Topological constructions of tensor fields on moduli spaces
Vladimir Turaev

TL;DR
This paper explores how the topological properties of a space can induce tensor fields on the smooth parts of its associated moduli spaces of the fundamental group, linking topology and geometric structures.
Contribution
It introduces a novel method for constructing tensor fields on moduli spaces using topological features of the underlying space.
Findings
Established a connection between topology and tensor fields on moduli spaces
Provided a new construction technique for tensor fields in geometric topology
Enhanced understanding of the structure of moduli spaces through topological methods
Abstract
We show how topology of a space may lead to tensor fields on (the smooth part of) moduli spaces of the fundamental group.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Topological constructions of tensor fields on moduli spaces
Vladimir Turaev
Vladimir Turaev
Department of Mathematics
Indiana University
Bloomington IN47405, USA
Abstract.
We show how topology of a space may lead to tensor fields on (the smooth part of) moduli spaces of the fundamental group.
1. Introduction
Moduli spaces of the fundamental groups of surfaces carry beautiful geometric structures, in particular, Poisson brackets, see [AB], [Wo], [FR]. These brackets were described by Goldman [Go1], [Go2] in terms of a Lie bracket in the module of loops in the surface, see [AKKN1], [AKKN2], [Ka], [LS] for recent work on Goldman’s bracket. Here we extend this line of study. Our starting point is the work of van den Bergh [VdB] and Crawley-Boevey [Cb] who derive from any algebra and an integer the (commutative) coordinate algebra of the affine scheme of -dimensional representations of . These authors also define a subalgebra which - under appropriate assumptions - is the coordinate algebra of the affine quotient scheme . We view the latter affine scheme as the moduli space of -dimensional representations of . Inspired by the interpretation of vector fields on a smooth manifold as derivations of the algebra of smooth functions on this manifold, we can define vector fields on as derivations of the algebra . More generally, for any integers , we can define -tensor fields on as -linear forms (or ) which are derivations in all variables. Despite a purely algebraic formulation, this approach may lead to smooth tensor fields on the smooth parts of moduli spaces, see [MT2]. To construct -linear forms in which are derivations in all variables, we use a method inspired by the work of Crawley-Boevey [Cb] on Poisson structures. Namely, we set and derive such -linear forms in from -linear forms satisfying certain assumptions. We call -linear forms in satisfying these assumptions -braces in .
Our main aim is a construction of braces in the group algebras of the fundamental groups of topological spaces. We give two such constructions. First, consider a topological space and let be the group algebra of . A gate in is a path-connected subspace such that all loops in are contractible in and has a cylinder neighborhood in . We show that a gate in gives rise to an -brace in for all . This “gate brace” induces -tensor fields on the moduli spaces of for all . For example, if is a surface with boundary, then any properly embedded segment in is a gate; so, it determines an -brace in and an -tensor field on the moduli space for all .
Our second construction of braces applies to so-called quasi-surfaces which we introduce here as generalizations of the usual surfaces with boundary. A quasi-surface, , is obtained by gluing a surface to an arbitrary topological space along a finite set of disjoint segments in . These segments give rise to gates in which split into the surface part (a copy of ) and the singular part (the rest). By the above, each gate induces an -brace in the group algebra of for all . For oriented , we use intersections of loops to define a skew-symmetric “intersection 2-brace” in generalizing the Goldman bracket of surfaces. Our main result is a Jacobi-type identity relating the intersection 2-brace to the gate 3-braces. This generalizes to quasi-surfaces the Jacobi identity for the Goldman bracket of surfaces. The intersection 2-brace induces skew-symmetric bilinear forms , and our Jacobi-type identity relates them to the trilinear forms in derived from the gate 3-braces. For completeness, we also define intersection pairings in 1-homology of quasi-surfaces generalizing the usual intersection pairings in 1-homology of surfaces.
Any surface with boundary may be viewed as a quasi-surface in multiple ways determined by a choice of disjoint properly embedded segments in splitting into the “surface part” and the “singular part”. For oriented , each such splitting determines a 2-brace in the group algebra of and the induced pairings . By the above, these brace and pairings satisfy Jacobi-type identities involving the 3-braces associated with the segments in question.
The first part of the paper (Sections 2–5) presents our algebraic methods and the second part (Sections 6–10) is devoted to topological constructions.
This work was supported by the NSF grant DMS-1664358.
2. Preliminries
We briefly recall representation schemes and trace algebras following [VdB], [Cb]. Then we discuss derivations in algebras.
2.1. Representation schemes
Throughout the paper we fix a commutative base ring . By a module we mean an -module and by an algebra we mean (unless explicitly stated to the contrary) an associative -algebra with unit. We associate with every algebra and an integer an affine scheme , the -th representation scheme of . For any commutative algebra , the set of -valued points of is the set of algebra homomorphisms . The coordinate ring, , of is generated (over ) by the symbols with and . These generators commute and satisfy the following relations: for all , where is the Kronecker delta; for all , , and ,
[TABLE]
The function on the set of -valued points of determined by assigns to a homomorphism the -entry of the matrix . That these functions satisfy the relations above is straightforward.
The action of the group on by conjugations induces an action of on the commutative algebra for all . Explicitly, for and any we have
[TABLE]
The set of invariant elements is a subalgebra of . This is the coordinate algebra of the affine quotient scheme which we view as the “moduli space” of -dimensional representations of .
2.2. The module and the trace
Given an algebra , let be the submodule of spanned by the commutators with . The quotient module is the zeroth Hochschild homology of . Now, for any integer , the linear map is called the trace and denoted . The trace annighilates all the commutators in and therefore . Thus, the trace induces a linear map also denoted .
