=2 as shown in Figures 5.1, 5.2, and 5.3 Recall that the map ρ is equivalent to removing the first strand of a string link. By induction hypothesis, the generators of PBn−1 (M ) are linkhomotopy classes of braids {Ai,r } ∪ {Ti,j }, 1 ≤ i ≤ n − 1, 1 ≤ r ≤ 2g 41  Chapter 5. PBn (S): surface with a puncture and group presentation Given a small annulus neighbourhood D that contains the puncture x0 on S and avoids the point-set P, any link-homotopy of a string link can be approximated by a link-homotopy that avoids the neighbourhood D × I in the cylinder S × I, as described in the course of the proof of Proposition 4.4. Add a trivial first strand to generators of PBn−1 (S), placing the strand within the neighbourhood D × I. We then obtain a set of well-defined linkhomotopy classes of pre-images in PBn (S). It is clear that the set obtained is {Ai,r } ∪ {Ti,j }, 2 ≤ i ≤ n, 1 ≤ r ≤ 2g Thus ρ has the effect of shifting the indices: ρ(Ai,r ) = Ai−1,r for 2 ≤ i ≤ n ρ(Ti,j ) = Ti−1,j−1 for 2 ≤ i ≤ n, 3 ≤ j ≤ n According to the procedure of Section 5.1 PBn (S) has a set of generators {a1,1 ...a1,2g ; t1,2 ...t1,n } ∪ {Ai,r , Ti,j , 2 ≤ i ≤ n, 1 ≤ r ≤ 2g} By using Lemmas 5.3 and 5.4 we transform this set into {ai,r , ti,j |1 ≤ i < j ≤ n, 1 ≤ r ≤ 2g} This is the image of the generating set of Γn under ϕ. Thus we see that ϕ is surjective. Proposition 5.8. ϕ : Γn → PBn (S) is injective. Proof. We show that the three types of relations in PBn (S) are a consequence of relations of Γn . Thus if ϕ(w) = 1 in PBn (S) then w = 1 in Γn . Type 1: images of relations of F(2g, n − 1) under i. [t1,j f , t1,j g ] = 1 for 2 ≤ j ≤ n correspond to a relation (LH1) in Γn with i = 1. Type 2: Given a relation in PBn−1 (S), its pre-image under ρ corresponds to the same relation in PBn (S) with index shifted. For example, relations of type (LH1) in PBn−1 (S) [tfi,j , tgi,j ] = 1 where f, g ∈ F(2g + n − i) for 1 ≤ i < j ≤ n − 1 become relations of type (LH1) in PBn (S) with a shift of index: [tfi,j , tgi,j ] = 1 where f, g ∈ F(2g + n − i), for 2 ≤ i < j ≤ n The corresponding relations in Γn are obvious. 42  Chapter 5. PBn (S): surface with a puncture and group presentation Type 3: Let x be the image under i of a generator x of F(2g, n − 1) in PBn (S). Let y be a chosen pre-image of a generator y of PBn−1 (S) under ρ in PBn (S). Let us denote by Ω the subset of Γn , consisting of all words in terms of the generators {a1,1 , ...a1,2g } ∪ {t1,1 ...t1,n }. We look at the pre-image of yxy −1 in Γn and find a word V in Ω , such that the expression yxy −1 = V holds in Γn . Looking at the image of yxy −1 = V under ϕ we obtain a relation of type 3 yxy −1 = ϕ(V ) in PBn (S). It then follows that every relation of type 3 is an image of a relation in Γn , making ϕ an injective map. Computations (1.) through (4.) below appear in [GM01] and are repeated here for the sake of completeness of argument. (1.) Relations Aj,s a1,r A−1 j,s = a1,r for r = s follow from (PR2) for i = 1. −1 −1 −1 Relations Aj,r a1,r A−1 j,r = (a1,r−1 ...a1,1 )t1,j (a1,1 ...a1,r ) : apply (PR2) to (PR3). −1 (2.) Relations Tk,l a1,r Tk,l = V , where V is a word in Ω follow from (PR6) when i = 1. −1 ∈ Ω follow from (PR4)-(PR5). (3.) Relations Tk,l T1,j Tk,l  (4.) Relations Aj,r T1,k A−1 j,r ∈ Ω for j > k follow from (PR6). Relations Aj,r T1,k A−1 j,r ∈ Ω for 1 < j ≤ k are computed as follows: −1 Relation (PR7) says that for s = 1, ..., 2g aj,s commutes with a−1 1,2g ...a1,1 T1,k a1,2g ...a1,1 . Therefore Aj,r commutes with the same element, so −1 −1 −1 −1 a−1 1,2g ...a1,1 T1,k a1,2g ...a1,1 = Aj,r (a1,2g ...a1,1 T1,k a1,2g ...a1,1 )Aj,r −1 −1 −1 −1 −1 −1 =(Aj,r a−1 1,2g Aj,r )...(Aj,r a1,1 Aj,r )(Aj,r T1,k Aj,r )(Aj,r a1,2g Aj,r )...(Aj,r a1,1 Aj,r )  Except for the middle term in the above product, we know how to write each term in the above product as some word in X. We can now easily compute conjugates of the remaining generators: (3.) above implies −1 −1 −1 −1 −1 −1 Tk,l t1,j Tk,l = Tk,l T1,j T1,j−1 Tk,l = Tk,l T1,j Tk,l Tk,l T1,j−1 Tk,l  is a word in Ω. (4.) above implies −1 −1 −1 Ai,r t1,j A−1 i,r = Ai,r T1,j Ai,r Ai,r T1,j−1 Ai,r  is also a word in Ω. 43  Chapter 5. PBn (S): surface with a puncture and group presentation We have now shown that the map ϕ : Γn → PBn (S) is a homomorphism that is surjective and injective. We see that the groups Γn and PBn (S) are isomorphic, and thus the group of link-homotopy classes of string links PBn (S) admits the presentation described in Theorem 1.  44  Chapter 5. PBn (S): surface with a puncture and group presentation  Figure 5.9: PR3  Figure 5.10: PR3  Figure 5.11: PR3  Figure 5.12: PR4  Figure 5.13: PR4 45  Chapter 5. PBn (S): surface with a puncture and group presentation  Figure 5.14: PR5: LHS  Figure 5.15: PR5: RHS  Figure 5.16: PR5: LSH and RHS are equivalent  46  Chapter 5. PBn (S): surface with a puncture and group presentation  Figure 5.17: PR6: 1  Figure 5.18: PR6: 2  Figure 5.19: PR7: braid β over Pj polygon  Figure 5.20: PR7 simplified  Figure 5.21: PR7 in Pi -polygon 47  Chapter 6  Homotopy string links over closed orientable surface For the remainder of this chapter let M be an orientable closed surface of genus g ≥ 1 and let S be the surface obtained by deleting a single point x0 from the surface M . By replacing the point x0 into S we induce a map p between the groups of homotopy braids over S and those over M : 1 → K → PBn (S)  p  PBn (M ) → 1  The goal of this chapter is to determine presentation of the group PBn (M ) of link-homotopy classes of string links over a closed orientable surface of genus g ≥ 1. The method is to compute the kernel K and combine it with presentation of PBn (S) obtained in the previous chapter. In the following sections we recall presentations of classical surface braid groups and give a new definition of link-homotopy. In the final section we build a diagram involving all the mentioned groups and compute presentation of PBn (M ).  