Celebratio Mathematica — Birman

[1] J. S. Birman: Braid groups and their relationship to mapping class groups. Ph.D. thesis, New York University, 1968. Advised by W. Magnus. MR 2617171 phdthesis

[2] J. S. Birman: “On braid groups,” Comm. Pure Appl. Math. 22 (January 1969), pp. 41–72. MR 0234447 Zbl 0157.30904 article

In 1962, R. H. Fox introduced the concept of a braid group associated with an arbitrary manifold, \( M \). It was shown by Fox and L. Neuwirth [7], and again by E. Fadell and J. Van Buskirk [6], that if the manifold was chosen to be the Euclidean plane, then Fox’s definition yielded the classical algebraic braid group of E. Artin [1], [2]. At the same time, the idea of braid groups on other manifolds seemed of interest in itself, and these groups were accordingly studied for the cases \( M = S^2 \) in [6] and \( M = P^2 \) in [10], with scattered results reported for manifolds of dimension \( > 2 \) in [5]. The present investigation begins with Fox’s definition, and concerns itslef with the algebraic, structural and geometric properties of these braid groups on arbitrary manifolds.

[3] J. S. Birman: “Automorphisms of the fundamental group of a closed, orientable 2-manifold,” Proc. Am. Math. Soc. 21 : 2 (May 1969), pp. 351–354. MR 0239593 Zbl 0175.50103 article

[4] J. S. Birman: “Mapping class groups and their relationship to braid groups,” Comm. Pure Appl. Math. 22 : 2 (March 1969), pp. 213–238. MR 0243519 Zbl 0167.21503 article

[5] J. S. Birman: “Non-conjugate braids can define isotopic knots,” Comm. Pure Appl. Math. 22 (March 1969), pp. 239–242. MR 0244985 Zbl 0187.45502 article

It is well-known that every knot or linkage can be obtained from a closed braid, and also that braids which represent conjugate elements in the braid group define isotopic knots or linkages [1]. It has, however, been an open question whether non-conjugate braids can define isomorphic knots or linkages. The question has been a difficult one to answer because of its close relationship to the conjugacy problem in the braid group, which until recently was an unsolved problem. In 1965 a partial answer was obtained by F. A. Garside [3], who derived a procedure for deciding when two braids are conjugate in the \( n \)-string braid group, and as an application showed that braids which are not conjugate could nevertheless define isotopic linkages. In the present paper we exhibit a very simple example which extends this result to knots, and discuss some of its consequences.

[6] J. S. Birman: “Abelian quotients of the mapping class group of a 2-manifold,” Bull. Am. Math. Soc. 76 : 1 (1970), pp. 147–150. MR 0249603 Zbl 0191.22401 article

[7] J. Birman and W. Magnus: “Discriminant and projective invariants of binary forms,” Comm. Pure Appl. Math. 23 (May 1970), pp. 269–275. A correction to this article was published in Comm. Pure Appl. Math. 26 (1973). MR 0259903 Zbl 0189.55002 article

[8] J. S. Birman: “Errata: ‘Abelian quotients of the mapping class group of a 2-manifold’,” Bull. Am. Math. Soc. 77 : 3 (1971), pp. 479. Errata to an article published in Bull. Am. Math. Soc. 76:1 (1970). MR 0267091 Zbl 0213.25001 article

[9] J. S. Birman: “On Siegel’s modular group,” Math. Ann. 191 (1971), pp. 59–68. MR 0280606 Zbl 0208.10601 article

Siegel’s modular group \( \Gamma_g \) is the group of all \( 2g \times 2g \) matrices with integral entries which satisfy the condition: \[ SJS^{\prime} = J \] where \( s \in \Gamma_g \), \( S^{\prime} = \) transpose of \( S \), and if \( I_g \) and \( 0_g \) are the \( g\times g \) unit and zero matrices respectively, then: \[ J = \left( \begin{array}{cc} 0_g & I_g \\ -I_g & 0_g \end{array} \right). \] Generators for \( \Gamma_g \) were first determined by Hua and Reiner [1949], and Klingen [1961] obtained a characterization of \( \Gamma_g \) for \( g\geq 2 \) by a finite system of defining relations. However, Klingen’s results have been of somewhat limited use because, while he gives a procedure for finding defining relations, he does not carry it through explicitly, and in fact the explicit determination of such a system involves a fairly tedious and lengthy calculation. Finding ourselves in the position of needing explicit information about \( \Gamma_g \), we set ourselves the task of reducing Klingen’s results to more useable form. The primary purpose of the paper is to give the results of this calculation (Theorem 1) and to describe the methodology behind our proof.

[10] J. S. Birman and H. M. Hilden: “On the mapping class groups of closed surfaces as covering spaces,” pp. 81–115 in Advances in the theory of Riemann surfaces (Stony Brook, NY, 1969). Edited by L. V. Ahlfors, L. Bers, H. M. Farkas, R. C. Gunning, I. Kra, and H. E. Rauch. Annals of Mathematics Studies 66. Princeton University Press, 1971. MR 0292082 Zbl 0217.48602 incollection

Irwin Kra	Related
Hershel Mark Farkas	Related
Harry Ernest Rauch	Related
Lipman Bers	Related
Lars Valerian Ahlfors	Related
Robert Clifford Gunning	Related
Hugh Michael Hilden	Related

[11] J. S. Birman: “A normal form in the homeotopy group of a surface of genus 2, with applications to 3-manifolds,” Proc. Am. Math. Soc. 34 : 2 (August 1972), pp. 379–384. MR 0295308 Zbl 0253.55001 article

[12] J. S. Birman and D. R. J. Chillingworth: “On the homeotopy group of a non-orientable surface,” Proc. Camb. Philos. Soc. 71 : 3 (May 1972), pp. 437–448. An erratum to this article was published in Math. Proc. Camb. Philos. Soc. 136:2 (2004). MR 0300288 Zbl 0232.57001 article

Let \( X \) be a closed, compact connected 2-manifold (a surface), which we will denote by \( O \) or \( N \) if we wish to stress that \( X \) is orientable or non-orientable. Let \( G(X) \) denote the group of all homeomorphisms \( X\to X \), \( D(X) \) the normal subgroup of homeomorphisms isotopic to the identity, and \( H(X) \) the factor group \( G(X)/D(X) \), i.e. the homeotopy group of \( X \). The problem of determining generators for \( H(O) \) was considered by Lickorish in [7,8], and the second of these papers specifies a finite set of generators of a particularly simple type. In [10] and [11] Lickorish considered the analogous problem for non-orientable surfaces, and, using Lickorish’s partial results, Chillingworth [4] determined a finite set of generators for \( H(N) \). While the generators obtained for \( H(O) \) and \( H(N) \) were strikingly similar, it was noteworthy that the techniques used in the two cases were different, and in particular that little use was made in the non-orientable case of the earlier results obtained on the orientable case. The purpose of this note is to show that the results of Lickorish and Chillingworth on non-orientable surfaces follow rather easily from the work in [7,8] by an application of some ideas from the theory of covering spaces [2]. Moreover, while Lickorish and Chillingworth sought only to find generators, we are able to show how in fact the entire structure of the group \( H(N) \) is determined by \( H(O) \), where \( O \) is an orientable double cover of \( N \). Finally, we are able to determine defining relations for \( H(N) \) for the case where \( N \) is the connected sum of 3 projective planes.

[13] J. S. Birman and H. M. Hilden: “Isotopies of homeomorphisms of Riemann surfaces and a theorem about Artin’s braid group,” Bull. Am. Math. Soc. 78 : 6 (November 1972), pp. 1002–1004. MR 0307217 Zbl 0255.57002 article

[14] J. S. Birman and H. M. Hilden: “Lifting and projecting homeomorphisms,” Arch. Math. (Basel) 23 : 1 (1972), pp. 428–434. MR 0321071 Zbl 0247.55001 article

Let \( X \) be a pathwise connected and locally pathwise connected topological space, \( G \) the group of all self-homeomorphisms of \( X \), and \( D \) the subgroup of maps isotopic to the identity. The homeotopy group of \( X \) is defined to be the group \( G/D \). Let \( \tilde{X} \) be a p.c., l.p.c. covering space of \( X \), with projection \( p \). The relationship between the homeotopy groups of \( \tilde{X} \) and \( X \) is studied. It is shown that under sufficiently strong restrictions on \( \tilde{X} \), \( X \) and \( p \) the homeotopy group of \( X \) is isomorphic to a factor group of the homeotopy group of \( \tilde{X} \), with weaker results as one weakens the restrictions on \( \tilde{X} \) and \( X \).

The situation studied here first came to the authors’ attention in an earlier investigation [Birman and Hilden 1971]. The homeotopy groups of 2-manifolds play an important role in the theory of Riemann surfaces, and also in the classification of 3-manifolds. It was shown in [Birman and Hilden 1971] that one could gain considerable insight into the structure of the homeotopy groups of surfaces by utilizing the fact that any closed compact orientable surface of genus \( g \) with \( (2g + 2) \) points removed can be regarded as a 2-sheeted covering of a \( (2g + 2) \)-punctured sphere, and making use of the known properties of the homeotopy group of the punctured sphere. The development of this relationship suggested that other coverings of more general spaces might also be of interest, thus motivating the present investigation. At the conclusion of this paper a new application to surface topology is discussed briefly. A detailed workingout of this application will be found in [Birman and Chillingworth 1969], which should appear concurrently with the present work.