The subalgebra of generated by is called the -th trace algebra of and is denoted . A direct computation shows that and therefore . If the ground ring is an algebraically closed field of characteristic zero and is a finitely generated algebra, then a theorem of Le Bruyn and Procesi [LBP] implies that so that is the coordinate algebra of .
2.3. Derivations
A derivation of an algebra is a linear map such that for all . We denote by the module of derivations of . Given , the commutator is a derivation of . This defines a Lie bracket in .
Any derivation carries into itself as
[TABLE]
for . Therefore induces a linear endomorphism of . A linear endomorphism of is a weak derivation if it is induced by a derivation .
By [Cb, Lemma 4.4], for any derivation and any integer , there is a unique derivation such that for all . Indeed, this formula defines on the generators of the algebra ; the compatibility with the defining relations is straightforward. Clearly, for all . Therefore and the restriction of to is a derivation of the algebra
2.4. Remark
If is the algebra of smooth -valued functions on a smooth manifold , then each smooth vector field on induces a derivation of carrying a function to the function . The map defines a Lie algebra isomorphism from the Lie algebra of smooth vector field on (with the Jacobi-Lie bracket) onto . Given an algebra and an integer , these results suggest to view the derivations of the trace algebra as vector fields on (the smooth part) of the affine quotient scheme . More generally, tensor fields on this affine scheme may be defined as maps which are derivations in all variables. Here for a set and an integer , we let be the direct product of copies of .
3. Braces and brackets
We define and study braces.
3.1. Braces
For an integer , an -brace in an algebra is a mapping which is a weak derivation in all variables: for any and , the map
[TABLE]
is a weak derivation. In particular, has to be linear in all variablres. For , an -brace in is just a weak derivation .
If is a commutative algebra, then , , and an -brace in is a mapping which is a derivation in all variables: for any and any , the map
[TABLE]
is a derivation.
The following lemma - inspired by W. Crawley-Boevey [Cb] - is our main tool producing braces in the trace algebras.
Lemma 3.1**.**
For any integers and any -brace in an algebra , there is a unique -brace in the algebra such that the trace carries to that is for all , we have
[TABLE]
Proof.
The uniqueness of is clear as generates the algebra . We first prove the existence of for . We need to show that given a weak derivation , there is a derivation such that for all . Pick a derivation inducing . By Section 2.3, the induced derivation restricts to a derivation of the algebra . The map satisfies the conditions of the lemma.
Suppose now that . Since the algebra is generated by the set , every has a (non-unique) finite expansion
[TABLE]
where the sum is over some finite sequences and the coefficients are in . Pick any and for pick an expansion of as in (3.1.2). If an -brace satisfies the conditions of the lemma, then using , the Leibnitz rule and (3.1.1), we obtain that where is a sum of products determined by the summands on the right-hand sides of the expansions . Each product involves factors of 3 types:
(I) the coefficients appearing in the summands in question;
(II) the traces where and runs over the indices determined by the summand of except one of these indices, say, ;
(III) the factor .
We claim that the element of does not depend on the choice of the expansion of . It is easy to reduce this claim to its special case where for and . (It is understood that we keep and use the formula as the expansion for ). Consider the projection . By the assumptions of the lemma, there is a derivation (possibly, depending on ) such that
[TABLE]
For any and , we have the following equalities in :
[TABLE]
[TABLE]
Using this formula to compute all factors of type (III) above, we easily deduce that . Since does not depend on the choice of , neither does .
Coming back to arbitrary , we similarly prove that the element of does not depend on the choice of the expansion for all . In other words, depends only on . We take as . The resulting map is easily seen to be an -brace in and to satisfy (3.1.1). ∎
3.2. Brackets
Given an integer , an -bracket in a module is a map which is linear in every variable. For , we say that is -symmetric if for all ,
[TABLE]
If , then -symmetric brackets are said to be cyclically symmetric. If , then -symmetric brackets are said to be skew-symmetric.
Lemma 3.2**.**
Given an algebra , integers , and an -symmetric -bracket in which is a weak derivation in the -th variable, there is a unique -brace in the algebra such that the trace carries to . The brace is -symmetric.
Proof.
Since is -symmetric and is a weak derivation in one variable, it is a weak derivation in all variables. Thus, is a brace. By Lemma 3.1, there is a unique -brace in such that the trace carries to . The -symmetry of implies that is -symmetric. ∎
4. Braces in group algebras
In this section, is the group algebra of a group . We construct braces in starting from Fox derivatives in .
4.1. Computation of
By definition, the module is generated by the set . Since generates , the module is generated by the set . Since for , the module is generated by the set . Thus, is the free module whose basis is the set of conjugacy classes of elements of .
4.2. Fox derivatives
A (left) Fox derivative in is a linear map such that for all . For any , we have then where is the linear map carrying all elements of to . For , we can uniquely expand where is non-zero for a finite set of . Consider the map
[TABLE]
and denote its linear extension by .
Lemma 4.1**.**
.
Proof.
It suffices to prove that for any . We have
[TABLE]
Therefore, by the definition of ,
[TABLE]
[TABLE]
The latter expression is invariant under the permutation . So, and . ∎
The linear map induced by is denoted by .
Theorem 4.2**.**
Let be the projection. For any and any Fox derivatives , the map defined by
[TABLE]
for is an -brace in .
Proof.