6.1  PBn (M ) and PBn (S): the group presentations  The pure braid group PBn (S) over the punctured surface S admits the same presentation as the group of homotopy braids PBn (S), if we ignore the relations induced by link-homotopy. Proposition 6.1. Let S be the surface obtained by deleting a single point x0 from a closed orientable surface of genus g ≥ 1. PBn (S) admits the presentation: Generators: {ai,r ; 1 ≤ i ≤ n, 1 ≤ r ≤ 2g} ∪ {tj,k ; 1 ≤ j < k ≤ n} Relations: (PR2)-(PR7) as defined in Theorem 5.1. Proof. The proof proceeds verbatim as the proof of Theorem 5.1, using the pure braid split exact sequence 1 → F(2g + n − 1) → PBn (S)  PBn−1 (S) → 1 48  Chapter 6. Homotopy string links over closed orientable surface We omit the relations (LH1) from the arguments of Chapter 5, which can be done without affecting proofs of any other relations. Pictures of the generators ai,r and tj,k are the same as in the previous chapter. The presentation for the pure braid group PBn (M ) over the closed orientable surface M was described in [GM01], whose construction and view of surface braids we closely followed in the previous chapters. Proposition 6.2. Let M be a closed orientable surface of genus g ≥ 1. PBn (M ) admits the presentation: Generators: {ai,r ; 1 ≤ i ≤ n, 1 ≤ r ≤ 2g} ∪ {tj,k ; 1 ≤ j < k ≤ n} Relations: (PR2)-(PR7) as defined in Theorem 5.1, and additional relations n−1  (PR1)  −1 −1 a−1 n,1 an,2 ...an,2g an,1 an,2 ...an,2g  −1 Ti,n−1 Ti,n  = i=1  j−1  (PR8)  −1 −1 −1 −1 (a−1 i,2g ...ai,1 Ti,j−1 Ti,j ai,1 ...aj,2g )aj,1 ...aj,2g )  Tj,n = ( i=1  Proof. There is a slight difference between the presentation as stated in this proposition and the original presentation of PBn (M ) as it appears in [GM01]. The original presentation uses braids {Ti,j }, 1 ≤ i < j ≤ n as part of the generating set. We recall Lemma 5.4 and use instead the braids {ti,j }, 1 ≤ i < j ≤ n, to easily match braids and link-homotopy string links in later constructions. The proof in [GM01] follows the same procedure described in Section 5.1, and uses the pure braid sequence for braid groups over the closed orientable surface M . We refer the reader to [GM01] for the complete proof, while here we demonstrate by pictures that relations (PR1) and (PR8) do indeed hold in PBn (M ). −1 (PR1) Let 1 ≤ i ≤ n − 1 be given. Draw the braid Ti,n−1 Ti,n in the Pn -polygon and notice that the braids in Figures 6.1 and 6.2 are in fact equivalent. Thus both LHS and RHS of (PR1) represent a braid in Figure 6.3. (PR8) Given i < j, use the Pi -polygon to draw the braid γ = ai,2g −1 ...ai,1 −1 Ti,j−1 Ti,j −1 ai,1 ...aj,2g . Figures 6.4, 6.5, and 6.6 show equivalent presentations of γ in the Pi - and Pj -polygons. Relation (PR8) becomes clear from drawing all factors in the Pj -polygon. 49  Chapter 6. Homotopy string links over closed orientable surface  −1 Figure 6.1: braid Ti,n−1 Ti,n  Figure 6.2: braid equivalent to −1 Ti,n−1 Ti,n in Pn -polygon  Figure 6.3: relation (PR1)  Figure 6.4: Braid γ in Pi -polygon...  Figure 6.5: ..becomes..  Figure 6.6: equivalent braid in Pj -polygon  We see that presentation of PBn (M ) can be obtained from the presentation of PBn (S) by adding the relations (PR1) and (PR8). An intuitive 50  Chapter 6. Homotopy string links over closed orientable surface  Figure 6.7: (PR8)...  All  components  of  Figure 6.8: ... in Pj -polygon  explanation for the new relations is clear: by attaching a copy of the unit interval x0 × I to the cylinder S × I, we ”close up” the hole and create more room for isotopy.  6.2  Isotopy braids  As we have seen in Chapter 2, link-homotopy involves isotopy, which may move and twist strands of a braid until the strands are no longer monotone. Consider a surface braid β in PBn (X), where X denotes a closed surface M or a surface S with a single puncture as above. We can increase the isotopy class of β to include all images of β under any ambient isotopy that fixes X × 0 and X × 1 in X × I. Let us call the resulting group the isotopy surface braid group, and its elements we shall call isotopy surface braids. It is clear that the isotopy braid group over X is isomorphic to the original surface braid group PBn (X). In the remaining sections of this chapter we shall work with objects that are isotopy braids, rather than braids in their classical monotone definition. To avoid excessive notation, let us still use the same symbols and denote the isotopy braid groups as PBn (S) and PBn (M ), with the understanding that the isotopy class of each braid has been increased.  51  Chapter 6. Homotopy string links over closed orientable surface  6.3  Link-homotopy: finger-loop moves  Recall that a link-homotopy from a string link σ to string link τ can be approximated by an ambient isotopy and a finite number of crossing changes. Each crossing change is confined to a small ball B, that contains two arcs of a given strand. (See Chapter 3.) We can think of B as a neighbourhood of an arc s that guides the crossing change. The endpoints of s lie on the strand, but otherwise s is disjoint from the string link as in Figure 6.9. Following the motion of the strands under ambient isotopy, the arc s and its neighbourhood B may bend, stretch, and wrap around strands of the string link, see Figure 6.10. If strands of σ and the arc s have moved under the isotopy, we can perform the crossing change via a finger-loop move, letting arcs of the strand approach each other along the stretched neighbourhood B, see Figure 6.11.  Figure 6.9: Arc s in a ball B  Figure 6.10: Motion of B and s under ambient isotopy  Figure 6.11: Finger-loop move along s The finger-loop approximation of link-homotopy has an important advantage: an ambient isotopy followed by a crossing change can be approximated by a crossing change followed by an ambient isotopy, provided that in the latter case the crossing change is performed by a finger-loop move along the B-neighbourhood of a guiding arc s. 52  Chapter 6. Homotopy string links over closed orientable surface  6.4  PBn (M ): group presentation  Putting together PBn (M ), PBn (S), PBn (S) and PBn (M ), we obtain the following commutative diagram. 