[15] J. S. Birman and H. M. Hilden: “The homeomorphism problem for \( S^3 \),” Bull. Am. Math. Soc. 79 : 5 (September 1973), pp. 1006–1010. MR 0319180 Zbl 0272.57001 article

Let \( M \) be a closed, orientable 3-manifold which is defined by a Heegaard splitting of genus \( g \). Each such Heegaard splitting may be associated with a self-homeomorphism of a closed, orientable surface of genus \( g \) (the surface homeomorphism is used to define a pasting map) and it will be assumed that this surface homeomorphism is given as a product of standard twist maps [Lickorish 1962] on the surface. We assert:

If \( M \) is defined by a Heegaard splitting of genus \( \leq 2 \), then an effective algorithm exists to decide whether \( M \) is topologically equivalent to the 3-sphere \( S^3 \). This algorithm also applies to a proper subset of all Heegaard splittings of genus \( > 2 \).

This result is of interest because it had not been known whether such an algorithm was possible for \( g\geq 2 \), and also because the algorithm has a possible application in testing candidates for a counterexample to the Poincaré conjecture.

In this note we will describe the algorithm, and sketch a brief proof. Related results about the connections between representations of 3-manifolds as Heegaard splittings, and as branched coverings of \( S^3 \), are summarized at the end of this paper. A detailed report will appear in another journal.

[16] J. Birman and W. Magnus: “Correction to: ‘Discriminant and projective invariants of binary forms’,” Comm. Pure Appl. Math. 26 (January 1973), pp. 85. Correction to an article published in Comm. Pure Appl. Math. 23 (1970). MR 0325533 Zbl 0252.55006 article

[17] J. S. Birman and H. M. Hilden: “On isotopies of homeomorphisms of Riemann surfaces,” Ann. Math. (2) 97 : 3 (May 1973), pp. 424–439. MR 0325959 Zbl 0237.57001 article

[18] J. S. Birman: “An inverse function theorem for free groups,” Proc. Am. Math. Soc. 41 (1973), pp. 634–638. MR 0330295 Zbl 0274.20032 article

Let \( F_n \) be a free group of rank \( n \) with free basis \( x_1,\dots \), \( x_n \). Let \( \{y_1,\dots \), \( y_k\} \) be a set of \( k \leq n \) elements of \( F_n \), where each \( y_i \) is represented by a word \[ Y_i(x_1, \dots, x_n) \] in the generators \( x_j \). Let \( \partial y_i/\partial x_j \) denote the free derivative of \( y_i \) with respect to \( x_j \), and let \[ J_{kn} = \|\partial y_i/\partial x_j\| \] denote the \( k\times n \) Jacobian matrix.

If \( k = n \), the set \( \{y_1,\dots \), \( y_n\} \) generates \( F_n \) if and only if \( J_{nn} \) has a right inverse. If \( k < n \), the set \( \{y_1,\dots \), \( y_k\} \) may be extended to a set of elements which generate \( F_n \) only if \( J_{kn} \) has a right inverse.

Several applications are given.

[19] J. S. Birman: “Plat presentations for link groups,” Comm. Pure Appl. Math. 26 (September 1973), pp. 673–678. Collection of articles dedicated to Wilhelm Magnus. MR 0336734 Zbl 0266.20027 article

[20] J. S. Birman: “Poincaré’s conjecture and the homeotopy group of a closed, orientable 2-manifold,” pp. 214–221 in Collection of articles dedicated to the memory of Hanna Neumann, VI, published as J. Austral. Math. Soc. 17 : 2. Australian National University (Canberra), March 1974. MR 0343255 Zbl 0282.55003 incollection

[21] J. S. Birman: Braids, links, and mapping class groups. Annals of Mathematics Studies 82. Princeton University Press, 1974. Based on lecture notes by James Cannon. An erratum to Theorem 2.7 is given in Can. J. Math. 34:6 (1982). MR 0375281 Zbl 0305.57013 book

[22] J. S. Birman: “Mapping class groups of surfaces: A survey,” pp. 57–71 in Discontinuous groups and Riemann surfaces (College Park, MD, 21–25 May 1973). Edited by L. Greenberg. Annals of Mathematics Studies 79. Princeton University Press, 1974. MR 0380762 Zbl 0297.57001 incollection

[23] J. S. Birman: “On the equivalence of Heegaard splittings of closed, orientable 3-manifolds,” pp. 137–164 in Knots, groups, and 3-manifolds: Papers dedicated to the memory of R. H. Fox. Edited by L. P. Neuwirth. Annals of Mathematics Studies 84. Princeton University Press, 1975. MR 0375318 Zbl 0337.57002 incollection

Ralph Hartzler Fox	Related
Lee Paul Neuwirth	Related

[24] J. S. Birman and H. M. Hilden: “Heegaard splittings of branched coverings of \( S^3 \),” Trans. Am. Math. Soc. 213 (November 1975), pp. 315–352. MR 0380765 Zbl 0312.55004 article

This paper concerns itself with the relationship between two seemingly different methods for representing a closed, orientable 3-manifold: on the one hand as a Heegaard splitting, and on the other hand as a branched covering of the 3-sphere. The ability to pass back and forth between these two representations will be applied in several different ways.

It will be established that there is an effective algorithm to decide whether a 3-manifold of Heegard genus 2 is a 3-sphere.
We will show that the natural map from 6-plat representations of knots and links to genus 2 closed oriented 3-manifolds is injective and surjective. This relates the question of whether or not Heegaard splittings of closed, oriented 3-manifolds are “unique” to the question of whether plat representations of knots and links are “unique”.
We will give a counterexample to a conjecture (unpublished) of W. Haken, which would have implied that \( S^3 \) could be identified (in the class of all simply-connected 3-manifolds) by the property that certain canonical presentations for \( \pi_1S^3 \) are always “nice”.

The final section of the paper studies a special class of genus 2 Heegard splittings: the 2-fold covers of \( S^3 \) which are branched over closed 3-braids. It is established that no counterexamples to the “genus 2 Poincaré conjecture” occur in this class of 3-manifolds.

[25] J. S. Birman: “A note on the construction of simply-connected 3-manifolds as branched covering spaces of \( S^3 \),” Proc. Am. Math. Soc. 55 : 2 (March 1976), pp. 440–442. MR 0394629 Zbl 0325.55003 article

Let \( K \) be a knot in \( S^3 \) and let \[ \omega:\pi_1(S^3 - K) \to \Sigma_n \] be a transitive representation into the symmetric group \( \Sigma_n \) on \( n \) letters. The pair \( (K, \omega) \) defines a unique closed, connected orientable 3-manifold \( M(K,\omega) \), which is represented as an \( n \)-sheeted covering space of \( S^3 \), branched over \( K \). A procedure is given for representing \( M(K,\omega) \) by a Heegard splitting, and a formula is given for computing the genus of that Heegard splitting of \( M(K,\omega) \). This formula is then applied to the 3-sheeted irregular covering spaces studied by Hilden (Bull. Amer. Math. Soc. 80 (1974), 1243–1244) and Montesinos (Bull. Amer. Math. Soc. 80 (1974), 845–846), and, also, Tesis (Univ. de Madrid, 1971) to show that these particular covering spaces cannot yield counterexamples to the Poincaré Conjecture if the branch set has bridge number \( < 4 \).

[26] J. S. Birman and R. Craggs: “On the \( \mu \)-invariant of \( Z \)-homology 3-spheres,” Bull. Am. Math. Soc. 82 : 2 (March 1976), pp. 253–255. MR 0397734 Zbl 0343.55001 article

[27] J. S. Birman: “On the stable equivalence of plat representations of knots and links,” Can. J. Math. 28 : 2 (1976), pp. 264–290. MR 0402715 Zbl 0339.55005 article

[28] J. S. Birman, F. González-Acuña, and J. M. Montesinos: “Heegaard splittings of prime 3-manifolds are not unique,” Michigan Math. J. 23 : 2 (1976), pp. 97–103. MR 0431175 Zbl 0321.57004 article

José María Montesinos Amilibia	Related
Francisco Javier González-Acuña	Related

[29] J. S. Birman: “The algebraic structure of surface mapping class groups,” pp. 163–198 in Discrete groups and automorphic functions (Cambridge, 28 July–15 August 1975). Edited by W. J. Harvey. Academic Press (London), 1977. MR 0488019 incollection

[30] J. S. Birman: “Special Heegaard splittings for closed, oriented 3-manifolds,” Topology 17 : 2 (1978), pp. 157–166. MR 0482764 Zbl 0412.57009 article

[31] J. S. Birman and R. Craggs: “The \( \mu \)-invariant of 3-manifolds and certain structural properties of the group of homeomorphisms of a closed, oriented 2-manifold,” Trans. Am. Math. Soc. 237 (March 1978), pp. 283–309. MR 0482765 Zbl 0383.57006 article

Let \( \mathcal{H}(n) \) be the group of orientation-preserving self-homeomorphisms of a closed oriented surface \( \operatorname{Bd} U \) of genus \( n \), and let \( \mathcal{H}(n) \) be the subgroup of those elements which induce the identity on \( H_1(\operatorname{Bd} U;\mathbf{Z}) \). To each element \( h \in \mathcal{H}(n) \) we associate a 3-manifold \( M(h) \) which is defined by a Heegard splitting. It is shown that for each \( h\in\mathcal{H}(n) \) there is a representation \( \rho \) of \( \mathcal{H}(n) \) into \( \mathbf{Z}/2\mathbf{Z} \) such that if \( k\in\mathcal{H}(n) \), then the \( \mu \)-invariant \( \mu(M(h)) \) is equal to the \( \mu \)-invariant \( \mu(M(kh)) \) if and only if \( k\in\operatorname{kernel} \rho \). Thus, properties of the 4-maniolds which a given 3-manifold bounds are related to group-theoretical structure in the group of homeomorphisms of a 2-manifold. The kernels of the homomorphisms from \( \mathcal{H}(n) \) onto \( \mathbf{Z}/2\mathbf{Z} \) are studied and are shown to constitute a complete conjugacy class of subgroups of \( \mathcal{H}(n) \). The class has nontrivial finite order.