We need to prove that is a weak derivation in all variables, i.e., for any and , the map
[TABLE]
is induced by a derivation in . Set
[TABLE]
For , we expand with . Then
[TABLE]
[TABLE]
where we use that . Thus, the map (4.2.2) is induced by the linear map carrying any to It remains to prove that for any , the linear map carrying any to is a derivation. Indeed, for , we have
[TABLE]
Also,
[TABLE]
and so
[TABLE]
Thus, is a derivation in . This completes the proof of the theorem. ∎
For , Theorem 4.2 may be rephrased by saying that for any Fox derivative in , the linear map induced by is also induced by a derivation . This derivation carries any to . In contrast to , the derivation may not annighilate .
Combining Theorem 4.2 with Lemma 3.1 we obtain the following.
Corollary 4.3**.**
For any integers and Fox derivatives in , there is a unique -brace in such that for all , we have
[TABLE]
If , then the -braces and are cyclically symmetric. This follows from the identities and for all .
4.3. Equivalence of Fox derivatives
Given a Fox derivative in and any , the linear map is also a Fox derivative denoted . We say that two Fox derivatives in are equivalent if there is such that . This is indeed an equivalence relation. Moreover, equivalent Fox derivatives induce the same braces in and . This follows from the identities
[TABLE]
for all and .
5. Quasi-Lie brackets and brace algebras
We define quasi-Lie brackets and brace algebras.
5.1. Quasi-Lie brackets
A quasi-Lie pair of brackets in a module is a pair formed by a skew-symmetric 2-bracket in and a cyclically symmetric 3-bracket in such that for any , we have
[TABLE]
Here the left-hand side is the usual Jacobiator of the 2-bracket. Both sides of (5.1.1) are cyclically symmetric. We call (5.1.1) the quasi-Jacobi identity. For the zero 3-bracket, we recover the standard Jacobi identity.
5.2. Examples
- Any bilinear pairing induces a quasi-Lie pair of brackets in with the 2-bracket and the 3-bracket
[TABLE]
for .
- For a quasi-Lie pair of brackets in a module and a 3-bracket in invariant under all permutations of the variables, the pair is also a quasi-Lie pair.
5.3. Brace algebras
A brace algebra is an algebra endowed with a quasi-Lie pair of brackets in the module such that both these brackets are braces in in the sense of Section 3.1. For example, a commutative brace algebra with zero 3-bracket is a Poisson algebra in the usual sense.
A brace homomorphism from a brace algebra to a brace algebra is a bracket-preserving linear map . Thus, should satisfy and for all .
Lemma 5.1**.**
Let be a commutative algebra carrying a skew-symmetric 2-brace and a cyclically symmetric 3-brace . If (5.1.1) holds for all elements of a generating set of , then it holds for all elements of .
Proof.
Let and be respectively the left and the right hand-sides of (5.1.1). Since and are linear in and cyclically symmetric, it suffices to verify the following: if (5.1.1) holds for the triples and , then it holds for the triple . Since is a brace,
[TABLE]
[TABLE]
[TABLE]
Similarly,
[TABLE]
Adding these three expansions and using the skew-symmetry of , we get
[TABLE]
Thus, satisfies the Leibnitz rule in the last variable. Since the bracket also satisfies this rule, so does . Consequently, if (5.1.1) holds for the triples and , then it holds for the triple . ∎
Recall the trace algebras associated with any algebra .
Theorem 5.2**.**
For any brace algebra and integer , there is a unique brace algebra structure on such that is a brace homomorphism.
Proof.
Let and be the brackets in forming a quasi-Lie pair. By Lemma 3.2, there are unique braces and in such that
[TABLE]
for all . Then (5.1.1) holds for all elements of the set . Since this set generates , Lemma 5.1 implies that (5.1.1) holds for all elements of . Also, since is -symmetric and is -symmetric, so are the braces and . Thus, these braces form a quasi-Lie pair. This turns into a brace algebra satisfying the conditions of the theorem. ∎
5.4. Remark
A bracket in a module is fully symmetric if it is invariant under all permutations of the variables. A quasi-Lie pair of brackets in a module gives rise to a fully symmetric 3-bracket by
[TABLE]
for any . The cyclic symmetry of is obvious and the invariance of under the permutation follows from (5.1.1). Conversely, if 2 is invertible in , then we can recover the 3-bracket from and via (5.4.1). Formula (5.1.1) follows then from the identity . This establishes a bijective correspondence between quasi-Lie pairs of brackets in and pairs (a skew-symmetric 2-bracket in , a fully symmetric 3-bracket in ).
6. Topological gates
We define gates in topological spaces and show how they give rise to braces.
6.1. Gates
A cylinder neighborhood of a subset of a topological space is a pair consisting of a closed set with and a homeomorphism carrying onto and carrying onto . Note that then and is closed in . A gate in is a path-connected subspace endowed with a cylinder neighborhood in and such that all loops in are contractible in . An example of a gate is provided by a simply connected codimension 1 proper submanifold of a manifold together with a suitable homeomorphism of a closed neighborhood of onto .
For the rest of this section, we fix a path-connected topological space , a gate , and its cylinder neighborhood which we identify with so that . Pick a point and set .
6.2. Gate derivatives
Here we associate with the gate an equivalence class of Fox derivatives in the algebra . We start with preliminaries on (continuous) paths. Let be the map which carries to for all and carries to . We say that a path is transversal to if and the map restricted to is transversal to . Then is a finite subset of . For a path , we denote the inverse path by . A path is a loop based in if . Such a loop represents an element of denoted .