1  1  ∨  ∨  (P R1), (P R8)  K  ∨  1  >  (LH1)  >  h  PBn (S)  ∨ >  PBn (S) p  p ∨  PBn (M )  h  >  >  1  ∨  PBn (M )  ∨  ∨  1  1  We now proceed to the main result of this work and compute a group presentation for the group of link-homotopy classes of string links PBn (M ) over the closed surface M . Given a braid β ∈ PBn (S) over the surface S with boundary, let [β] denote the link-homotopy class of β in PBn (S), i.e. h(β) = [β] ∈ PBn (S). We determine the structure of the kernel K of the map p by showing that if [β] is in the kernel K, then [β] must be in the normal closure (P R1), (P R8) in PBn (S). For the sake of completeness the text of the following theorem repeats the statement of relations (LH1), (PR1)-(PR8), which remain unchanged from the Theorem 5.1 and Proposition 6.2. Theorem 6.3. Let M be a closed orientable surface of genus g ≥ 1. The group of homotopy string links PBn (M ) admits the presentation: Generators: {ai,r ; 1 ≤ i ≤ n, 1 ≤ r ≤ 2g} ∪ {tj,k ; 1 ≤ j < k ≤ n} Relations: (LH1)  [tfi,j , tgi,j ] = 1 where f, g ∈ F(2g + n − i) n−1  (PR1)  −1 −1 a−1 n,1 an,2 ...an,2g an,1 an,2 ...an,2g  −1 Ti,n−1 Ti,n  = i=1  53  Chapter 6. Homotopy string links over closed orientable surface (PR2)  ai,r Aj,s = Aj,s ai,r for 1 ≤ i < j ≤ n;  (PR3)  r=s  −1 −1 −1 (ai,1 ...ai,r )Aj,r (a−1 i,j ...ai,1 )Aj,r = Ti,j Ti,j−1  for 1 ≤ i < j ≤ n; (PR4)  1 ≤ r, s ≤ 2g; 1 ≤ r ≤ 2g  Ti,j Tk,l = Tk,l Ti,j for 1 ≤ i < j < k < l ≤ n or 1 ≤ i < k < l ≤ j ≤ n  (PR5)  −1 −1 −1 −1 Tk,l Ti,j Tk,l = Ti,k−1 Ti,k Ti,j Ti,l Ti,k Ti,k−1 Ti,l  for 1 ≤ i < k ≤ j < l ≤ n (PR6)  ai,r Tj,k = Tj,k ai,r for 1 ≤ i < j < k ≤ n or 1 ≤ j < k < i ≤ n,  (PR7)  −1 ai,r (a−1 j,2g ...aj,1 Tj,k aj,2g ...aj,1 )  =  1 ≤ r ≤ 2g  −1 (a−1 j,2g ...aj,1 Tj,k aj,2g ...aj,1 )ai,r  for 1 ≤ j < i ≤ k ≤ n j−1  (PR8)  −1 −1 −1 −1 (a−1 i,2g ...ai,1 Ti,j−1 Ti,j ai,1 ...aj,2g )aj,1 ...aj,2g )  Tj,n = ( i=1  where F(2g + n − i) is the free group on generators {ai,r ; 1 ≤ r ≤ 2g} ∪ {ti,j ; i < j ≤ n}, −1 Aj,s = aj,1 ...aj,s−1 a−1 j,s+1 ...aj,2g ,  Ti,j = ti,j ...ti,i+1 Proof. Let β be an isotopy-braid in PBn (S), and suppose that [β] is in the kernel K of p. There is thus a link-homotopy H in M × I that takes β to In , the trivial braid on n strands. H can be separated into two stages. First we perform crossing changes via finger-loop moves along small neighbourhoods of guiding arcs and obtain an isotopy-braid β . The crucial point is that the guiding arcs can be moved by isotopy to avoid the neighbourhood of x0 × I, and so all the crossing changes take place in S × I. Therefore isotopy-braid β belongs to group PBn (S), while [β] and [β ] represent the same homotopy string link in S × I. The second stage of H is an isotopy which happens in M × I and takes the isotopy-braid β to the trivial braid In . We thus see that β is in the kernel of map p : PBn (S) PBn (M ), i.e. β is in the normal closure (P R1), (P R8) in PBn (S). Since β is a product of conjugates of relators of (P R1) and (P R8), we see that [β ] is a product of conjugates of relators (P R1) and (P R8) under link-homotopy, i.e. [β ] ∈ (P R1), (P R8) in PBn (S). 54  Chapter 7  Invariants of homotopy string links over a disk In his search for invariants to classify links up to link-homotopy, John Milnor defined a certain quotient of the free group, which we shall call reduced free group, following [HL90]. Milnor defined an expansion µ of the reduced free group into a certain polynomial ring with integer coefficients, and showed µ to be injective (see [Mil54]). Deborah Goldsmith showed that the group of homotopy string links PBn (D) over a disk has a structure of an iterated semi-direct product of reduced free groups ([Gol74]). Habegger and Lin observed in [HL90] that the integral coefficients of Milnor’s injective expansion µ gave a set of complete invariants of PBn (D). In this chapter we give a new proof of the injectivity of the expansion µ using an alternative presentation of the reduced free group which reveals its interesting structure.  7.1  Two definitions of reduced free group  Let F denote a free group on the set of generators {x1 , ..., xk }. Definition 7.1. Take a quotient of F by relations that say each xi commutes with its conjugates. The resulting group is the classical reduced free group RF (k) as defined [HL90] and [Mil54]. Let J1 denote the subgroup of F generated by commutators of conjugates of xi : J1 =< [xi , xi ], 1 ≤ i ≤ k > The same computation as in Remark 3.2 shows that RF (k) has the presentation F/J1 . There is an alternative presentation of RF (k), first given by Jerome Levine in [Lev88], which shall be very useful in the construction of invariants. Definition 7.2. The reduced free group RF (k) is obtained as a quotient of the free group F by relations which set to 1F every commutator C in {xi } with repeats. 55  Chapter 7. Invariants of homotopy string links over a disk To make precise the meaning of a ”commutator with repeats” we first recall the definitions of commutators and lower central series. We define the commutators in {xi } recursively: 1. Commutators of weight 1 are x1 , ..., xk . 2. Commutators of weight n are words [C1 , C2 ], where C1 , C2 are distinct commutators of weight < n, and n = wt C1 + wt C2 . Commutators of weight ≥ n generate a normal subgroup Fn . The series F = F1 ⊇ F2 ⊇ F3 ⊇ ... is called the lower central series of F . It is a well-known fact that a free group F is residually nilpotent, which means that the intersection of all its lower central series subgroups is the identity: ∞ i=1 Fi = {1F }. Following Levine [Lev88] we say that xi occurs r times in a commutator C as follows 1. if C = xj then r = 1 if i = j and r = 0 otherwise. 2. if C = [C1 , C2 ] then r = r1 + r2 where xi occurs r1 times in C1 and r2 times in C2 . We say that a commutator C has repeats if some xi occurs at least twice in C. Finally, let J2 denote the normal subgroup generated by commutators with repeats. Levine shows that J1 = J2 . and we see that the definitions 7.1 and 7.2 of the reduced free group RF (k) are indeed equivalent. From now on we denote J1 = J2 = J. To prove that J1 = J2 Levine uses P. Hall’s construction of basic commutators in a free group, which we recall in the next section.  7.2  Basic commutators  Definition 7.3. [Fen83] Basic commutators in a set {xi } are defined inductively as follows: 1. Each basic commutator C has a weight wt C, taking one of the values 1,2,3... . 56  Chapter 7. Invariants of homotopy string links over a disk 2. The basic commutators of weight 1 are the elements of {xi }. A basic commutator of weight > 1 is a word of the form C = [C1 , C2 ], where C1 , C2 are previously defined basic commutators and wt C =wt C1 +wt C2 . 