[32] J. S. Birman and J. Powell: “Special representations for 3-manifolds,” pp. 23–51 in Geometric topology (Athens, GA, 1–12 August 1977). Edited by J. C. Cantrell. Academic Press (New York), 1979. MR 537723 Zbl 0471.57002 incollection

Using techniques from the link calculus, it is shown that each closed, oriented 3-manifold admits a very special type of Heegaard decomposition which is also a very special surgery. The associated sewing map is the restriction to the Heegaard surface of a homeomorphism of a handlebody which fixes a complete system of meridian discs. The groups of isotopy classes of all such maps is closely related to the classical pure braid group. The associated Heegaard diagram consists of a pair of curves \( (\mathbf{x},\mathbf{y}) \) which have the following property: there is an associated diagram \( (\mathbf{x},\mathbf{z}) \) for \( S^3 \) such that \( (\mathbf{y},\mathbf{z}) \) also defines \( S^3 \) and the intersection matrices \( \mathbf{x} \cap \mathbf{z} \) and \( \mathbf{y} \cap \mathbf{z} \) are each the identity matrix. These diagrams have a great deal of interesting structure, including certain self-duality and finiteness properties.

James Cecil Cantrell	Related
Jerome E. Powell	Related

[33] J. S. Birman: “A representation theorem for fibered knots and their monodromy maps,” pp. 1–8 in Topology of low-dimensional manifolds: Proceedings of the second Sussex conference, 1977 (Chelwood Gate, UK, 8–11 July 1977). Edited by R. Fenn. Lecture Notes in Mathematics 722. Springer (Berlin), 1979. MR 547448 Zbl 0406.57004 incollection

[34] H. Seifert and W. Threlfall: Seifert and Threll: A textbook of topology and Seifert: Topology of 3-dimensional fibred spaces. Edited by J. S. Birman and J. Eisner. Pure and Applied Mathematics 89. Academic Press (New York), 1980. Two German texts combined into a single volume. MR 0575168 Zbl 0469.55001 book

William Richard Maximilian Hugo Threlfall	Related
Herbert Karl Johannes Seifert	Related
Julian Eisner	Related

[35] J. S. Birman and J. M. Montesinos: “On minimal Heegaard splittings,” Michigan Math. J. 27 : 1 (1980), pp. 47–56. MR 555836 Zbl 0436.57003 article

In 1936 J. H. C. Whitehead developed an algorithm for transforming a set of words in a free group to a new set which has minimum length among all systems which are equivalent to the given system under automorphisms of the free group (see [Whitehead 1936] and also [Higgins and Lyndon 1974]). Whitehead’s stated motivation for studying this question was related (but different) topological question: if a Heegard diagram for a 3-manifold \( M \) does not have minimal genus, is there an algorithm for systematically reducing the genus and then further modifying the diagram to some “simplest” possible diagram for that \( M \)? (See section 4 of [Whitehead 1936].) Such an algorithm would in certain cases solve the homeomorphism problem for 3-manifolds. The results of Whitehead were later sharpened by Zieschang [1970], with further positive contributions by Waldhausen [1968, 1978].

[36] J. S. Birman: “Errata: On the conjugacy problem in the braid group,” Can. J. Math. 34 : 6 (1982), pp. 1396–1397. Errata for a theorem in the book Braids, links, and mapping class groups (Princeton Univ. Press, 1974). MR 678679 article

[37] J. S. Birman and M. E. Kidwell: “Fixed points of pseudo-Anosov diffeomorphisms of surfaces,” Adv. in Math. 46 : 2 (November 1982), pp. 217–220. MR 679909 Zbl 0508.55001 article

This article supplements Richard Miller’s paper [1982] by establishing a result on the minimum number of fixed points in the homotopy class of a pseudo-Anosov surface transformation. We thus verify one of the many far-reaching results that William Thurston announced in a preprint [1977] dealing with the classification up to homotopy of diffeomorphisms of compact surfaces. Thurston stated his theorems without proofs, but with many hints at techniques. Fathi, Laudenbach, Poénaru, and other workers at an intensive seminar at the University of Orsay gave detailed proofs of some (but not all) of Thurston’s assertions and published their findings in a monograph [1979]. More recently, Miller [1982] and Gilman [1981] have verified Thurston’s claim that his Theorem 4 [1977, p. 81] follows from the classical investigations of Jakob Nielsen [1927, 1932]. Using Miller’s exposition of the Nielsen–Thurston theory, we establish Thurston’s Theorem 6 [1977, p. 12]:

An orientation-preserving pseudo-Anosov diffeomorphism of a closed, orientable surface has the minimum number of periodic points of every period among all maps in its homotopy class.

[38] J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical systems, I: Lorenz’s equations,” Topology 22 : 1 (1983), pp. 47–82. Part II was published in Low-dimensional topology (1983). MR 682059 Zbl 0507.58038 article

[39] J. S. Birman and R. F. Williams: “Knotted periodic orbits in dynamical system, II: Knot holders for fibered knots,” pp. 1–60 in Low-dimensional topology. Edited by S. J. Lomonaco\( Jr. \). Contemporary Mathematics 20. American Mathematical Society (Providence, RI), 1983. Part I was published in Topology 22:1 (1983). MR 718132 Zbl 0526.58043 incollection

Robert Fones Williams	Related	Volume
Samuel James Lomonaco, Jr.	Related

[40] J. S. Birman, A. Lubotzky, and J. McCarthy: “Abelian and solvable subgroups of the mapping class groups,” Duke Math. J. 50 : 4 (December 1983), pp. 1107–1120. MR 726319 Zbl 0551.57004 article

Let \( M \) be an orientable, compact Riemann surface of genus \( g \) with \( b \) boundary components and \( c \) connected components. Assume each connected component of \( M \) has negative Euler characteristic. The mapping class group, \( \mathscr{M}(M) \), or \( M \) is the group of isotopy classes of orientation preserving self-homeomorphisms of \( M \), where if \( \partial M \neq \varnothing \), admissible isotopies fix each component of \( \partial M \) setwise. (Thus, in particular, the isotopy class of a Dehn twist about a curve which is parallel to a component of \( \partial M \) is considered to be trivial.) The reader is referred to [Birman 1977], [Harvey 1979], [Thurston 1988], [Fathi et al. 1975], and [Gilman 1983] for background concerning this group. The main results of this paper will be two theorems about the algebraic structure of \( \mathscr{M}(M) \):

Let \( G \) be an abelian subgroup of \( \mathscr{M}(M) \). Then \( G \) is finitely generated with torsion free rank bounded by \( 3g + b - 3c \).

Every solvable subgroup of \( \mathscr{M}(M) \) is virtually abelian. Furthermore, if \( G \) is a virtually solvable subgroup of \( \mathscr{M}(M) \), then \( G \) contains an abelian subgroup, \( A \), such that the index of \( A \) in \( G \) is bounded by \( V(M) \), where \( V(M) \) is a positive integer depending only upon \( M \).

Alexander Lubotzky	Related
John David McCarthy	Related

[41] J. S. Birman and C. Series: “An algorithm for simple curves on surfaces,” J. London Math. Soc. (2) 29 : 2 (1984), pp. 331–342. MR 744104 Zbl 0507.57006 article

Let \( M \) be a compact orientable surface with non-empty boundary and with \( \chi(M) < 0 \), and let \( \Gamma = \pi_1 M \). Let \( \hat{C} \) be the free homotopy class of a closed loop on \( M \) and let \( W = W(\hat{C}) \) be a word in a fixed set of generators \( \overline{\Gamma} \) which represents \( \hat{C} \). In this paper we give an algorithm to decide, starting with \( W \), whether \( \hat{C} \) has a simple representative, that is a representative without self-intersections. Such a word will be said to be simple. As an application, we begin a study of simple words in \( \overline{\Gamma} \). Our results also apply to infinite geodesics on \( M \), corresponding to biinfinite words in \( \overline{\Gamma} \), where now we ask which finite blocks appear in such a word when the corresponding infinite homotopy class has a simple representative.