Pick a path such that and . Consider a loop based in and transversal to . For , we let be the path in obtained as the product of the path with any path in from to , and finally with . Then a loop based in . Since all loops in are contractible in , the homotopy class does not depend on the choice of . Set if at the loop crosses upwards (i.e., from to ), and otherwise. Set
[TABLE]
Lemma 6.1**.**
Formula (6.2.1) defines a map whose linear extension , denoted , is a Fox derivative. If is another path from to , then the Fox derivatives and are equivalent in the sense of Section 4.3.
Proof.
It is clear that all elements of can be represented by loops based in and transversal to . We claim that if two such loops are homotopic, then . There is a homotopy from to such that the loop is based in and transversal to except for a finite set of near which the homotopy pushes a branch of across creating or destroying a pair of transversal crossings with . It is easy to see that the contributions of these two crossings to cancel each other. Therefore, Formula (6.2.1) yields a well-defined map which extends by linearity to a map .
If are loops in based in and transversal to , then so is their product, and it follows directly from the definitions that . Consequently, is a Fox derivative in .
Given two paths from to , we let be the homotopy class of the loop obtained as the product of with a path in from to , and with . It is easy to see that . Thus, the Fox derivatives and are equivalent. ∎
6.3. Gate braces
Let be the set of free homotopy classes of loops in and let be the free module with basis . The map carrying the homotopy classes of loops to their free homotopy classes induces a bijection where is the set of conjugacy classes of elements of . By Section 4.1, so that we can identify with .
By Lemma 6.1, a sequence of gates in (not necessarily disjoint or distinct) determines a sequence of equivalence classes of Fox derivatives in . By Section 4, the latter induces -braces in the algebras and . In particular, the sequence of copies of a gate determines a cyclically symmetric -brace in . We compute in geometric terms as follows. Consider loops based in and transversal to . Pick a point . For and , let be the loop based in and obtained as the product of a path in from to , the loop based in , and the path . In the next lemma, the free homotopy class of a loop in is denoted .
Lemma 6.2**.**
Under the assumptions above,
[TABLE]
Proof.
We claim that both sides of (6.3.1) are preserved when the loop is replaced by the loop for a path transversal to and such that . The invariance of the left-hand side is obvious since . We prove the invariance of the right-hand side which we denote by . If the path misses , then the claim is obvious because the loops and meet in the same points which contribute the same to and . Otherwise, the path expands as a product of a finite number of paths each of which is transversal to and intersects in one point. Thus, it suffices to prove our claim in the case where intersects in one point, say, . Then meets at the crossings of with and two additional crossings at the point which is traversed first by and then by . Let be the corresponding values of the parameter, so that are respectively the first and the last crossings of with . It is easy to see that and the corresponding loops and are homotopic. Therefore the terms of associated with and cancel each other, while the remaing terms yield . Thus, .
Replacing by for a path transversal to and running from to , we obtain a loop transversal to and based in . By the previous paragraph, it suffices to prove (6.3.1) with this new loop instead of . By a similar argument, it suffices to prove (6.3.1) in the case where all the loops are based in .
Now, pick a path from to . By the definition of the Fox derivative , for any ,
[TABLE]
where . Therefore
[TABLE]
[TABLE]
Setting for and substituting in Formula (4.2.1) the above expression for , we obtain a formula equivalent to (6.3.1). ∎
6.4. The dual map
The gate determines a linear map “dual” to . This map carries the homology class of a loop transversal to to . It is clear that the sum of the coefficients of the expression (6.3.1) is equal to .
7. Quasi-surfaces
7.1. Generalities
By a surface we mean a smooth 2-dimensional manifold with boundary. A quasi-surface is a topological space obtained by gluing a surface to a topological space along a continuous map where is a union of a finite number of disjoint segments in . Note that and . Here we impose no conditions on and do not require to be compact or connected or maximal among surfaces in .
The quasi-surface has path-connected components of 3 types: (i) components of disjoint from ; (ii) path-connected components of disjoint from ; (iii) path-connected components of meeting both and . For components of type (i) our results below are standard in the topology of surfaces. For components of type (ii), all our operations are identically zero. The novelty of this work concerns the components of type (iii).
7.2. Examples
In the following examples, is a surface.
-
When is a family of disjoint segments in , the unique map from to a 1-point space determines a quasi-surface. For , it is a copy of . As a consequence, any surface with non-void boundary is a quasi-surface.
-
Given disjoint finite subsets of , we obtain a quasi-surface by collapsing each of these subsets into a point. Here is an -point set with discrete topology and is a small closed neighborhood in of the union of our finite sets.
-
Given disjoint segments in and points in a topological space , we obtain a quasi-surface by gluing to along the map carrying to for .
-
Let be a surface with boundary and let be a union of a finite number of disjoint proper embedded segments in . Suppose that splits into two subsurfaces (possibly disconnected) and so that . Then is homeomorphic to the quasi-surface determined by the tuple where is the inclusion.
7.3. Conventions
Fix for the rest of the paper a tuple as in Section 7.1. We assume that is path-connected, , and is oriented. We will identify a closed neighborhood of in with so that and
[TABLE]
We will often use the surface
[TABLE]
which is a copy of embedded in . It is called the surface part of . We provide with the orientation induced from that of .
For , denote by the corresponding component of . Set
[TABLE]
Clearly, is an embedded segment in . Endowing with the cylinder neighborhood we turn into a gate in in the sense of Section 6.1. This is the -th gate of . The gates split into the surface part and the singular part which is the mapping cylinder of the gluing map . All paths from a point of to a point of have to cross a gate.