3. Basic commutators are ordered so as to satisfy the following: (a) Basic commutators of the same weight are ordered arbitrarily. (b) If wt C >wt C then C > C 4. (a) If wt C > 1 and C = [C1 , C2 ] then C1 < C2 (b) If wt C > 2 and C = [C1 , [C2 , C3 ]] then C1 ≥ C2 The next theorem due to P. Hall illustrates the main purpose of basic commutators: to serve as a basis for quotients of free group by its lower central series subgroups. Theorem 7.4. [MKS66] There exists a set of basic commutators, for any m. Given a set of basic commutators C1 < C2 < ..., then every element of of F/Fq has a unique representative as a monomial C1 e1 C2 e2 ...Cn en , where C1 , ..., Cn are all the basic commutators of weight < q.  7.3  Magnus expansion and basic commutators with repeats  In the remainder of this chapter we make use of a well-know expansion of a free group F generated by {x1 , ..., xk }. Let Λ denote a ring of non-commuting power series in the variables {X1 , ..., Xk }, with integral coefficients. The Magnus Expansion µ is an injective homomorphism of F into the group of units U of Λ, defined by µ : F −→ Λ xi → 1 + Xi xi −1 → 1 − Xi + (Xi )2 − (Xi )3 + (Xi )4 − ... Every element of U can be written uniquely as ±1 + ρ + h, where ρ is homogeneous of degree > 0, and every term of h has degree higher than the degree of ρ. We call ρ the principal part. The following lemma is proved by induction in [Lev88]. Lemma 7.5. If C is a commutator of weight n in {xi } then 57  Chapter 7. Invariants of homotopy string links over a disk - The principal part ρ of µ(C) is of degree n, and - Each variable Xi appears in every term of ρ with a total degree equal to the number of occurrences of xi in C Example 7.6. A simple computation shows that the Magnus Expansion µ takes the commutator [x2 , [x1 , x2 ]] of weight 3 to an element of U 1 + 2x2 x1 x2 − x22 x1 − x1 x22 + O(4), where O(4) denotes terms of order 4 and higher. Note that if we assume the ordering x1 < x2 < ... < xk , then [x2 , [x1 , x2 ]] is a basic commutator. The following proposition and its proof appear in [Lev88] as well. We repeat the proof here to illustrate the techniques which shall be of use in our subsequent proofs as well. Proposition 7.7. Let {Ci } be a set of basic commutators in {xi } Suppose C1 < C2 < ... < Cn are those of weight ≤ q. If an element of J has representation C1 e1 ...Cn en modulo Fq+1 then whenever ei = 0, Ci must be a commutator with repeats. Proof. Let g ∈ J ∩ Fq , then g has the representation g = Cr er ...Cs es modulo Fq+1 , where Cr , ..., Cs are basic commutators of weight q. The principal part of µ(g) is ρ = er ρr + ... + es ρs , where ρi is the principal part of µ(Ci ). It is one of the more important properties of basic commutators that the principal parts {ρr , ...ρs } form a linearly independent set (see [MKS66]). Therefore if ei = 0 for some Ci without repeats, then by Proposition 7.5 ρ shall have terms in which no variable has total degree > 1. Now let Ni denote the normal closure generated by xi in F . If f ∈ Ni , then by a simple computation every term of µ(f ) − 1 contains Xi with total degree > 0. J is the subgroup generated by all [Ni , Ni ], therefore if g ∈ J then every term of µ(g) − 1 must contain some variable Xi with total degree > 1 - a contradiction. An easy calculation establishes the result. The following corollary shall be useful in the next section.  58  Chapter 7. Invariants of homotopy string links over a disk Corollary 7.8. Let {Ci } be a set of basic commutators in {xi } Suppose C1 < C2 < ... < Cn are those of weight ≤ k. Let f ∈ F have representation C1e1 ...Cnen modulo Fk+1 , where k is the rank of the free group F . Then f ∈ J if and only if each Ci is a commutator with repeats. Proof. (⇒) Follows directly from the previous lemma. (⇐) Since F is a free group of rank k, it follows that Fk+1 ⊂ J, since any commutator of weight > k must have repeats.  7.4  Injective expansion of RF(k)  We now return to the Magnus expansion µ of a free group F of rank k into the group of units U of the power series ring Λ in non-commuting variables {X1 , ...Xk }. Let R denote the subset of Λ generated by monomials with repeats of a variable. An example of an element of R is 4X1 X2 X1 + X33 X1 . R is an additive subgroup of Λ and R is closed under right and left multiplication by elements of Λ. It follows that R is a two-sided ideal of Λ. Let 1 + R denote the set of elements of the form ”1+ monomials with repeat of a variable”. We also recall a special fact about the commutator subgroup F2 of the free group F and a multiplicative subgroup D2 of U that consists of elements of the form ”1+ terms of degree two and higher”. The following proposition is proved in [MKS66] Proposition 7.9. D2 equals the image of F2 under the Magnus expansion µ. We now go on to prove a rather technical proposition which shall be crucial in the proof of the main theorem of this chapter. Proposition 7.10. If each term of Magnus expansion µ(f ) has repeats then f is an element of J, i.e. if µ(f ) ∈ 1 + R then f ∈ J Proof. Let f ∈ F , and suppose that µ(f ) ∈ 1 + R. Since by Proposition 7.9 D2 = µ(F2 ), we can conclude that f ∈ F2 . By Theorem 7.4 f can be expressed as a product of powers of basic commutators of weight ≤ k, modulo Fk+1 , where k is the rank of F . Let us say that the first non-zero 59  Chapter 7. Invariants of homotopy string links over a disk exponents shall occur for basic commutators of some weight i. We obtain the following expression for f f = Ki Ki+1 ...Kq ,  (∗)  where each Kj is a product of powers of basic commutators of weight j: Kj = Cj,1 ej,1 ...Cj,rj ej,rj . For a contradiction let us suppose that f does not belong to J. In such case Corollary 7.8 implies the expression (∗) must have non-zero exponents for some basic commutators with no repeats. Let us say that the first basic commutator with no repeats occurs in (∗) for commutators of weight n. Now apply Magnus expansion µ to the expression (∗) µ(f ) = µ(Ki )...µ(Kn−1 )µ(Kn )µ(Kn+1 )...