[42] J. H. Rubinstein and J. S. Birman: “One-sided Heegaard splittings and homeotopy groups of some 3-manifolds,” Proc. London Math. Soc. (3) 49 : 3 (1984), pp. 517–536. MR 759302 Zbl 0527.57003 article

[43] J. S. Birman and C. Series: “Geodesics with bounded intersection number on surfaces are sparsely distributed,” Topology 24 : 2 (1985), pp. 217–225. MR 793185 Zbl 0568.57006 article

Let \( M \) be a surface of negative Euler characteristic, possibly with boundary, which is either compact or obtained from a compact surface by removing a finite set of points. Let \( D \) be the Poincaré disc. Choose any representation of \( M \) as \( U/\Gamma \), where \( U \subseteq D \) is the universal covering space of \( M \) and \( \Gamma \subset \operatorname{Isom}(D) \). Then the Poincaré metric on \( D \) induces a metric of constant negative curvature on \( M \) and geodesics in \( U \) project to geodesics on \( M \). A geodesic on \( M \) is said to be complete if it is either closed and smooth, or open and of infinite length in both directions. complete geodesics coincide with those which never intersect \( \partial M \). Note that if \( M \) is obtained from a compact surface by removing a finite number of points to form cusps then a complete open geodesic on \( M \) might tend toward infinity along a cusp. In this paper we study the family \( G_k \) of complete geodesics which have at most \( k \) transversal self-intersections, \( k\geq 0 \).

[44] J. S. Birman: “On the Jones polynomial of closed 3-braids,” Invent. Math. 81 : 2 (June 1985), pp. 287–294. MR 799267 Zbl 0588.57005 article

[45] J. S. Birman and B. Wajnryb: “3-fold branched coverings and the mapping class group of a surface,” pp. 24–46 in Geometry and topology (College Park, MD, 1983–1984). Edited by J. C. Alexander and J. L. Harer. Lecture Notes in Mathematics 1167. Springer (Berlin), 1985. MR 827260 Zbl 0589.57009 incollection

James Crew Alexander	Related
Dov Bronislaw Wajnryb	Related
John L. Harer	Related

[46] J. S. Birman: “Knotted periodic orbits in dynamical systems,” pp. 49–58 in Group theoretical methods in physics: Proceedings of the 14th ICGTMP (Seoul, 26–30 August 1985). Edited by Y. M. Cho. World Scientific (Singapore), 1986. MR 852702 Zbl 0709.58029 incollection

[47] J. S. Birman and B. Wajnryb: “Markov classes in certain finite quotients of Artin’s braid group,” Israel J. Math. 56 : 2 (1986), pp. 160–178. MR 880289 Zbl 0621.20025 article

[48] J. S. Birman and C. Series: “Geodesics with multiple self-intersections and symmetries on Riemann surfaces,” pp. 3–11 in Low-dimensional topology and Kleinian groups (Coventry and Durham, UK, 1984). Edited by D. B. A. Epstein. London Mathematical Society Lecture Note Series 112. Cambridge University Press, 1986. MR 903857 Zbl 0618.57001 incollection

David Bernard Alper Epstein	Related
Caroline Mary Series	Related

[49] J. S. Birman and C. Series: “Dehn’s algorithm revisited, with applications to simple curves on surfaces,” pp. 451–478 in Combinatorial group theory and topology (Alta Lodge, UT, 15–18 July 1984). Edited by S. M. Gersten and J. R. Stallings. Annals of Mathematics Studies 111. Princeton University Press, 1987. MR 895628 Zbl 0624.20033 incollection

John Robert Stallings, Jr.	Related
Stephen M. Gersten	Related
Caroline Mary Series	Related

[50] J. S. Birman: “Book reviews: Gerhard Burde and Heiner Zieschang, ‘Knots’, Louis H. Kauffman, ‘On knots’,” Bull. Am. Math. Soc. (N.S.) 19 : 2 (1988), pp. 550–558. MR 1567720 article

Gerhard Burde	Related
Louis H. Kauffman	Related
Heiner Zieschang	Related

[51] J. S. Birman and T. Kanenobu: “Jones’ braid-plat formula and a new surgery triple,” Proc. Am. Math. Soc. 102 : 3 (March 1988), pp. 687–695. MR 929004 Zbl 0643.57007 article

[52] J. S. Birman and C. Series: “Algebraic linearity for an automorphism of a surface group,” J. Pure Appl. Algebra 52 : 3 (July 1988), pp. 227–275. MR 952081 Zbl 0646.57007 article

Let \( M \) be a compact surface, \( \chi(M) < 0 \), and let \( \Gamma = \pi_1M \). Let \( S(M) \) be the set of isotopy classes of multiple simple loops on \( M \). Each \( \Lambda \in S(M) \) determines a family of cyclic words \( W = W(\Lambda) \), with associated coinitial graph \( \tau \). The finite set of coinitial graphs, obtained as \( \Lambda \) ranges over \( S(M) \), is interpreted as a set of ‘\( \pi_1 \)-train tracks’ on \( M \). The linearity theorem asserts that if a topologically induced automorphism \( \phi \) of \( \Gamma \) maps the set of weights \( W \) supported on \( \tau \) to a set supported on \( \tau^{\prime} \), then, with appropriate restrictions, the action is linear on the positive linear span of the \( W \)’s.

[53] J. S. Birman: “Mapping class groups of surfaces,” pp. 13–43 in Braids (Santa Cruz, CA, 13–26 July 1986). Edited by J. S. Birman and A. Libgober. Contemporary Mathematics 78. American Mathematical Society (Providence, RI), 1988. MR 975076 Zbl 0663.57008 incollection

[54] Braids: Proceedings of the AMS-IMS-SIAM joint summer research conference on Artin’s braid group (Santa Cruz, CA, 13–26 July 1986). Edited by J. S. Birman and A. Libgober. Contemporary Mathematics 78. American Mathematical Society (Providence, RI), 1988. Zbl 0651.00010 book

[55] J. S. Birman and H. Wenzl: “Braids, link polynomials and a new algebra,” Trans. Am. Math. Soc. 313 : 1 (1989), pp. 249–273. MR 992598 Zbl 0684.57004 article

[56] J. S. Birman and W. W. Menasco: “Studying links via closed braids, IV: Composite links and split links,” Invent. Math. 102 : 1 (December 1990), pp. 115–139. An erratum for this article was published in Invent. Math. 160:2 (2005); Parts I, III and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part II was published in Topology Appl. 40:1 (1991); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1069243 Zbl 0711.57006 article

The main result concerns changing an arbitrary closed braid representative of a split or composite link to one which is obviously recognizable as being split or composite. Exchange moves are introduced; they change the conjugacy class of a closed braid without changing its link type or its braid index. A closed braid representative of a composite (respectively split) link is composite (split) if there is a 2-sphere which realizes the connected sum decomposition (splitting) and meets the braid axis in 2 points. It is proved that exchange moves are the only obstruction to representing composite or split links by composite or split closed braids. A special version of these theorems holds for 3 and 4 braids, answering a question of H. Morton. As an immediate Corollary, it follows that braid index is additive (resp. additive minus 1) under disjoint union (resp. connected sum).

@article {key1069243m,
	AUTHOR = {Birman, Joan S. and Menasco, William
		 W.},
	TITLE = {Studying links via closed braids, {IV}:
		 {C}omposite links and split links},
	JOURNAL = {Invent. Math.},
	FJOURNAL = {Inventiones Mathematicae},
	VOLUME = {102},
	NUMBER = {1},
	MONTH = {December},
	YEAR = {1990},
	PAGES = {115--139},
	DOI = {10.1007/BF01233423},
	NOTE = {An erratum for this article was published
		 in \textit{Invent. Math.} \textbf{160}:2
		 (2005); Parts I, III and VI were published
		 in \textit{Pac. J. Math.} \textbf{154}:1
		 (1992), \textbf{161}:1 (1993) and \textbf{156}:2
		 (1992); Part II was published in \textit{Topology
		 Appl.} \textbf{40}:1 (1991); Part V
		 was published in \textit{Trans. Am.
		 Math. Soc.} \textbf{329}:2 (1992). MR:1069243.
		 Zbl:0711.57006.},
	ISSN = {0020-9910},
	CODEN = {INVMBH},
}

[57] J. S. Birman: On the work of Vaughan F. R. Jones, 1990. from 60m VHS videocassette “Addresses on the works of the 1990 Fields medalists and the Rolf Nevanlinna Prize winner: ICM-90”. Video of the address published in Proceedings of the ICM, I (1991) and Fields Medallists’ lectures (1997). MR 1175179 misc

[58] J. S. Birman: “Recent developments in braid and link theory,” Math. Intell. 13 : 1 (1991), pp. 52–60. MR 1084535 Zbl 0713.57002 article

[59] J. S. Birman and W. W. Menasco: “Studying links via closed braids, II: On a theorem of Bennequin,” Topology Appl. 40 : 1 (June 1991), pp. 71–82. Parts I, III, and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1114092 Zbl 0722.57001 article

[60] J. S. Birman, D. Haimo, S. Landau, B. Srinivasan, V. S. Pless, and J. E. Taylor: “In her own words: Six mathematicians comment on their lives and careers,” Notices Am. Math. Soc. 38 : 7 (1991), pp. 702–706. MR 1125373 Zbl 0741.01032 article

Deborah Tepper Haimo	Related
Susan Eva Landau	Related
Vera Stephen Pless	Related
Bhama Srinivasan	Related
Jean Ellen Taylor	Related

[61] J. S. Birman: “The work of Vaughan F. R. Jones,” pp. 9–18 in Proceedings of the International Congress of Mathematicians (Kyoto, 21–29 August 1990), vol. 1. Edited by I. Satake. Mathematical Society of Japan (Tokyo), 1991. Reprinted in Fields Medallists’ lectures (1997). MR 1159199 Zbl 0743.01023 incollection