7.4. Gate orientations
A gate orientation of is an orientation of all the gates of . Gate orientations of canonically correspond to orientations of the 1-manifold . Given a gate orientation of and points , we say that lies on the -left of and lies on the -right of if and the -orientation of leads from to . We write then or . We set if the -orientation of is compatible with the orientation of , i.e., if the pair (a -positive tangent vector of , a vector directed inside ) is positively oriented in . Otherwise, . Also, we let be the gate orientation obtained from by inverting the direction of while keeping the directions of the other gates. We let denote the gate orientation of opposite to on all gates.
7.5. Generic loops
In the rest of the paper by a loop in we mean a circular loop, i.e., a continuous map . The intersection of the set with the -th gate is denoted . A generic loop in is a loop in such that (i) all branches of in are smooth immersions meeting transversely at a finite set of points lying in the interior of the gates, and (ii) all self-intersections of in are double transversal intersections lying in . The set of self-intersections in (= double points) of a generic loop is denoted by . This set is finite and lies in .
A generic loop in never traverses a point of a gate more than once, and the set is finite. The sign of at a point is if goes near from to and otherwise.
We define six local moves on a generic loop in keeping its free homotopy class. The move is a deformation of in the class of generic loops. This move preserves the number . The moves modify in a small disk in and are modeled on the Reidemeister moves on knot diagrams (with over/under-data dropped). The move adds a small curl to and increases by . The move pushes a branch of across another branch of increasing by . The move pushes a branch of across a double point of keeping . The move pushes a branch of across a gate keeping . The move pushes a double point of across a gate decreasing by . Graphically, the moves are similar to . We call the moves and their inverses loop moves. It is clear that generic loops in are freely homotopic if and only if they can be related by a finite sequence of loop moves.
A finite family of loops in is generic if all these loops are generic and all their mutual crossings in are double transversal intersections in . In particular, these loops can not meet at the gates. We will use the following notation. For generic loops in , consider the set of triples
[TABLE]
Given a gate orientation of , we define a set by
[TABLE]
Clearly,
[TABLE]
8. Homological intersection forms
As a prelude to more sophisticated operations, we define here intersection forms in 1-homology of .
8.1. First homological intersection form
Given a gate orientation of , we define a bilinear form
[TABLE]
called the first homological intersection form of . The idea is to properly position the loops near the gates and then to count intersections of the loops in the surface part of with signs. We say that an (ordered) pair of loops in is -admissible if this pair is generic and so that the crossings of with every gate lie on the -left of the crossings of with this gate. Taking a generic pair of loops in and pushing the branches of crossing the gates to the -left and pushing the branches of crossing the gates to the -right, we obtain an -admissible pair of loops (possibly, with more crossings than the initial pair). Thus, any pair of loops in may be deformed into an -admissible pair.
For a generic pair of loops in , the set of crossings of with in is a finite subset of denoted . For a point , set if the (positive) tangent vectors of and at form an -positive basis in the tangent space of at and set otherwise.
Lemma 8.1**.**
For any -admissible pair of loops in , the “crossing number”
[TABLE]
depends only on the homology classes of in . The formula defines a bilinear form (8.1.1).
Proof.
For each , one endpoint of the gate lies on the -left of the other endpoint. Pick disjoint closed segments and containing these two endpoints respectively. Clearly, for all and . We say that a loop in is -left (respectively, -right) if it is generic and meets the gates of only at points of (respectively, of ). Given an -admissible pair of loops in , we can push the branches of crossing the gates to the left and push the branches of crossing the gates to the right without creating or destroying intersections between and . Consequently, is homotopic (in fact, isotopic) to an -left loop and is homotopic to an -right loop such that . Since is a deformation retract of for all , any -left loops homotopic in are homotopic in the class of -left loops in . Similarly, any -right loops homotopic in are homotopic in the class of -right loops in . Such homotopies of obviously preserve . Therefore the number depends only on the (free) homotopy classes of in . Moreover, since this number linearly depends on both loops and , it depends only on their homology classes. This implies the claim of the lemma. ∎
We emphasize that the crossings of loops in do not contribute to the crossing number. For loops in the surface , the crossing number is the usual homological intersection number. The crossing numbers of loops in with arbitrary loops in are equal to zero.
For an -admissible pair of loops in , the pair is -admissible. Using these pairs to compute and , we obtain two sums which differ only in the signs of the terms. Hence, for any ,
[TABLE]
Lemma 8.2**.**
For any homology classes represented by a generic pair of loops in ,
[TABLE]
Proof.
Consider an -admissible pair of loops obtained from by pushing the branches of meeting the gates to the -left of the branches of meeting the gates. This modifies in a small neighborhood of the gates; we can assume that have the same intersections in as plus one additional intersection for each triple . Observe that . Consequently,
[TABLE]
[TABLE]
∎
Formula (8.1.4) generalizes (8.1.2) because for an -admissible pair of loops . We describe next the dependence of on . We will use the linear map “dual” to the -th gate. This map carries the homology class of any generic loop to . In the notation of Section 6.4, .
Theorem 8.3**.**
For any and ,
[TABLE]
Proof.
Pick an -admissible pair of loops representing respectively . We compute from the definition and compute from Lemma 8.2. The resulting expressions differ in the sum associated with . Since the pair is -admissible, the set consists of all triples . Therefore
[TABLE]
[TABLE]
∎
8.2. Second homological intersection forms
Pick a gate orientation of and define a skew-symmetric bilinear form by
[TABLE]
for . This form does not depend on because, by (8.1.5),
[TABLE]
for any and . We call the second homological intersection form of . Both the first and the second homological intersection forms generalize the standard intersection form in 1-homology of a surface. Indeed, the value of the form (8.1.1) (respectively, (8.2.1)) on any pair of homology classes of loops in is equal to the usual intersection number of these loops in (respectively, twice this number).
