Lefschetz fibrations, intersection numbers, and representations of the framed braid group
We examine the action of the fundamental group of a Riemann surface with punctures on the middle dimensional homology of a regular fiber in a Lefschetz fibration, and describe to what extent this action can be recovered from the intersection numbers of vanishing cycles. Basis changes for the vanishing cycles result in a nonlinear action of the framed braid group on strings on a suitable space of matrices. This action is determined by a family of cohomologous -cocycles parametrized by distinguished configurations of embedded paths from the regular value to the critical values. In the case of the disc, we compare this family of cocycles with the Magnus cocycles given by Fox calculus and consider some abelian reductions giving rise to linear representations of braid groups. We also prove that, still in the case of the disc, the intersection numbers along straight lines, which conjecturally make sense in infinite dimensional situations, carry all the relevant information.
Picard–Lefschetz theory can be viewed as a complexification of Morse theory with the stable and unstable manifolds replaced by vanishing cycles and the count of connecting orbits in the Morse–Witten complex replaced by the intersection numbers of the vanishing cycles along suitable paths in the base. Relevant topological information that can be recovered from these data includes, in Morse theory, the homology of the underlying manifold and, in Picard–Lefschetz theory, the monodromy action of the fundamental group of the base on the middle dimensional homology of a regular fiber. That the monodromy action on the vanishing cycles can be recovered from the intersection numbers follows from the Picard–Lefschetz formula
Here is a Lefschetz fibration over a Riemann surface with fibers of complex dimension , meaning that is a complex manifold and the map is holomorphic and has only nondegenerate critical points. We assume moreover that each singular fiber contains exactly one critical point, and denote by a regular fiber over . In equation (1), is an oriented vanishing cycle associated to a curve from to a singular point , is the Dehn–Arnold–Seidel twist about obtained from the (counterclockwise) monodromy around the singular fiber along the same curve, is the induced action on and denotes the intersection form. Equation (1) continues to hold when is a symplectic Lefschetz fibration as introduced by Donaldson [7, 8]. In either case the vanishing cycles are embedded Lagrangian spheres and so their self-intersection numbers are when is even and when is odd. See [2, Chapter I] and [3, §2.1] for a detailed account of Picard–Lefschetz theory and an exhaustive reference list.
The object of the present paper is to study an algebraic setting, based on equation (1), which allows one to describe the monodromy action of the fundamental group in terms of intersection matrices. This requires the choice of a distinguished basis of vanishing cycles and the ambiguity in this choice leads to an action of the braid group on distinguished basis , which in turn determines an action on a suitable space of matrices. In the case of the disc, this action was previously considered by Bondal  in the context of mirror symmetry. Our motivation is different and comes from an attempt of two of the authors (A.O. and D.S.) to understand complexified Floer homology in the spirit of Donaldson–Thomas theory . In this theory the complex symplectic action or Chern–Simons functional is an infinite dimensional analogue of a Lefschetz fibration. While there are no vanishing cycles, one can (conjecturally) still make sense of their intersection numbers along straight lines and build an intersection matrix whose orbit under the braid group might then be viewed as an invariant. Another source of inspiration for the present paper is the work of Seidel [20, 21] about vanishing cycles and mutations.
We assume throughout this paper that the base of our Lefschetz fibration is a compact Riemann surface, possibly with boundary, not diffeomorphic to the -sphere. (In particular, is oriented.) Let be the set of critical values and be a regular value. If is diffeomorphic to the unit disc we assume that . To assemble the intersection numbers into algebraic data it is convenient to choose a collection of ordered embedded paths from a regular value to the critical values (see Figure 1). Following [2, 20] we call such a collection a distinguished configuration and denote
by the set of homotopy classes of distinguished configurations. Any distinguished configuration determines an ordering of the set of critical values by . It also determines vanishing cycles as well as special elements of the fundamental group
(obtained by encircling counterclockwise along ). The orientation of is not determined by the path and can be chosen independently. However, when is even, the monodromy along changes the orientation of . Thus, to choose orientations consistently, we fix nonzero tangent vectors for , choose orientations of the vanishing cycles in the directions of these vectors, and consider only distinguished configurations that are tangent to the vectors at their endpoints. We call these marked distinguished configurations and denote by the set of homotopy classes of marked distinguished configurations. The oriented vanishing cycles determine homology classes, still denoted by
These data give rise to a monodromy character via
Here denotes the monodromy action of the fundamental group. Any such function satisfies the conditions
The part of that is generated by the vanishing cycles under the action of can be recovered as the quotient
Here is the group ring of , whose elements are thought of as maps with finite support and whose multiplication is the convolution product ; the map is regarded as an endomorphism by the convolution product . The -module is equipped with an intersection pairing, a -action , and special elements , defined by
Here acts on by
denotes the standard basis, and is defined by and for . With these structures is isomorphic to the submodule of generated by the vanishing cycles modulo the kernel of the intersection form. The isomorphism is induced by the map that assigns to every the homology class .