Vaughan F. R. Jones	Related	Volume
Ichirō Satake	Related

[62] J. S. Birman: “A progress report on the study of links via closed braids,” pp. 869–895 in Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische Gesellschaf in Hamburg, Teil 4 [Mathematical sciences yesterday and today: 300 years of the Hamburg Mathematical Society, part 4], published as Mitt. Math. Ges. Hamburg 12 : 4. Issue edited by R. Carlsson. 1991. MR 1175012 Zbl 0890.57010 incollection

[63] J. S. Birman: “Book review: F. M. Goodman, P. de la Harpe and V. F. R. Jones, ‘Coxeter graphs and towers of algebras’,” Bull. Am. Math. Soc. (N.S.) 25 : 1 (1991), pp. 195–199. MR 1567945 article

Frederick Michael Goodman	Related
Pierre de la Harpe	Related
Vaughan F. R. Jones	Related	Volume

[64] J. S. Birman and W. W. Menasco: “Studying links via closed braids, V: The unlink,” Trans. Am. Math. Soc. 329 : 2 (February 1992), pp. 585–606. Parts I, III and VI were published in Pac. J. Math. 154:1 (1992), 161:1 (1993) and 156:2 (1992); Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990). MR 1030509 Zbl 0758.57005 article

The main result is a version of Markov’s Theorem which does not involve stabilization, in the special case of the \( r \)-component link. As a corollary, it is proved that the stabilization index of a closed braid representative of the unlink is at most 1. To state the result, we need the concept of an “exchange move”, which modifies a closed braid without changing its link type or its braid index. For generic closed braids exchange moves change conjugacy class. Theorem 1 shows that exchange moves are the only obstruction to reducing a closed \( n \)-braid representative of the \( r \)-component unlink to the standard closed \( r \)-braid representative, through a sequence of braids of nonincreasing braid index.

[65] J. S. Birman and W. W. Menasco: “Studying links via closed braids, I: A finiteness theorem,” Pacific J. Math. 154 : 1 (May 1992), pp. 17–36. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1154731 Zbl 0724.57001 article

This paper is the first in a series which study the closed braid representatives of an oriented link type \( \mathscr{L} \) in oriented 3-space. A combinatorial symbol is introduced which determines an oriented spanning surface \( F \) for a representative \( L \) of \( \mathscr{L} \). The surface \( F \) is in a special position in 3-space relative to the braid axis \( A \) and the fibers in a fibration of the complement of \( A \). The symbol simultaneously describes \( F \) as an embedded surface and \( L \) as a closed braid. Therefore it is both geometrically and algebraically meaningful. Using it, a complexity function is introduced. It is proved that \( \mathscr{L} \) is described by at most finitely many combinatorial symols, and thus by finitely many conjugacy classes in each braid group \( B_n \) when the complexity is minimal.

[66] J. S. Birman: “On Shafarevich’s essay ‘Russophobia’,” Math. Intell. 14 : 2 (1992), pp. 3–4. Letter to the editor in response to to S. Zdravkovska’s article “Listening to Igor Rostislavovich Shafarevich” in Math. Intell. 11:2 (1989), with a reply by Sheldon Axler. MR 1160698 article

Smilka Zdravkovska	Related
Sheldon Jay Axler	Related
Igor Rostislavovich Shafarevich	Related

[67] J. S. Birman and W. W. Menasco: “A calculus on links in the 3-sphere,” pp. 625–631 in Knots 90 (Osaka, 15–19 August 1990). Edited by A. Kawauchi. de Gruyter (Berlin), 1992. MR 1177450 Zbl 0764.57005 incollection

Akio Kawauchi	Related
William Wyatt Menasco	Related

[68] J. S. Birman and W. W. Menasco: “Studying links via closed braids, VI: A nonfiniteness theorem,” Pacific J. Math. 156 : 2 (December 1992), pp. 265–285. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1186805 Zbl 0739.57002 article

Exchange moves were introduced in an earlier paper by the same authors. They take one closed \( n \)-braid representative of a link to another, and can lead to examples where there are infinitely many conjugacy classes of \( n \)-braids respresenting a single links type.

If a link type has infinitely many conjugacy classes of closed \( n \)-braid representatives, then \( n\geq 4 \) and the infinitely many classes divide into finitely many equivalence classes under the equivalence relation generated by exchange moves.

This theorem is the last of the preliminary steps in the authors’ program for the development of a calculus on links in \( S^3 \).

Choose integers \( n \), \( g\geq 1 \). Then there are at most finitely many link types with braid index \( n \) and genus \( g \).

[69] J. S. Birman: A new look at knot polynomials, 1992. 60 minute VHS videocassette. MR 1201149 Zbl 0821.57001 misc

[70] J. S. Birman: “New points of view in knot theory,” Bull. Am. Math. Soc. (N.S.) 28 : 2 (1993), pp. 253–287. MR 1191478 Zbl 0785.57001 article

In this article we shall give an account of certain developments in knot theory which followed upon the discovery of the Jones polynomial in 1984 [Jones 1985]. The focus of our account will be recent glimmerings of understanding of the topological meaning of the new invariants. A second theme will be the central role that braid theory has played in the subject. A third will be the unifying principles provided by representations of simple Lie algebras and their universal enveloping algebras. These choices in emphasis are our own. They represent, at best, particular aspects of the far-reaching ramifications that followed the discovery of the Jones polynomial.

[71] J. S. Birman and X.-S. Lin: “Knot polynomials and Vassiliev’s invariants,” Invent. Math. 111 : 2 (1993), pp. 225–270. MR 1198809 Zbl 0812.57011 article

[72] J. S. Birman and W. W. Menasco: “Studying links via closed braids, III: Classifying links which are closed 3-braids,” Pacific J. Math. 161 : 1 (November 1993), pp. 25–113. Part II was published in Topology Appl. 40:1 (1991); Part IV was published in Invent. Math. 102:1 (1990); Part V was published in Trans. Am. Math. Soc. 329:2 (1992). MR 1237139 Zbl 0813.57010 article

A complete solution is given to the classification problem for oriented links which are closed three-braids. The Classification Theorem asserts that, up to a finite list of exceptional cases, links which can be represented by closed 3-braids are represented by a unique conjugacy class in the group of 3-braids. The exceptional cases are the expected ones (links of braid index 1 and 2) and an unexpected infinite family of invertible links, each member of which has two 3-braid axes. The two axes correspond to diagrams which are related by “braid-preserving flypes”.

An algorithm is given which begins with an arbitrary closed 3-braid (or alternatively any link diagram with 3 Seifert circles), and converts it into a normal form which characterizes its oriented link type in oriented 3-space. One can decide from the normal form whether the link is prime or composite, split or irreducible, amphicheiral and or invertible. One can decide if the braid index is 3, 2 or 1. Using related results of P. J. Xu, one may determine the genus and construct a surface of maximum Euler characteristic with boundary the given link.

It is proved that the stabilization index of a link which is represented by a closed 3-braid is \( \leq 1 \), i.e. any two 3-braid representatives of the same link type become conjugate after a single stabilization to \( B_4 \).

[73] J. S. Birman and W. W. Menasco: “Special positions for essential tori in link complements,” Topology 33 : 3 (July 1994), pp. 525–556. An erratum for this article was published in Topology 37:1 (1998). MR 1286930 Zbl 0833.57004 article

The decomposition of links into non-split components, by cutting along essential 2-spheres, is a fundamental step in any attempt to understand the link problem. In an earlier paper [1990] the authors studied that problem from the point of view of braid theory. A somewhat more subtle decomposition of a link complement involves splitting along essential tori. By a fundamental theorem which is due to Alexander (see p. 107 of [Rolfsen 1976]) every embedded torus \( T \) in \( S^3 \) is the boundary of a solid torus \( V \) on at least one side. The solid torus \( V \) may, however, be knotted, and this makes the study of embedded tori much more difficult than embedded 2-spheres, since the latter cannot be knotted. The first major attempt to understand embedded tori in link complements was a groundbreaking paper by Schubert [1953]. The seminal role which is played in the topology and geometry of link complements by embedded tori was later underscored in the important work of Jaco and Shalen [1979], Johansson [1975] and Thurston [1982], who showed that if \( M^3 \) is a 3-manifold, then there is a finite collection \( \Omega \) of essential, non-peripheral tori \( T_1,\dots \),\( T_q \), in \( M^3 \) such that each component of \( M^3 \) split open along the tori in \( \Omega \) is either Seifert-fibered or hyperbolic. Our goal in this paper is to apply the techniques of [Birman and Menasco 1990] to the study of essential tori in link complements.