Theorem 8.4**.**
For any gate orientation and any , we have
[TABLE]
Proof.
Applying (8.1.5) consequtively to all elements of , we get
[TABLE]
Substituting , we get
[TABLE]
This formula and the equality imply (8.2.2). ∎
Formula (8.2.2) shows that if , then is a sum of and terms associated with the gates.
8.3. Remark
For , the definitions in this section and below do not depend on the orientation of and extend to non-orientable quasi-surfaces.
9. Homotopy intersection forms
We define homotopy intersection forms of refining the homological forms above. In this section and below, is the set of free homotopy classes of loops in and is the free -module with basis . By Sections 4 and 6, for each , the gate determines a cyclically symmetric -bracket in . It is denoted .
9.1. First homotopy intersection form
Pick a gate orientation of . Any pair can be represented by an -admissible pair of loops in , cf. Section 8.1. For a point , consider the loops which are reparametrizations of , respectively, starting and ending in . Consider the product loop based in and set
[TABLE]
where for a loop in , we let be its free homotopy class. The sum on the right-hand side of (9.1.1) is an algebraic sum of all possible ways to graft and . This sum is preserved under all loop moves on keeping this pair -admissible. Hence, does not depend on the choice of in the homotopy classes . Extending the map by bilinearity, we obtain a bilinear pairing
[TABLE]
We call this pairing the first homotopy intersection form of . The proof of Formula (8.1.3) applies here and shows that for any ,
[TABLE]
For a generic (non-double) point of a generic loop , we let be the loop which starts at and goes along until coming back to . Having two generic loops and points , on the same gate, we can multiply the loops using an arbitrary path in connecting their base points . The resulting loop determines a well-defined element of denoted .
Lemma 9.1**.**
Let be represented by a generic pair of loops . Then
[TABLE]
The proof repeats the proof of Lemma 8.2 with obvious modifications. If , then (9.1.4) simplifies to
[TABLE]
Theorem 9.2**.**
For any and ,
[TABLE]
Proof.
It suffices to handle the case . Then proceed as in the proof of Corollary 10.5.17 replacing by and using Formula (6.3.1) to compute . ∎
Applying (9.1.6) consecutively to all and using (9.1.3), we get
Corollary 9.3**.**
For any ,
[TABLE]
9.2. Second homotopy intersection form
We define a 2-bracket in by for all . This skew-symmetric bracket does not depend on because, by (9.1.6),
[TABLE]
for all . (Here we use the symmetry of the brackets .) We call the 2-bracket the second homotopy intersection form of . Both the first and the second homotopy intersection forms generalize Goldman’s [Go1], [Go2] bracket: the value of (respectively, ) on any pair of free homotopy classes of loops in is equal to their Goldman’s bracket (respectively, twice this bracket).
Theorem 9.1 allows us to compute for from any generic pair of loops representing and any gate orientation of . Namely,
[TABLE]
[TABLE]
where if and if . Note also the identity
[TABLE]
which can be easily deduced from (9.1.7). Consequently, if , then the form expands as a sum of and terms associated with the gates.
9.3. Remark
Other algebraic operations associated with surfaces may be extended to quasi-surfaces. This includes algebraic intersections of loops (see [Tu1]), Lie cobrackets (see [Tu2], [Ha]), double brackets and generalized Dehn twists (see [MT1]), and quasi-Poisson structures on the representation spaces (see [MT2]). In a sequel to this paper, the author plans to discuss natural cobrackets appearing in the study of quasi-surfaces.
10. Main theorem
10.1. Statement
We state our main result concerning the quasi-surface . Let with . Consider the group algebra and the second homotopy intersection form in . The next theorem computes the Jacobiator of this 2-form via the 3-bracket
[TABLE]
This theorem shows that the failure of the intersection form to satisfy the Jacobi identity is entirely due to the presence of the gates.
Theorem 10.1**.**
The brackets and are braces in forming a quasi-Lie pair.
This theorem can be rephrased by saying that the pair turns into a brace algebra. Combining Theorems 10.1 and 5.2, we conclude that for all , the -th trace algebra of carries a unique structure of a brace algebra such that the trace is a brace homomorphism.
By Example 7.2.1 (case ), every surface is a quasi-surface with a single gate which separates the surface part (a copy of ) from a cone over a segment in . Here since each loop in this quasi-surface may be deformed into the complement of the gate. Theorem 10.1 yields then the usual Jacobi relation for . On the other hand, in Example 7.2.4 it may well happen that .
The proof of Theorem 10.1 occupies the rest of the section.
10.2. Proof of Theorem 10.1: beginning
Throughout the proof we fix a gate orientation of . By Section 6.3, the brackets in are cyclically symmetric braces in (independent of ). Therefore so is their sum . The skew-symmetry of the 2-bracket in is obvious. We now prove that this 2-bracket is a brace in . Since it is skew-symmetric, it suffices to prove that is a weak derivation in the second variable. Pick any , and represent the pair by an -admissible pair of loops in where is based in . For each point , consider the loop obtained by reparametrization of so that its starts and ends in . We have where is the path in going from to along and is the path in going from to along . The product path in is a loop based in ; consider its homotopy class . Recall the crossing sign and set
[TABLE]
The sum on the right-hand side is an algebraic sum of all possible ways to graft to . (For surfaces, the pairing (10.2.1) was first introduced by Kawazumi and Kuno [KK1], [KK2].) It is easy to see from the definitions that (i) the expression depends only on and does not depend on the choice of and (ii) the linear extension of the map is a derivation of the algebra . Next, denote the projection by and note that for each , its image is the conjugacy class of in . Comparing Formulas (9.1.1) and (10.2.1), we obtain that . By (9.1.3),
[TABLE]
[TABLE]
Consequently, the linear endomorphism of is induced by the linear endomorphism of . Since the maps and are derivations of , so is their sum . This implies that the bracket is a weak derivation in the second variable and is a brace.