µ(Kq ) Recall that for each j, µ(Kj ) = 1 + ρj + hj , where the principal part ρj is the sum of the principal parts of µ((K)j,k ) = µ(Cj,1 ej,1 ...Cj,rj ej,rj ), i.e. ρj = ej,1 ρj,1 + ... + ej,rj ρj,rj . Recall that ρj is the sum of homogeneous monomials of degree j, while hj is the sum of terms of degree higher than j. For i ≤ j < n each term of µ(Kj ) has repeats and therefore each term of the product µ(Ki )...µ(Kn−1 ) also has repeats. Now consider µ(Kn ) By Proposition7.5 ρn contains monomials of degree n without repeats. Note that monomials with repeats may also be present. Write the product of the remaining terms µ(Kn+1 )...µ(Kq ) as 1+O(n+1), where O(n + 1) denotes terms of degree n + 1 and higher. We now multiply out all terms of µ(Ki )...µ(Kn )...µ(Kq ) and obtain Magnus expansion µ(f ). The resulting sum contains two sets of terms of degree n: a sum of terms with repeats coming from µ(Ki )...µ(Kn−1 ) and possibly from µ(Kn ) and a sum of terms without repeats coming from µ(Kn ). These terms do not cancel each other, thus µ(f ) contains terms with no repeats of a variable and cannot be an element of 1 + R - contradiction. Now let us finally define an expansion of RF (k) and pave the way for the invariants of PBn (D) Let Λ denote a quotient ring of Λ by the two-sided ideal R, generated by monomials with repeats. In words we can describe Λ as a polynomial ring  60  Chapter 7. Invariants of homotopy string links over a disk in non-commutative variables {X1 , ..., Xk }, such that any monomial with repeat of variable Xi , for some 1 ≤ i ≤ k, is set to zero. Define the reduced Magnus expansion µ µ : RF (k) −→ Λ xi  xi −1  −→ 1 + Xi −→ 1 − Xi  where xi for 1 ≤ i ≤ k are the generators of RF (k). As a consequence of Proposition 7.10 we have the following theorem Theorem 7.11. The reduced Magnus expansion µ : RF (k) → Λ is injective. Proof. RF (k) is defined as a quotient group of free group F by J. Consider the following diagram with exact rows 1  >  J  >  F  p  >  RF (k)  µ  µ ∨  1  >  R  >  Λ  π  >  >  1  >  1  ∨  Λ  The diagram is commutative: given a generator xi of F , µ ◦ p(xi ) = µ(xi ) = 1 + Xi π ◦ µ(xi ) = π(1 + Xi ) = 1 + Xi −1 µ ◦ p(x−1 i ) = µ(xi ) = 1 − Xi 2 3 π ◦ µ(x−1 i ) = π(1 − Xi + Xi − Xi ...) = 1 − Xi  The rest of the proof is a case of standard diagram-chasing. Let f ∈ RF (k) and suppose that µ(f ) = 1Λ . Let f ∈ F be a pre-image of f . By commutativity of the diagram and by definition of Λ = Λ/R we see that µ(f ) is an element of 1 + R. Proposition 7.10 implies that f is an element of J ⊂ F , hence f = p(f ) = 1RF (k) . Therefore µ is injective.  7.5  Integer invariants of homotopy string links  In [Gol74] Deborah Goldsmith showed that the group of homotopy string links PBk (D) is isomorphic to an iterated semi-direct product of reduced  61  Chapter 7. Invariants of homotopy string links over a disk free groups, reminiscent of the Artin normal form of the pure braid group PBk (D) which we discussed in Section 2.3 PBk (D)  RF (k − 1)  RF (k − 2)  ...  RF (1).  Each homotopy string link can be written uniquely as a product τ = τ1 ...τk−1 , where τi ∈ RF (k − i), for 1 ≤ i ≤ k − 1. To obtain integer invariants of τ we consider the image of each τi under the reduced Magnus expansion µ of RF (k − i) into the ring Λ on k − i variables. We then take the integer coefficients of each µ(τi ) for 1 ≤ i ≤ k − 1. Since the reduced Magnus expansion µ is injective, the total set of integers is bound to be unique for each homotopy string link τ . Example 7.12. The group of homotopy string links on two strands PB2 (D) is isomorphic to RF (1), which is in fact simply a copy of integers Z. The trivial homotopy string link on two strands has integer invariant of 0 associated to it, while the string link pictured in the Figure 3.1 has integer invariant −1. Example 7.13. Consider a link-homotopically trivial pure braid τ on 3 strands given by −1 −1 −1 t1,3 t1,2 t−1 1,3 t1,2 t1,3 t1,2 t1,3 t1,2 as shown in the Figure 1.1 of the Introduction. A simple computation shows that under the reduced Magnus expansion of RF (2), the image of τ is an element of the form 1 + monomials with repeats. Setting monomials with repeats to zero, we see that the invariants, associated with τ are {(0, 0); (0)}. In [HL90] Nathan Habegger and Xiao-Song Lin observed the correspondence between the above invariants of homotopy string links and the Milnor µ-invariants of [Mil54]. In his work [Mil54] John Milnor attempted to describe homotopy classes of ordered oriented n-component links in the Euclidean space by integer invariants. Milnor injected a quotient of the fundamental group of the complement of a sublink on n − 1 components into a certain polynomial ring with integral coefficients. The remaining component of the link played the role of an element of this quotient group. 62  Chapter 7. Invariants of homotopy string links over a disk Milnor obtained link-homotopy invariants of links by taking the integer coefficients of the expansion modulo indeterminacies given by greatest common divisors of lower order integer invariants. Milnor’s µ-invariants give a complete classification of 3-component links. Links whose lower order invariants vanish are also classified completely, for example n-component links whose every sublink is trivial. For illustrations please see the images in the introduction chapter of our work. Milnor’s construction closely parallels the semi-direct product decomposition of PBk (D) by Goldsmith. As a group PBk (D) is in fact torsion-free and nilpotent ([HL90]), and its rank precisely equals the number of Milnor’s µ-invariants needed to describe, up to indeterminacy, a link with k components. Thus while Milnor’s integers are not link-homotopy invariants of links, they are link-homotopy invariants of homotopy string links over a disk. For a more detailed discussion please see [HL90].  63  Chapter 8  PBn(D) is bi-orderable. In this chapter, we consider the orderability properties of the groups of classical homotopy string links PBn (D) over a disk. In [HL90] Nathan Habegger and Xiao-Song Lin have shown that PBn (D) is torsion-free and nilpotent, and it is well-known that torsion-free nilpotent groups are bi-orderable (see [MR77] p.37). However, we prefer to describe an explicit, calculable ordering of homotopy string links, which is compatible with the semidirect product structure discussed in the previous chapter.  