[74] The mathematical legacy of Wilhelm Magnus: Groups, geometry and special functions (Brooklyn, NY, 1–3 May 1992). Edited by W. Abikoff, J. S. Birman, and K. Kuiken. American Mathematical Society (Providence, RI), 1994. MR 1292894 Zbl 0801.00023 book

William Abikoff	Related
Wilhelm Magnus	Related
Kathryn F. Kuiken	Related

[75] J. S. Birman, D. D. Long, and J. A. Moody: “Finite-dimensional representations of Artin’s braid group,” pp. 123–132 in The mathematical legacy of Wilhelm Magnus: Groups, geometry and special functions (Brooklyn, NY, 1–3 May 1992). Edited by W. Abikoff, J. S. Birman, and K. Kuiken. Contemporary Mathematics 169. American Mathematical Society (Providence, RI), 1994. MR 1292900 Zbl 0847.20035 incollection

William Abikoff	Related
Kathryn F. Kuiken	Related
Wilhelm Magnus	Related
Darren David Long	Related
John Atwell Moody	Related

[76] J. S. Birman and B. Wajnryb: “Errata: ‘Presentations of the mapping class group’,” Israel J. Math. 88 : 1–3 (October 1994), pp. 425–427. Errata for two articles, the first being “3-fold branched coverings and the mapping class group of a surface” published in Geometry and topology (1985), the other by Wajnryb alone. MR 1303506 Zbl 0811.57019 article

[77] J. S. Birman: “Knots, links and braids,” Arkhimedes 46 : 4 (1994), pp. 311–326. MR 1336648 Zbl 1231.57004 article

[78] J. S. Birman: “On the combinatorics of Vassiliev invariants,” pp. 1–19 in Braid group, knot theory and statistical mechanics, II. Edited by C. N. Yang and M. L. Ge. Advanced Series in Mathematical Physics 17. World Scientific (River Edge, NJ), 1994. MR 1338597 Zbl 0938.57004 incollection

M. L. Ge	Related
Chen-Ning Franklin Yang	Related

[79] J. Birman: “Studying links via closed braids,” pp. 1–67 in Lecture notes of the ninth KAIST mathematics workshop (Taejon, Korea, 1–13 August 1994), vol. 1. Edited by S. H. Bae, G. T. Jin, and K. H. Ko. Korea Advanced Institute of Science and Technology (Taejon), 1994. Zbl 0835.57002 incollection

Ki Hyoung Ko	Related
Sunghan Bae	Related
Gyo Taek Jin	Related

[80] J. S. Birman: “Book review: Charles Livingston, ‘Knot theory’,” Am. Math. Monthly 102 : 8 (October 1995), pp. 755–757. MR 1542746 article

[81] “The braid groups, knots and algebraic geometry,” pp. 1–4 in Special issue on braid groups and related topics (Jerusalem, May 1995), published as Topology Appl. 78 : 1–2. Issue edited by J. S. Birman and M. Teicher. Elsevier (Amsterdam), July 1997. Editorial. incollection

[82] Special issue on braid groups and related topics (Jerusalem, May 1995), published as Topology Appl. 78 : 1–2. Issue edited by J. S. Birman and M. Teicher. Elsevier (Amsterdam), July 1997. Papers from the special session of an AMS-IMU meeting. MR 1465021 Zbl 0872.00016 book

[83] J. S. Birman: “The work of Vaughan F. R. Jones,” pp. 435–445 in Fields Medallists’ lectures. Edited by M. F. Atiyah and D. Iagolnitzer. World Scientific Series in 20th Century Mathematics 5. World Scientific (River Edge, NJ), 1997. Reprinted from Proceedings of the International Congress of Mathematicians (1991). MR 1622915 incollection

M. F. Atiyah	Related	Volume
Daniel Iagolnitzer	Related
Vaughan F. R. Jones	Related	Volume

[84] J. S. Birman and W. W. Menasco: “Erratum: ‘Special positions for essential tori in link complements’,” Topology 37 : 1 (1998), pp. 225. Erratum for an article published in Topology 33:3 (1994). MR 1480888 article

[85] J. S. Birman and R. Trapp: “Braided chord diagrams,” J. Knot Theor. Ramif. 7 : 1 (1998), pp. 1–22. MR 1611967 Zbl 0896.57003 article

[86] J. S. Birman and E. Finkelstein: “Studying surfaces via closed braids,” J. Knot Theor. Ramif. 7 : 3 (1998), pp. 267–334. MR 1625362 Zbl 0907.57006 article

[87] J. Birman, K. H. Ko, and S. J. Lee: “A new approach to the word and conjugacy problems in the braid groups,” Adv. Math. 139 : 2 (November 1998), pp. 322–353. MR 1654165 Zbl 0937.20016 article

Sang Jin Lee	Related
Ki Hyoung Ko	Related

[88] J. S. Birman and M. D. Hirsch: “A new algorithm for recognizing the unknot,” Geom. Topol. 2 (1998), pp. 175–220. MR 1658024 Zbl 0955.57005 article

The topological underpinnings are presented for a new algorithm which answers the question: “Is a given knot the unknot?” The algorithm uses the braid foliation technology of Bennequin and of Birman and Menasco. The approach is to consider the knot as a closed braid, and to use the fact that a knot is unknotted if and only if it is the boundary of a disc with a combinatorial foliation. The main problems which are solved in this paper are: how to systematically enumerate combinatorial braid foliations of a disc; how to verify whether a combinatorial foliation can be realized by an embedded disc; how to find a word in the the braid group whose conjugacy class represents the boundary of the embedded disc; how to check whether the given knot is isotopic to one of the enumerated examples; and finally, how to know when we can stop checking and be sure that our example is not the unknot.

[89] J. S. Birman and N. C. Wrinkle: “Holonomic and Legendrian parametrizations of knots,” J. Knot Theor. Ramif. 9 : 3 (May 2000), pp. 293–309. MR 1753797 Zbl 1001.57015 article

Holonomic parametrizations of knots were introduced in 1997 by Vassiliev, who proved that every knot type can be given a holonomic parametrization. Our main result is that any two holonomic knots which represent the same knot type are isotopic in the space of holonomic knots. A second result emerges through the techniques used to prove the main result: strong and unexpected connections between the topology of knots and the algebraic solution to the conjugacy problem in the braid groups, via the work of Garside. We also discuss related parametrizations of Legendrian knots, and uncover connections between the concepts of holonomic and Legendrian parametrizations of knots.

[90] J. S. Birman and N. C. Wrinkle: “On transversally simple knots,” J. Diff. Geom. 55 : 2 (2000), pp. 325–354. MR 1847313 Zbl 1026.57005 article

This paper studies knots that are transversal to the standard contact structure in \( \mathbb{R}^3 \) bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type \( \mathcal{TK} \) is transversally simple if it is determined by its topological knot type \( \mathcal{K} \) and its Bennequin number. The main theorem asserts that any \( \mathcal{TK} \) whose associated \( \mathcal{K} \) satisfies a condition that we call exchange reducibility is transversally simple.

As a first application, we prove that the unlink is transversally simple, extending the main theorem in [Eliashberg 1991]. As a second application we use a new theorem of Menasco [2001] to extend a result of Etnyre [1999] to prove that all iterated torus knots are transversally simple. We also give a formula for their maximum Bennequin number. We show that the concept of exchange reducibility is the simplest of the constraints that one can place on \( \mathcal{K} \) in order to prove that any associated \( \mathcal{TK} \) is transversally simple. We also give examples of pairs of transversal knots that we conjecture are not transversally simple.

[91] J. S. Birman: “Scientific publishing: A mathematician’s viewpoint,” Notices Am. Math. Soc. 47 : 7 (August 2000), pp. 770–774. Zbl 1003.00502 article

[92] J. S. Birman, K. H. Ko, and S. J. Lee: “The infimum, supremum, and geodesic length of a braid conjugacy class,” Adv. Math. 164 : 1 (December 2001), pp. 41–56. MR 1870512 Zbl 1063.20039 article

Algorithmic solutions to the conjugacy problem in the braid groups \( B_n \), \( n = 2,\,3,\,4,\dots \) were given in earlier work. This xlinconcerns the computation of two integer class invariants, known as “inf” and “sup.” A key issue in both algorithms is the number \( m \) of times one must “cycle” (resp. “decycle”) in order to either increase inf (resp. decrease sup) or to be sure that it is already maximal (resp. minimal) for the class. Our main result is to prove that \( m \) is bounded above by \[ \frac{n^2-n}{2}-1 \] in the situation stated by E. A. Elrifai and H. R. Morton (1994, Quart. J. Math. Oxford 45, 479–497) and by \( n-2 \) in the situation stated by authors (1998, Adv. Math. 139, 322–353). It follows immediately that the computation of inf and sup is polynomial in both word length and braid index, in both algorithms. The integers inf and sup determine (but are not determined by) the shortest geodesic length for elements in a conjugacy class, and so we also obtain a polynomial-time algorithm for computing this length.