It remains to verify the Jacobi-type identity (5.1.1). This is done in the next three subsections.
10.3. Preliminaries on simple loops
We say that a finite family of loops in the quasi-surface is simple if these loops meet the gates of transversely and have no mutual crossings or self-crossings in the surface part of . A simple family of loops is generic.
Lemma 10.2**.**
Any finite family of loops in can be deformed in into a simple family of loops.
Proof.
Consider first a single loop in . Since is path-connected and contains a gate, we can deform our loop into a generic loop which meets a gate at least once. If the set of double points of in is empty, then we are done. Otherwise, pick a point . Starting at and moving along (in the given direction of ), we meet several double points with and then come to a point of a certain gate . The segment of connecting to is embedded in and meets only at its endpoint . Let be the branch of transversal to at . Push the branch towards along while keeping and transversal and eventually push across at . This transformation of decreases by 1 and increases by 2. Continuing by induction, we deform our loop into a generic loop without self-intersections in . If the original family of loops contains loops, then we first deform it into a generic family of loops which all meet some gates. Then pushing branches at crossings and self-crossings as above, we deform the latter family into a simple family of loops. ∎
10.4. Preliminaries on sign functions
We define two functions used in the proof. The first function, , is defined on the set by and . The second function, also denoted , is defined on the set of all triples by the formula
[TABLE]
This function is invariant under all permutations of as easily follows from the identity for all . The same identity implies another useful equality: for all , we have
[TABLE]
10.5. Proof of (5.1.1)
Any (possibly, non-associative) algebra carries the bracket . For , set
[TABLE]
A direct computation shows that
[TABLE]
We apply these observations to the algebra with multiplication . In view of (10.5.1), the identity (5.1.1) is equivalent to the identity
[TABLE]
for all . Since both sides are linear in , it suffices to handle the case . Set
[TABLE]
Formula (9.1.3) implies that
[TABLE]
Thus,
[TABLE]
In our computations, we represent by loops in , respectively.
If the loops lie in then Goldman’s results imply that for all so that . It is also clear that , and (10.5.2) follows. For completeness, we check the identity in this case (it is also included in the general case treated below). Deforming if necessary , we can assume that the triple is generic in the sense of Section 7.5. Then is computed by (9.1.1). To compute , consider all intersections of the loop with the loops . At such an intersection, say , the loop meets either or . Thus, where
[TABLE]
[TABLE]
Here is the loop obtained by grafting the loops to at the points . More precisely, this loop goes along starting and ending in , then along from to , then along starting and ending in , and finally returns along to . Note that the inclusions , ensure that so that the loop is well-defined. The loop is defined similarly grafting the loops to at the points . Therefore
[TABLE]
Note that as directly follows from the definitions and the identity for . Permuting , we get and . Summing up, we obtain .
Consider now the general case where the loops do not necessarily lie in . By Lemma 10.2, deforming in , we can ensure that this triple of loops is simple. By (9.1.5),
[TABLE]
where are loops reparametrizing and based respectively in while is a path connecting and in . We deform the loop by slightly pushing its subpaths “behind the gate”, i.e., into . (The endpoints of these subpaths are pushed into along , respectively.) The resulting loop is denoted by . Thus,
[TABLE]
Note that the loop is simple; moreover, the pair formed by this loop and is simple. Applying (9.1.5) again, we get
[TABLE]
For any , the set is a disjoint union of the sets and . Therefore . By (10.5.7),
[TABLE]
where
[TABLE]
and
[TABLE]
Combining (10.5.6) with (10.5.8), we obtain
[TABLE]
To compute the latter sum, we rewrite as follows. For , the homotopy class is represented by the loop obtained by grafting and to via a path in from to and a path in from to . To give a precise description of this loop, we separate two cases.
Case 1: so that the points are pairwise distinct (possibly, ). In this case the loop starts at and goes: along the gate to , then along the full loop back to , then along back to , then along to , then along the gate to , then along the full loop back to , then along back to , and finally along back to .
Case 2: . Then and are three distinct points on the gate . If , then the loop starts at and goes: along to , then along the full loop back to , then along to , then along the full loop back to , then along to , and finally along the full loop back to . If , then the loop starts at and goes: along to , then along the full loop back to , then along to , then along the full loop back to , then along to , then along the full loop back to , and finally along to .
In both cases,
[TABLE]
We call the summands corresponding to the triples with the 4-tuple terms. The summands with (and ) are called 3-tuple terms.
Similarly, for , the homotopy class is represented by the loop obtained by grafting and to via a path in from to and a path in from to . A precise description of this loop also includes two cases determined by whether or not ; we leave the details to the reader. Thus,
[TABLE]
We call the summands corresponding to the triples with the 4-tuple terms. The summands with (and ) are called 3-tuple terms.