The monodromy character depends on the choice of a marked distinguished configuration and this dependence gives rise to an action of the framed braid group of on strings, based at , on the space of monodromy characters. More precisely, our distinguished configuration determines special elements obtained by encircling counterclockwise along . Denote by
the space of monodromy characters on . The framed braid group , interpreted as the mapping class group of diffeomorphisms in that preserve the set and the collection of vectors , acts freely and transitively on the space of homotopy classes of marked distinguished configurations (see Sections 3 and 4). Here denotes the identity component of the group of diffeomorphisms of that fix . The framed braid group also acts on the fundamental group and, for every and every , the isomorphism maps to . This action actually determines the (unframed) braid (see Section 4).
Our first theorem asserts that there is a canonical family of isomorphisms
which extends the geometric correspondence between monodromy characters associated to different choices of the distinguished configuration. It is an open question if every element can be realized by a (symplectic) Lefschetz fibration with critical fibers over . This question is a refinement of the Hurwitz problem of finding a branched cover with given combinatorial data. The isomorphisms are determined by a family of cocycles
with values in the group of invertible matrices over the group ring. To describe them we denote by the permutation associated to the action of on the ordering determined by . Then the entry of the matrix is
(first , second ) for and it is zero for . We emphasize that for . See Figure 2 below for some examples.
In the next theorem we think of an element as a function with finite support, denote , and multiply matrices using the convolution product (see Section 2).
Theorem A. (i) The maps are injective and satisfy the cocycle and coboundary conditions
(ii) The maps in (i) determine bijections
(iii) Given a Lefschetz fibration and elements , , we have
In terms of Serre’s definition of non-abelian cohomology [22, Appendix to Chapter VII] the first equation in (5) asserts that is a cocycle for every . The second equation in (5) asserts that the cocycles are all cohomologous and hence define a canonical -cohomology class
We call it the Picard–Lefschetz monodromy class.
In the case there is another well known cocycle arising from a topological context, namely the Magnus cocycle
Here denotes the usual braid group (with no framing), which we view as a subgroup of using the framing determined by a vector field on whose only singularity is an attractive point at and such that for all (see Section 6). The Magnus cocycle is also related to an intersection pairing [26, 19] and its dependence on the choice of the distinguished configuration is similar to the dependence of the Picard–Lefschetz cocycle. The cohomology classes and are distinct and nontrivial in . After reduction of to the infinite cyclic group, both cocycles define linear representations of the braid group with coefficients in . In the case of the Magnus cocycle, this is the famous Burau representation [4, 17]. In the case of the Picard–Lefschetz cocycle, this representation was first discovered by Tong–Yang–Ma  and is a key ingredient in the classification of -dimensional representations of the braid group on strings . For pure braids, i.e braids which do not permute the elements of , the Tong–Yang–Ma representation is determined by linking numbers (see Section 6).
We continue our discussion of the planar case . The group is then isomorphic to the free group generated by , and it is convenient to switch from the geometric picture in Theorem A to generators and relations. We denote by the abstract group generated by , with relations
This gives rise to a contravariant free and transitive action of on denoted by , and to an action of on via
for , and .
The third equation in (3) shows that, in the case of the disc, we can switch from matrix valued functions to actual matrices. More precisely, a monodromy character is uniquely determined by the single matrix . The latter satisfies
We denote by the space of matrices satisfying (9). The map is explicitly given by , the homomorphism being defined on generators by
Here is the matrix with the -th entry on the diagonal equal to one and zeroes elsewhere. The representation induces an action of on the quotient module which preserves the intersection form . Moreover, the triple becomes isomorphic to : see Section 2. The next result rephrases Theorem A for the particular case of the disc, and strengthens it with an additional uniqueness statement. The first part of (ii) has been already proved by Bondal [5, Proposition 2.1] for upper triangular matrices instead of (skew-)symmetric ones. For we denote by the permutation matrix of the transposition and, for , we denote by the diagonal matrix with -th entry on the diagonal equal to and the other diagonal entries equal to .
Theorem B. (i) There is a unique function satisfying the following conditions.
(Cocycle) For all and we have
(Normalization) For all we have and
(ii) The function in (i) determines a contravariant group action of on via
for and . This action is compatible with the action of on the space of marked distinguished configurations in the sense that, for every Lefschetz fibration with singular fibers over , every , and every , we have
(iii) For every and every the matrix induces an isomorphism from to which preserves the bilinear pairings and satisfies
By Theorem B every symplectic Lefschetz fibration with critical fibers over determines a -equivariant map
which can be viewed as an algebraic invariant of . Our next theorem asserts that this invariant is uniquely determined by the matrix
of intersection numbers of vanishing cycles along straight lines. Here we assume that no straight line connecting two points in contains another element of ; such a set is called admissible. Denote by the space of matrices satisfying
We define a map by
where is the ordering of given by , is identified with , and
Here the right hand side denotes the based loop obtained by first traversing , then moving clockwise near until reaching the straight line from to , then following , then moving counterclockwise near until reaching , and finally traversing in the opposite direction
(see Figure 3). Geometrically, this means that the intersection matrix assigns to a pair the intersection number of the vanishing cycles along the straight line from to , where the orientations at the endpoints are determined by moving the oriented vanishing cycles in the directions and clockwise towards the straight line. (Given a marked distinguished configuration and the loop as above, the straight line from to corresponds to the curve .)