Sang Jin Lee	Related
Ki Hyoung Ko	Related

[93] Knots, braids, and mapping class groups: Papers dedicated to Joan S. Birman (New York, 14–15 March 1998). Edited by J. Gilman, W. W. Menasco, and X.-S. Lin. AMS/IP Studies in Advanced Mathematics 24. American Mathematical Society and International Press (Providence, RI and Somerville, MA), 2001. Proceedings of a conference in low-dimensional topology in honor of Joan S. Birman’s 70th birthday. MR 1873102 Zbl 0980.00048 book

William Wyatt Menasco	Related
Xiao-Song Lin	Related
Jane Piore Gilman	Related

[94] “Joan S. Birman: Publications and Ph.D. theses supervised,” pp. xiii–xvi in Knots, braids, and mapping class groups: Papers dedicated to Joan S. Birman (New York, 14–15 March 1998). Edited by J. Gilman, W. W. Menasco, and X.-S. Lin. AMS/IP Studies in Advanced Mathematics 24. American Mathematical Society (Providence, RI), 2001. MR 1873103 Zbl 0988.01500 incollection

William Wyatt Menasco	Related
Xiao-Song Lin	Related
Jane Piore Gilman	Related

[95] J. S. Birman and W. W. Menasco: “On Markov’s theorem,” pp. 295–310 in Knots 2000 Korea (Volume 1) (Yongpyong, Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 3. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), 2002. MR 1905686 Zbl 1059.57002 incollection

Let \( \chi \) be an oriented link type in the oriented 3-sphere \( S^3 \) or \[ \mathbb{R}^3 = S^3 - \{\infty\} .\] A representative \( X \in \chi \) is said to be a closed braid if there is an unknotted curve \[ \mathbf{A} \subset S^3 - X \] (the axis) and a choice of fibration \( \mathscr{H} \) of the open solid torus \( S^3 - \mathbf{A} \) by meridian discs \[ \{H_{\theta}: \theta\in [0, 2\pi]\} ,\] such that whenever \( X \) meets a fiber \( H_{\theta} \) the intersection is transverse.

Closed braid representations of \( \chi \) are not unique, and Markov’s well-known theorem asserts that any two are related by a finite sequence of elementary moves. The main result in this paper is to give a new proof of Markov’s theorem. We hope that our new proof will be of interest because it gives new insight into the geometry.

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
William Wyatt Menasco	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related

[96] J. S. Birman, M. Rampichini, P. Boldi, and S. Vigna: “Towards an implementation of the B–H algorithm for recognizing the unknot,” pp. 601–645 in Knots 2000 Korea (Volume 2) (Yongpyong, Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 4. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Hackensack, NJ), 2002. MR 1915499 Zbl 1007.57004 incollection

In the manuscript [Birman and Hirsch 1998] the first author and Michael Hirsch presented a then-new algorithm for recognizing the unknot. The first part of the algorithm required the systematic enumeration of all discs which support a ‘braid foliation’ and are embeddable in 3-space. The boundaries of these ‘foliated embeddable discs’ (FEDs) are the collection of all closed braid representatives of the unknot, up to conjugacy, and the second part of the algorithm produces a word in the generators of the braid group which represents the boundary of the previously listed FEDs. The third part tests whether a given closed braid is conjugate to the boundary of a FED on the list.

In this paper we describe implementations of the first and second parts of the algorithm. We also give some of the data which we obtained. The data suggests that FEDs have unexplored and interesting structure. Open question are interspersed throughout the manuscript.

The third part of the algorithm was studied in [Birman, Ko and Lee 1998] and [Birman, Ko and Lee 2001], and implemented by S. J. Lee [preprint]. At this writing his algorithm is polynomial for \( n\leq 4 \) and exponential for \( n\geq 5 \).

Ki Hyoung Ko	Related
Paolo Boldi	Related
Sebastiano Vigna	Related
Yukio Matsumoto	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Marta Rampichini	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related

[97] Knots 2000 Korea (Volume 1) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 3. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), 2002. Zbl 0995.00510 book

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related

[98] Knots 2000 Korea (Volume 3) (Yongpyong, South Korea, 31 July–5 August 2000), published as J. Knot Theor. Ramif. 11 : 6. Issue edited by J. S. Birman, C. M. Gordon, G. T. Jin, L. H. Kauffman, A. Kawauchi, K. H. Ko, J. P. Levine, and Y. Matsumoto. World Scientific (Singapore), September 2002. Zbl 1018.00508 book

Ki Hyoung Ko	Related
Yukio Matsumoto	Related
Cameron McAllan Gordon	Related	Volume
Akio Kawauchi	Related
Louis H. Kauffman	Related
Jerome Paul Levine	Related
Gyo Taek Jin	Related

[99] J. S. Birman and D. R. J. Chillingworth: “Erratum: ‘On the homeotopy group of a non-orientable surface’,” Math. Proc. Camb. Philos. Soc. 136 : 2 (2004), pp. 441. Erratum to an article published in Proc. Cambridge Philos. Soc. 71:3 (1972). MR 2040584 article

[100] J. S. Birman and J. A. Moody: “Obstructions to trivializing a knot,” Israel J. Math. 142 (2004), pp. 125–162. MR 2085713 Zbl 1074.57002 article

[101] J. S. Birman and W. W. Menasco: “Erratum: ‘Studying links via closed braids, IV: Composite links and split links’,” Invent. Math. 160 : 2 (2005), pp. 447–452. Erratum to an article published in Invent. Math. 102:1 (1990). MR 2138073 article

[102] J. S. Birman and T. E. Brendle: “Braids: A survey,” Chapter 2, pp. 19–103 in Handbook of knot theory. Edited by W. Menasco and M. Thistlethwaite. Elsevier B. V. (Amsterdam), 2005. MR 2179260 Zbl 1094.57006 incollection

This article is about Artin’s braid group \( \mathbf{B}_n \) and its role in know theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid group generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literature is extensive, and for reasons of space our coverage necessarily omits many very interesting developments. Open problems are noted and so-labeled, as we encounter them. A guide to computer software is given together with an extensive bibliography.

Morwen Bernard Thistlethwaite	Related
William Wyatt Menasco	Related
Tara Elise Brendle	Related

[103] J. S. Birman and W. W. Menasco: “Stabilization in the braid groups, I: MTWS,” Geom. Topol. 10 : 1 (2006), pp. 413–540. MR 2224463 Zbl 1128.57003 article

Choose any oriented link type \( \mathscr{X} \) and closed braid representatives \( X_+ \), \( X_- \) of \( \mathscr{X} \), where \( X_- \) has minimal braid index among all closed braid representatives of \( \mathscr{X} \). The main result of this paper is a ‘Markov theorem without stabilization’. It asserts that there is a complexity function and a finite set of ‘templates’ such that (possibly after initial complexity-reducing modifications in the choice of \( X_+ \) and \( X_- \) which replace them with closed braids \( X^{\prime}_+ \), \( X^{\prime}_- \)) there is a sequence of closed braid representatives \[ X^{\prime}_+ = X^1 \to X^2 \to \cdots \to X^r = X^{\prime}_- \] such that each passage \( X^i\to X^{i+1} \) is strictly complexity reducing and non-increasing on braid index. The templates which define the passages \( X^i \to X^{i+1} \) include 3 familiar ones, the destabilization, exchange move and flype templates, and in addition, for each braid index \( m\geq 4 \) a finite set \( \mathscr{T}(m) \) of new ones. The number of templates in \( \mathscr{T}(m) \) is a non-decreasing function of \( m \). We give examples of members of \( \mathscr{T}(m) \), \( m\geq 4 \), but not a complete listing. There are applications to contact geometry, which will be given in a separate paper.

[104] J. S. Birman and W. W. Menasco: “Stabilization in the braid groups, II: Transversal simplicity of knots,” Geom. Topol. 10 : 3 (2006), pp. 1425–1452. MR 2255503 Zbl 1130.57005 ArXiv math.GT/0310280 article

[105] J. S. Birman: “The topology of 3-manifolds, Heegaard distance and the mapping class group of a 2-manifold,” pp. 133–149 in Problems on mapping class groups and related topics. Edited by B. Farb. Proceedings of Symposia Pure Mathematics 74. American Mathematical Society (Providence, RI), 2006. MR 2264538 Zbl 1304.57033 incollection

We have had a long-standing interest in the way that structure in the mapping class group of a surface reflects corresponding structure in the topology of 3-manifolds, and conversely. We find this area intriguing because the mapping class group (unlike the collection of closed orientable 3-manifolds) is a group which has a rich collection of subgroups and quotients, and they might suggest new ways to approach 3-manifolds. (For example, it is infinite, non-abelian and residually finite [14]). In the other direction, 3-manifolds have deep geometric structure, for example the struture that is associated to intersections between 2-dimensional submanifolds, and that sort of inherently geometric structure that might bring new tools to bear on open questions regarding the mapping class group. That dual theme is the focus of this article.

[106] A. Jackson and L. Traynor: “Interview with Joan Birman,” Notices Am. Math. Soc. 54 : 1 (January 2007), pp. 20–29. Interview conducted in May 2006. MR 2275922 Zbl 1128.01303 article

Allyn Jackson	Related
Lisa Traynor	Related

[107] J. S. Birman, V. Gebhardt, and J. González-Meneses: “Conjugacy in Garside groups, I: Cyclings, powers and rigidity,” Groups Geom. Dyn. 1 : 3 (2007), pp. 221–279. Part III was published in J. Algebra 316:2 (2007). MR 2314045 Zbl 1160.20026 article

In this paper a relation between iterated cyclings and iterated powers of elements in a Garside group is shown. This yields a characterization of elements in a Garside group having a rigid power, where ‘rigid’ means that the left normal form changes only in the obvious way under cycling and decycling. It is also shown that, given \( X \) in a Garside group, if some power \( X^m \) is conjugate to a rigid element, then \( m \) can be bounded above by \( \|\Delta\|^3 \). In the particular case of braid groups \( \{B_n; n \in \mathbb{N}\} \), this implies that a pseudo-Anosov braid has a small power whose ultra summit set consists of rigid elements. This solves one of the problems in the way of a polynomial solution to the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in braid groups. In addition to proving the rigidity theorem, it will be shown how this paper fits into the authors’ program for finding a polynomial algorithm to the CDP/CSP, and what remains to be done.