8.1  Orderability: definitions and examples  Let G be a group, and < a strict total ordering of its elements. We say that (G; <) is a left-ordered group if the ordering < is invariant under multiplication on the left, i.e. if g < h then f g < f h for all f , g, h in G. If the ordering is also invariant under multiplication on the right, we say that (G; <) is a bi-ordered group. An alternative definition of orderability can be given by defining a subset of ”positive elements”. G is left-orderable if there exists a subset P ⊂ G described by two properties below. We call elements in P positive, those in P −1 - negative, and P itself is denoted a positive cone. 1. P is closed under multiplication, i.e. a product of positive elements is again positive. 2. G = P {1} P −1 , i.e. P partitions G, and so every element in G is either positive, negative, or identity. Given a left-ordering < we define positive cone P as the set of all positive elements of G: P = {g ∈ G|1 < g}. Given a positive cone P , we define a left-ordering < by saying g < f if and only if g −1 f is positive, i.e. g −1 f ∈ P . (G, <) is a bi-ordering, if P satisfies an additional property 3. P is closed under conjugation by elements of G: gP g −1 ⊂ P for any g in G. 64  Chapter 8. PBn (D) is bi-orderable. Example 8.1. The integers Z, the rationals Q, the reals R under operation of addition are bi-orderable. Example 8.2. Given a line with irrational slope that passes through the origin, we define an ordering on Z2 . We can choose the positive cone PZ2 to be the set of all points that lies above the line. It is clear that Z2 has uncountably many orderings. Ordered groups possess interesting properties that make the study of orderability worth-while: Proposition 8.3. If a group G is left-orderable, then G is torsion-free. Proposition 8.4. If a group G is bi-orderable, then G has unique roots: g n = hn implies g = h for all g, h in G. Proposition 8.5. If a group G is left-orderable, then its integral group ring ZG has no zero-divisors. Recently the question of orderability of braid groups came into view, as illustrated in the following set of examples. Example 8.6. Classical braid group Bn (D) is left-orderable, as first proved by Dehornoy [DDRW02]. Some alternative proofs of the left-orderability of Bn (D) followed, two being topological in nature. The details can be found under the same reference [DDRW02]. Example 8.7. Surface pure braid groups are bi-orderable, with the exception of braid groups over the sphere S2 and the projective plane RP2 , as proved by Gonz´ alez-Meneses in [GM02] Example 8.8. The classical pure braid group over disk PBn (D) is biorderable. In [KR02] Kim and Rolfsen use the fact that PBn (D) can be presented as an iterated semi-direct product of free groups. They construct a bi-ordering of a finitely generated free group by using the Magnus expansion of a free group F into a ring of formal power series. The fact that PBn (D) is isomorphic to a certain subgroup of automorphisms of the free group provides a necessary technical condition to impose a bi-order on PBn (D). We now state the main goal of this section. We make use of the similarity between pure braids and string links and follow the pattern of Example 8.8 to construct a bi-order on PBn (D). The main difference lies in the details 65  Chapter 8. PBn (D) is bi-orderable. of the proofs, the case of PBn (D) being more technically challenging. As an example of such a challenge recall the proof of injectivity of the reduced Magnus expansion of RF (k) into the polynomial ring Λ (see Chapter 7) versus the elegant and straightforward proof of the injectivity of the original Magnus expansion of F [MKS66].  8.2  Ordering semi-direct products  Let H and Q be bi-ordered groups. It is easy to see that the direct product H × Q is bi-orderable, under the lexicographic ordering hq < h q if and only if q < q , or q = q and h < h Note: to accommodate semi-direct product written as , the lexicographic ordering becomes ”eastern”, with comparison starting on the right. For the rest of the chapter, the terms lexicographic ordering refers to the ”eastern” lexicographic ordering as defined above. However, a semi-direct product H Q is not necessarily bi-ordered under lexicographic ordering. Example 8.9. The fundamental group G of the Klein bottle can be written as a semi-direct product Z Z of two infinite cyclic groups. Now write the presentation of this group in terms of the generators and relations: G =< x, y|yxy −1 = x−1 > . Consider x = 1 and suppose that x is positive. If G is bi-ordered, then yxy −1 is also positive - which is a contradiction to the fact that x−1 must be negative. If we assume that x = 1 is negative, then x−1 must be positive, and we arrive at the same contradiction again. Given elements h in H, and q in Q, we let hq denote h under the action of q. Recall that the multiplication in a semi-direct product H Q is given by the formula (h, q)(h , q ) = (hh q , qq ), where hq = qhq −1 . The following technical lemma provides a necessary and sufficient condition for lexicographic ordering of a semi-direct product to be a bi-order. Lemma 8.10. Let H and Q be bi-ordered groups. Then the lexicographic order on H Q is a bi-ordering if and only if the action of Q on H preserves the order on H. Equivalently, q(PH )q −1 ⊂ PH for all q ∈ Q. 66  Chapter 8. PBn (D) is bi-orderable. Proof. =⇒ Let h be an element of the positive cone PH , and hence (h, 1) is in the positive cone PH Q . The action on h by any q ∈ Q is equivalent to conjugation in the semi-direct product group H Q. (1, q)(h, 1)(1, q −1 ) = (1hq , q)(1, q −1 ) = (hq , q)(1, q −1 ) = (hq , 1) Since H Q is a bi-ordered group, we know that (hq , 1) is in PH Q , i.e. (1, 1) < (hq , 1). Therefore we know that 1 <H hq , and thus hq is in the positive cone PH . ⇐= We proceed to check the three conditions required for an ordering to be a bi-order. 1. Let (h, q) and (h , q ) be positive elements in H  Q, then  (a) if q = 1Q or q = 1Q , then 1Q < qq (b) if both q and q equal 1Q , then 1H < hh q = hh In either case 1 < (h, q)(h , q, ) = (hh q , qq ), so positive cone PH closed under multiplication.  