Theorem C. The map defined by (15) is invariant under the diagonal action of on . Moreover, for every , there is a unique equivariant map
for every .
Let be a symplectic Lefschetz fibration with critical fibers over an admissible set . Let and , be the associated critical points of . Then the number is the algebraic count of negative gradient flow lines
from to . According to Donaldson–Thomas  this count of gradient flow lines is (conjecturally) still meaningful in suitable infinite dimensional settings. It thus gives rise to an intersection matrix and hence, by Theorem C, also to an equivariant map . A case in point, analogous to symplectic Floer theory, is where is the complex symplectic action on the loop space of a complex symplectic manifold.
The paper is organized as follows. In Section 2 we explain an algebraic setting for monodromy representations, Section 3 discusses the framed braid group , and in Section 4 we prove that acts freely and transitively on the space of distinguished configurations. Theorem A is contained in Theorem 5.5 from Section 5. We compare in Section 6 the Picard–Lefschetz cocycle with the Magnus cocycle, and discuss some related linear representations of the braid group. Theorem B is proved in Section 7, in Section 8 we introduce monodromy groupoids, in Section 9 we prove Theorem C, and Section 10 illustrates the monodromy representation by an example. We include a brief discussion of some basic properties of Lefschetz fibrations in Appendix A, summarizing relevant facts from [2, Chapter I].
2 Monodromy representations
We examine algebraic structures that are relevant in the study of monodromy representations associated to Lefschetz fibrations.
Fix a positive integer . Let be a group and be pairwise distinct elements of . A monodromy character on is a matrix valued function satisfying
for all and . A monodromy representation of is a tuple consisting of a -module with a nondegenerate bilinear pairing, a representation that preserves the bilinear pairing, and elements , satisfying
for all and . The automorphisms are called Dehn twists and the are called vanishing cycles.
Every monodromy character satisfies
Every monodromy representation satisfies
Equation (23) follows from (17) and (19) by replacing with and interchanging and . To prove the second equation in (24) use equation (19) with and ; then use (18). The proofs of (25) and (26) are similar.
Every monodromy representation gives rise to a monodromy character and vice versa. If is a monodromy representation then the associated monodromy character assigns to every the intersection matrix
Conversely, every monodromy character gives rise to a monodromy representation as follows.
Denote by the group ring of . One can think of an element either as a function with finite support or as a formal linear combination . With the first viewpoint, the multiplication in is the convolution product
The group acts on the group ring by the formula for and . In the formal sum notation we have
For any function we introduce the -module
where is the convolution product. This abelian group is equipped with a bilinear pairing
and a group action defined by
which preserves the bilinear pairing. The special elements are given by
These structures are well defined for any function . The next lemma asserts that the tuple is a monodromy representation whenever is a monodromy character.
(ii) Let be a monodromy representation and be its character (27). Then the map
induces an isomorphism of monodromy representations from to , where denotes the submodule generated by the vanishing cycles and is the kernel of the intersection form.
The -module is isomorphic to the image of the homomorphism and hence is torsion free. That the bilinear pairing in (29) is nondegenerate follows directly from the definition. That it satisfies (20) follows from (17) and that it satisfies (21) follows from (18) and the identity . To prove (22) fix an index and an element . Abbreviate
and define by
This proves (i).
Our geometric motivation for Lemma 2.3 is the following. Let be the function associated to a Lefschetz fibration and a distinguished configuration via (2). Let be the submodule generated by the vanishing cycles and be the kernel of the intersection form on . Then the homomorphism descends to a -equivariant isomorphism of -modules that identifies the pairing in (4) with the intersection form and maps the element defined by (4) to the equivalence class .
Assume that is generated freely by . In this case the function in Definition 2.1 is completely determined by . This matrix satisfies
It determines a monodromy representation
with special elements associated to the standard basis of and the -action uniquely determined by
Here denotes the matrix with the -th entry on the diagonal equal to and zeroes elsewhere. The function can be recovered from the matrix via
where is defined by equation (33). Moreover, the monodromy representations and are isomorphic. The isomorphism assigns to each the equivalence class of the function with value at and zero elsewhere. Its inverse is induced by the map
The -module of -matrices with entries in the group ring is naturally an algebra over the group ring. One can think of an element also as a function with finite support. Then the product is given by
and this formula continues to be meaningful when only one of the factors has finite support. The group ring is equipped with an involution given by The conjugate transpose of a matrix is defined by
and it satisfies .
Let be a group that acts covariantly on and denote the action by In the intended application is the fundamental group of a Riemann surface with punctures and is the braid group on strings in the same Riemann surface with one puncture. The action of on extends linearly to an action on by algebra automorphisms given by
This action extends to the -module of formal sums of elements of with integer coefficients. These correspond to arbitrary integer valued functions on . So acts on componentwise, or equivalently by