Juan González-Meneses	Related
Volker Gebhardt	Related

[108] J. S. Birman, V. Gebhardt, and J. González-Meneses: “Conjugacy in Garside groups, III: Periodic braids,” J. Algebra 316 : 2 (October 2007), pp. 746–776. Parts I and II were published in Groups Geom. Dyn. 1:3 (2007) and Groups Geom. Dyn. 2:1 (2008). MR 2358613 Zbl 1165.20031 article

An element in Artin’s braid group \( B_n \) is said to be periodic if some power of it lies in the center of \( B_n \). In this paper we prove that all previously known algorithms for solving the conjugacy search problem in \( B_n \) are exponential in the braid index \( n \) for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group \( B_n \) and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms.

This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in \( B_n \), which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin–Tits groups of spherical type.

Juan González-Meneses	Related
Volker Gebhardt	Related

[109] J. S. Birman, V. Gebhardt, and J. González-Meneses: “Conjugacy in Garside groups, II: Structure of the ultra summit set,” Groups Geom. Dyn. 2 : 1 (2008), pp. 13–61. Part III was published in J. Algebra 316:2 (2007). MR 2367207 Zbl 1163.20023 article

This paper is the second in a series in which the authors study the conjugacy decision problem (CDP) and the conjugacy search problem (CSP) in Garside groups.

The ultra summit set \( \mathrm{USS}(X) \) of an element \( X \) in a Garside group \( G \) is a finite set of elements in \( G \), which is a complete invariant of the conjugacy class of \( X \) in \( G \). A fundamental question, if one wishes to find bounds on the size of \( \mathrm{USS}(X) \), is to understand its structure. In this paper we introduce two new operations on elements \( Y \in \mathrm{USS}(X) \), called ‘partial cycling’ and ‘partial twisted decycling’, and prove that if \( Y \), \( Z \in \mathrm{USS}(X) \), then \( Y \) and \( Z \) are related by sequences of partial cyclings and partial twisted decyclings. These operations are a concrete way to understand the minimal simple elements whose existence follows from the convexity property of ultra summit sets.

Using partial cycling and partial twisted decycling, we investigate the structure of a directed graph \( \Gamma_X \) determined by \( \mathrm{USS}(X) \), and show that \( \Gamma_X \) can be decomposed into ‘black’ and ‘grey’ subgraphs. There are applications relating to the authors’ program for finding a polynomial solution to the CDP/CSP in the case of braids, which is outlined in the first paper of this series. A different application is to give a new algorithm for solving the CDP/CSP in Garside groups which is faster than all other known algorithms, even though its theoretical complexity is the same as that of the established algorithm using ultra summit sets. There are also applications to the theory of reductive groups.

Juan González-Meneses	Related
Volker Gebhardt	Related

[110] J. S. Birman and G. Tian: “Preface,” Commun. Contemp. Math. 10 : supplement 1 (November 2008), pp. v–vii. In memory of Xiao-Song Lin, to whom this supplementary issue is dedicated. MR 2468363 article

Gang Tian	Related
Xiao-Song Lin	Related

[111] J. S. Birman and W. W. Menasco: “A note on closed 3-braids,” Commun. Contemp. Math. 10 : supplement 1 (November 2008), pp. 1033–1047. MR 2468377 Zbl 1158.57006 article

[112] J. S. Birman, T. E. Brendle, and N. Broaddus: “Calculating the image of the second Johnson–Morita representation,” pp. 119–134 in Groups of diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday (Tokyo, 11–15 September 2006). Edited by R. C. Penner. Advanced Studies in Pure Mathematics 52. Mathematical Society of Japan (Tokyo), 2008. MR 2509709 Zbl 1183.57016 ArXiv 0708.3861 incollection

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus \( g \) with one boundary component to \( \bigwedge^3H \), the third exterior product of the homology of the surface. Morita then extended Johnson’s homomorphism to a homomorphism from the entire mapping class group to \[ \tfrac{1}{2} \bigwedge^3 H \rtimes \operatorname{Sp}(H) .\] This Johnson–Morita homomorphism is not surjective, but its image is finite index in \( \frac{1}{2} \) \( \bigwedge^3 H \rtimes \) \( \operatorname{Sp}(H) \) [Morita 1993]. Here we give a description of the exact image of Morita’s homomorphism. Further, we compute the image of the handlebody subgroup of the mapping class group under the same map.

Tara Elise Brendle	Related
Nathan Darrell Broaddus	Related
Robert Clark Penner	Related
Shigeyuki Morita	Related

[113] J. S. Birman, D. Johnson, and A. Putman: “Symplectic Heegaard splittings and linked abelian groups,” pp. 135–220 in Groups of diffeomorphisms: In honor of Shigeyuki Morita on the occasion of his 60th birthday (Tokyo, 11–15 September 2006). Edited by R. C. Penner. Advanced Studies in Pure Mathematics 52. Mathematical Society of Japan (Tokyo), 2008. MR 2509710 Zbl 1170.57018 ArXiv 0712.2104 incollection

Robert Clark Penner	Related
Dennis Lee Johnson	Related	Volume
Shigeyuki Morita	Related
Thomas Andrew Putman	Related

[114] W.-T. Lam: The charm of topology: Dr. Joan Birman: Mathematics is very beautiful, 2009. Grand Prize winner of the 2009 Association of Women Mathematicians Essay Contest. misc

[115] J. Birman and I. Kofman: “A new twist on Lorenz links,” J. Topol. 2 : 2 (2009), pp. 227–248. MR 2529294 Zbl 1233.57001 article

Twisted torus links are given by twisting a subset of strands on a closed braid representative of a torus link. T-links are a natural generalization given by repeated positive twisting. We establish a one-to-one correspondence between positive braid representatives of Lorenz links and T-links, so Lorenz links and T-links coincide. Using this correspondence, we identify over half of the simplest hyperbolic knots as Lorenz knots. We show that both hyperbolic volume and the Mahler measure of Jones polynomials are bounded for infinite collections of hyperbolic Lorenz links. The correspondence provides unexpected symmetries for both Lorenz links and T-links, and establishes many new results for T-links, including new braid index formulas.

[116] J. S. Birman: “Book reviews: Christian Kassel and Vladimir Turaev, ‘Braid groups’ and Patrick Dehornoy, ‘Ordering braids’,” Bull. Am. Math. Soc. (N.S.) 48 : 1 (2011), pp. 137–146. MR 2731658 Zbl 1292.00013 article

Christian Kassel	Related
Vladimir G. Turaev	Related
Patrick Dehornoy	Related

[117] J. Birman, P. Brinkmann, and K. Kawamuro: “Polynomial invariants of pseudo-Anosov maps,” J. Topol. Anal. 4 : 1 (2012), pp. 13–47. MR 2914872 Zbl 1268.57002 article

Peter Brinkmann	Related
Keiko Kawamuro	Related

[118] J. S. Birman: “The mathematics of Lorenz knots,” pp. 127–148 in Topology and dynamics of chaos: In celebration of Robert Gilmore’s 70th birthday. Edited by C. Letellier and R. Gilmore. Nonlinear Science Series A 84. World Scientific (Hackensack, NJ), 2013. MR 3289735 Zbl 1270.37001 incollection

Robert Gilmore	Related
Christophe Letellier	Related

[119] J. Birman, N. Broaddus, and W. Menasco: “Finite rigid sets and homologically nontrivial spheres in the curve complex of a surface,” J. Topol. Anal. 7 : 1 (2015), pp. 47–71. MR 3284389 Zbl 1308.57009 article

Aramayona and Leininger have provided a “finite rigid subset” \( \mathfrak{X}(\Sigma) \) of the curve complex \( \mathscr{C}(\Sigma) \) of a surface \( \Sigma = \Sigma_g^n \), characterized by the fact that any simplicial injection \[ \mathfrak{X}(\Sigma) \to \mathscr{C}(\Sigma) \] is induced by a unique element of the mapping class group \( \operatorname{Mod}(\Sigma) \). In this paper we prove that, in the case of the sphere with \( n \geq 5 \) marked points, the reduced homology class of the finite rigid set of Aramayona and Leininger is a \( \operatorname{Mod}(\Sigma) \)-module generator for the reduced homology of the curve complex \( \mathscr{C}(\Sigma) \), answering in the affirmative a question posed in [Aramayona and Leininger 2013]. For the surface \( \Sigma = \Sigma_g^n \) with \( g \geq 3 \) and \( n \in \{0,1\} \) we find that the finite rigid set \( \mathfrak{X}(\Sigma) \) of Aramayona and Leininger contains a proper subcomplex \( X(\Sigma) \) whose reduced homology class is a \( \operatorname{Mod}(\Sigma) \)-module generator for the reduced homology of \( \mathscr{C}{\Sigma} \) but which is not itself rigid.

Nathan Darrell Broaddus	Related
William Wyatt Menasco	Related

[120] J. S. Birman and W. W. Menasco: “The curve complex has dead ends,” Geom. Dedicata 177 (August 2015), pp. 71–74. MR 3370023 Zbl 1335.57031 article

[121] J. S. Birman and H. M. Hilden: “Erratum to ‘Isotopies of homeomorphisms of Riemann surfaces’,” Ann. of Math. 185 : 1 (2017), pp. 345. MR 3583359 Zbl 06686591 article

Complete Bibliography