Q  is  2. Let (h, q) be some element of H Q, and suppose q = 1Q . Since Q is bi-ordered, q must be either positive on negative in Q, correspondingly making (h, q) either positive or negative in H Q. If q equals 1Q we determine the sign of (h, q) according to the sign of h, since H is biordered. We thus see that any element of H Q is exactly one of positive, negative, or the identity. 3. Let (h, q) be an element in the positive cone PH Q , i.e. (h, q) > (1, 1). We show that PH Q is closed under conjugation by any (a, b) in H Q: −1  (a, b)(h, q)(a, b)−1 = (a, b)(h, q)((a−1 )b , b−1 ) −1  = (ahb , bq)((a−1 )b , b−1 ) −1  = (ahb [(a−1 )b ]bq , bqb−1 ) = (ahb (a−1 )q , bqb−1 ) If q = 1, we see that bqb−1 is in the positive cone PQ , since Q is bi-ordered. If q = 1, then ahb (a−1 )q = ahb a−1 is in PH since PH is closed under conjugation by elements of Q. Thus in either case (PH Q )(a,b) ⊂ PH Q for all (a, b) in H Q.  67  Chapter 8. PBn (D) is bi-orderable.  8.3  Ordering RF (k)  To build an ordering on RF (k) we first recall several useful constructions that we developed in the previous chapter. In Chapter 7 we gave a definition of the reduced free group RF (k) as a quotient of the free group F on k variables x1 , ..., xk by relations which set to 1F every commutator C in {xi } with repeats. We also described in detail the reduced Magnus expansion of RF (k) into the polynomial ring Λ on k non-commuting variables X1 , ..., Xk , with with integral coefficients: µ : RF (k) −→ Λ xi  xi −1  −→ 1 + Xi −→ 1 − Xi  The main Theorem 7.11 of Chapter 7 showed that the reduced Magnus expansion µ is in fact injective. In this section we build an ordering on the polynomial ring Λ and restrict this ordering to a certain subgroup of Λ that contains the image of RF (k) under the reduced Magnus expansion µ. We then proceed to compare elements of RF (k) by looking at their images under µ Definition 8.11. Let us define an ordering on Λ, which we shall call the reduced Magnus ordering. Let f and g be polynomials in Λ. We first arrange the monomials within f and g by degree. Let us now assume an alphabetical order on the variables X1 , ..., Xk , for example X1 is the first letter, X2 is the second letter, etc. Within each degree we arrange the monomials lexicographically. We now compare f and g degree by degree, term by term. We find the first term at which f and g differ and look at its leading coefficients f and g . We declare that f > g if f > g . Example 8.12. Let f = 1 + X2 and g = 1 + X1 − X2 , then f = 1 + 0X1 + X2 < 1 + X1 − X2 = g Note that reduced Magnus ordering does not define a left-order on Λ. By definition, order must be invariant under left multiplication, however (−1)(1 + X1 ) = −1 − X1 < 0, while 1 + X1 > 0. Let H denote a set of elements of Λ of the form (1+ higher order terms). It is easy to see that H is a multiplicative subgroup of Λ. Consider an element g of H of the form g = 1 + G, where G consists of terms of non-zero degree. Then the inverse element g −1 is given by g −1 = 1 − G + G2 − G3 + ... + (−1)k Gk , 68  Chapter 8. PBn (D) is bi-orderable. because gg −1 = (1 + G)(1 − G + G2 − G3 + ... + (−1)k Gk ) = 1 + (−1)k+1 Gk+1 =1 Note that every monomial of total degree greater than k is bound to have repeats of some variable and hence will be set to zero in Λ. Proposition 8.13. The reduced Magnus ordering induces a bi-ordering on the subgroup H, and hence on the classical reduced free group RF (k). Proof. We define a positive cone PH under the reduced Magnus ordering as follows. Let f be an element of H. Arrange the monomials in f as described in the definition of the reduced Magnus ordering, and look at the terms of non-zero total degree. We say that f is positive, i.e. f is an element of PH , if the coefficient of the leading monomial is positive. −1 partitions H, since the leading coefficient of We see that PH ∪ 1 ∪ PH an element must be either positive or negative. To see that PH is closed under multiplication, let f and g be two elements of PH , where f = 1 + F and g = 1 + G. We see that f g = (1 + F )(1 + G) = 1 + F + G + F G > 0. The following simple case-by-case argument demonstrates that f g is positive. If f = g then their respective leading coefficients f and g are equal and the leading coefficient f g equals to the sum f + g > 0. If f = g, then one of the elements is bigger than the other under the ordering <. Let us say that g < f . If g contains monomials of smaller degree than those of f , then g is the leading coefficient of f g. If the smallest degree is the same for both f and g then two possibilities may occur. If the lexicographically smallest monomial is the same for both f and g, then f g = f + g > 0, otherwise f g = g > 0 is the leading coefficient of f g. Finally, we demonstrate the invariance of PH under conjugation by any  69  Chapter 8. PBn (D) is bi-orderable. element g of H. Let us assume that g is positive. gf g −1 = (1 + G)(1 + F )(1 − G + G2 − G3 + ... + (−1)k Gk ) = (1 + F + G + GF )(1 − G + G2 − G3 + ... + (−1)k Gk ) = 1 − G + G2 + ... + (−1)k Gk + + F − F G + F G2 + ... + F (−1)k Gk + G − G2 + ... + G(−1)k Gk + GF − GF G + GF G2 + ... + GF (−1)k Gk = 1 + F − F G + F G2 + ... + F (−1)k Gk + GF − GF G + GF G2 + ... + GF (−1)k Gk Multiplying F by any sum of monomials of non-zero degree, we raise degree of every monomial in F . It is then clear that in the above expression for gf g −1 monomials of the smallest degree come from F . f is positive, hence the coefficient of the monomial of smallest degree in F is also positive. It follows that gf g −1 is an element of the positive cone PH . Proposition 8.14. The reduced Magnus ordering is preserved under any automorphism of RF (k) that induces the identity on the abelianization of RF (k) which we denote by RF (k)ab =  RF (k) [RF (k), RF (k)]  Proof. We first perform an auxilary computation. Let O(2) stand for terms of degree 2 and higher. H2 denotes the subgroup of H whose elements are of the form 1 + O(2). The following computation shows that the reduced Magnus expansion µ maps [RF (k), RF (k)] into the subgroup H2 . Let a and b be elements of RF (k), and let 1+A and 1+B denote their respective images under µ. We see that µ takes commutator aba−1 b−1 into the subgroup H2 µ(aba−1 b−1 ) = (1 + A)(1 + B)(1 − A + ... + (−1)n An )(1 − B + ... + (−1)n B n ) = (1 + A + B + O(2))(1 − A + O(2))(1 − B + O(2)) = (1 + A + B + O(2) − A + O(2))(1 − B + O(2)) = (1 + B + O(2))(1 − B + O(2)) = 1 + B + O(2) − B + O(2) = 1 + O(2)  70  Chapter 8. PBn (D) is bi-orderable. We now return to the proof of our proposition. We let ϕ be an automorphism that induces the identity on RF (k)ab and explore the action of ϕ on the generators {xi }ki=1 of RF (k). Since ϕ induces the identity on RF (k)ab , we see that ϕ(xi ) = xi C, where C is an element of the commutator subgroup [RF (k), RF (k)] of the reduced free group RF (k). The image of ϕ(xi ) under the expansion µ becomes µ(ϕ(xi )) = µ(xi C) = (1 + Xi )(1 + O(2)) = 1 + Xi + O(2). To observe the effect of ϕ on the word f in RF (k) we replace each instance of xi by ϕ(xi ). Under the reduced Magnus expansion µ each occurrence of Xi is replaced by Xi + O(2), which implies that the first non-zero non-constant terms of µ(f ) and µ(ϕ(f )) are equal. It follows that the image of a positive element f under ϕ remains positive.  8.4  Automorphisms of RF (k) and order on PBn (D)  The following elegant construction appears in [HL90] and describes the connection between a string link over a disk and an automorphism of RF (k). We return to the geometric description of string links over a disk. Recall from Chapter 3 that a string link σ over the disk D is a piece-wise linear proper embedding of a disjoint union of unit intervals into a cylinder D × I, such that the ith strand of σ starts at the point Pi × {0} and ends at the point Pi × {1}. Let us introduce some notation. Given a string link σ, let Y denote the complement of the strings of σ in the cylinder D×I. Let Y0 and Y1 denote the boundaries of Y , where Y0 stands for the punctured disk (D\{P1 , ..., Pk })×0, and Y1 stands for (D \ {P1 , ..., Pk }) × 1. Let Gn denote the nth lower central series subgroup of a group G. A theorem of Stallings [Sta65] implies that the inclusion maps Y0 → Y ← Y1 induce isomorphisms of the quotients of the fundamental groups by the lower central series subgroups π1 (Y0 ) (π1 (Y0 ))m  π1 (Y ) (π1 (Y ))m  π1 (Y1 ) (π1 (Y1 ))m  (∗)  for all finite m. 71  Chapter 8. PBn (D) is bi-orderable. It is clear that π1 (Y0 ) and π1 (Y1 ) are free groups on k generators. Since strands of a string link may be knotted on themselves and on other strands, we see that π1 (Y ) may not be a free group on k generators. However, it is true that a given generator of π1 (Y ) is a conjugate of a meridian xi , circling the ith strand, for some 1 ≤ i ≤ k. In order to ignore ”knotting” phenomenon, we introduced the reduced group G(σ) of Y , in which commutators of conjugates of meridians are set to 1. (See Chapter 3.) Recall the alternative definition of RF (k) as a quotient of a free group F on k generators by its normal subgroup J, generated by commutators with repeats of a variable (see Chapter 7). RF (k) is clearly nilpotent, since any commutator of length greater than k is bound to have repeats and therefore shall be set to 1. Now we re-write expression (*) in terms of reduced groups: RF (k) −→ G(σ) ←− RF (k) Thus given a string link σ we obtain an automorphism of generators {x1 , ...xk } of RF (k) that satisfy (1.) φ(xi ) is conjugate to xi for 1 ≤ i ≤ k. (2.) φ(x1 ...xk ) = x1 ...xk . Let Ak denote the group of automorphisms φ of RF (k) satisfying conditions (1.) and (2.) above. Directly from the definition of Ak we obtain Lemma 8.15. Each automorphism ϕ ∈ Ak induces the identity on RF (k)ab . The following theorem is proved in [HL90] Theorem 8.16. The map PBk (D) → Ak σ → φσ is an isomorphism Proof. The proof cleverly combines method of induction together with the reduced Magnus expansion µ of RF (k). Note that the above theorem is similar to a well-known result on classical pure and full braid groups over a disk, which associates PBk (D) with the group of automorphisms of a free group F [Bir74].  72  Chapter 8. PBn (D) is bi-orderable. We now combine the definition of lexicographic ordering from page 66, Lemma 8.10, Propositions 8.13 and 8.14, Theorem 8.16 and the semi-direct product decomposition of PBn (D) from [Gol74] and obtain Theorem 8.17. The ”eastern” lexicographical ordering on PBn (D)  RF (n − 1)  RF (n − 2)  ...  RF (1)  is a bi-ordering. To compare elements τ1 ...τn−1 and τ1 ...τn−1 we compare τn−i and τn−i via the reduced Magnus ordering of RF (i) for i = 1, ...n − 1, i.e. with comparison starting on the right.  73  Bibliography [Ada94]  Colin C. Adams. The knot book. W. H. Freeman and Company, New York, 1994.  [Art47]  Emil Artin. Theory of braids. Ann. Math, 48(1):101–126, 1947.  [Bel04]  Paolo Bellingeri. On presentations of surface braid groups. J. Algebra, 274:543–563, 2004.  [Bir69]  Joan S. Birman. On braid groups. Comm. Pure Appl. Math., 22:41–72, 1969.  [Bir74]  Joan S. Birman. Braids, Links and Mapping Class Groups. Number 82 in Annals of Mathematics Studies, Princeton University Press, Princeton, 1974.  [DDRW02] Patrick Dehornoy, Ivan Dynnikov, Dale Rolfsen, and Bert Wiest. Why are braids orderable?, volume 14 of Panoramas et Synth`eses [Panoramas and Syntheses]. Soci´et´e Math´ematique de France, Paris, 2002. [Fen83]  Roger A. Fenn. Techniques of geometric topology, volume 57 of LMS Lecture Notes. 1983.  [FN62]  E. Fadell and L. Neuwirth. Configuration spaces. Math. Scand., 10:111–118, 1962.  [FVB62]  E. Fadell and J. Van Buskirk. The braid groups of E2 and S2 . Duke Math. J., 29:243–257, 1962.  [GG04]  D.L. Gon¸calves and J. Guaschi. On the structure of surface pure braid groups. J. Pure Appl. Algebra, 186:187–218, 2004.  [GM01]  Juan Gonz´ alez-Meneses. New presentations of surface braid groups. J. of Knot Theory Ramifications, 10(3):431–451, 2001.  74  Bibliography [GM02]  Juan Gonz´ alez-Meneses. Ordering pure braid groups on compact, connected surfaces. Pacific J. of Math., 203(2):369–378, 2002.  [Gol74]  Deborah Louise Goldsmith. Homotopy of braids - In answer to a question of E. Artin. In Topology Conference, volume 375 of Lecture Notes in Mathematics, pages 91–96. 1974.  [HL90]  Nathan Habegger and Xiao-Song Lin. The classification of links up to link-homotopy. J. Amer. Math. Soc., 3(2):389–419, 1990.  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