Substituting these expressions for in (10.5.9), we expand as a linear combination of 4-tuple and 3-tuple terms. Then Formula (10.5.3) yields such an expansion of and Formula (10.5.5) yields such an expansion of . The (total) contribution of the 4-tuple terms to is denoted by , and the (total) contribution of the 3-tuple terms to is denoted by . We stress that .
We prove next that . Note first that each point in a 4-tuple (or a 3-tuple) term is traversed by exactly one of the loops . We will write for the corresponding signs . For example, , etc. Also set . In this notation, the contribution of 4-tuple terms to is equal to where
[TABLE]
and
[TABLE]
To compute and , we cyclically permute and in the formulas above via . It is convenient to simultaneously permute the indices and permute the labels via . This gives
[TABLE]
and
[TABLE]
Applying the same permutations again, we get
[TABLE]
To compute , we apply to the permutation and the following permutation of the indices: , . Thus,
[TABLE]
We conclude that the contribution of 4-tuple terms to is equal to
[TABLE]
for
[TABLE]
Then
[TABLE]
We now compute all the ’s. Comparing the expansions of and above, we observe that their summands are defined by the same formula; here we use the obvious fact that the loops and are freely homotopic provided . The summation in these two expansions goes over complementary sets of indices as the inclusion holds if and only if . Therefore
[TABLE]
Analogously,
[TABLE]
and
[TABLE]
Using that , we deduce from these expressions that
[TABLE]
[TABLE]
[TABLE]
As a consequence, the permutation of keeps and transforms and into each other. Hence, and so
[TABLE]
where is the sum of the 3-tuple terms in the expansion of . Therefore to prove (10.5.2) we need to check that
[TABLE]
Observe that each 3-tuple term in the expansion of is associated with an element of and pairwise distinct points . For each such triple of points, set
[TABLE]
and
[TABLE]
Here is the function of 3 signs defined in Section 10.4. The loop is the product of three loops formed via connecting their base points by arbitrary paths in . (In other words, we treat as a big base point for these loops.) The loop is defined similarly. Note that by the cyclic symmetry of free homotopy classes of loops, we have
[TABLE]
Let be the sum of 3-tuple terms associated with . Clearly,
[TABLE]
We prove below that for all ,
[TABLE]
In this and similar sums run respectively over the sets . We first explain that this formula implies (10.5.12). Indeed, using the invariance of under permutations and (10.5.15), we deduce from (10.5.16) that
[TABLE]
Therefore
[TABLE]
[TABLE]
[TABLE]
Adding up these equalities over all , we get (10.5.12).
It remains to prove (10.5.16). Fix . Observe that Formulas (10.5.9)–(10.5.11) simplify for 3-tuple terms. Indeed, as . Also, if , then ; if , then . By the computations above, the contribution of the 3-tuple terms (with fixed ) to is equal to where
[TABLE]
[TABLE]
It is understood that the sum runs over satisfying the indicated inequalities. The description of the loop above shows that it is freely homotopic to if and to if . Thus,
[TABLE]
and
[TABLE]
Similarly,
[TABLE]
Cyclically permuting , , , we get
[TABLE]
[TABLE]
[TABLE]
[TABLE]
The contribution of 3-tuple terms (with given ) to is the sum of 6 terms (10.5.17)–(10.5.22). Then, by (10.5.5), is the sum of these 6 terms and 6 similar terms obtained by replacing with . Under this replacement, the only change on the right-hand sides of Formulas (10.5.17)–(10.5.22) concerns the summation domain. For example, the summation domain in (10.5.17) changes from the set of triples such that to the set of triples such that . The latter condition may be rewritten as .
For , consider the 12 terms as above and pick their -summands (some of the -summands may be zero). We claim that the sum of these 12 summands is equal to for all . This clearly implies Formula (10.5.16). To prove our claim, we consider possible positions of the points on . Replacing, if necessary, by , we can assume that . This leaves us with 3 cases: (a) ; (b) , and (c) . In Case (a), only the -summands of , , and may be non-zero and their sum is
[TABLE]
[TABLE]
[TABLE]
where the last equality follows from (10.4.1) and (10.4.2). In Case (b), only the -summands of , , and may be non-zero and their sum is equal to
[TABLE]
[TABLE]
[TABLE]
In Case (c), only the -summands of , , and may be non-zero and their sum is equal to
[TABLE]
[TABLE]
[TABLE]
This proves the claim above and completes the proof of the theorem.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
- 1[AKKN 1] A. Alekseev, N. Kawazumi, Y. Kuno, F. Naef, The Goldman-Turaev Lie bialgebra in genus zero and the Kashiwara-Vergne problem. Adv. Math. 326 (2018), 1–53.
- 2[AKKN 2] A. Alekseev, N. Kawazumi, Y. Kuno, F. Naef, The Goldman-Turaev Lie bialgebra and the Kashiwara-Vergne problem in higher genera. ar Xiv:1804.09566.
- 3[AB] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. Ser. A 308, (1983), 523–615
- 4[Cb] W. Crawley-Boevey, Poisson structures on moduli spaces of representations. J. Algebra 325 (2011), 205–215.
- 5[FR] V. V. Fock, A. A. Rosly, Poisson structure on moduli of flat connections on Riemann surfaces and the r 𝑟 r -matrix. (Russian) Moscow Seminar in Math. Physics. English translation: Amer. Math. Soc. Transl. Ser. 2, 191, 67–86 (1999).
- 6[Go 1] W. M. Goldman, The symplectic nature of fundamental groups of surfaces. Adv. in Math. 54 (1984), no. 2, 200–225.
- 7[Go 2] W. M. Goldman, Invariant functions on Lie groups and Hamiltonian flows of surface group representations. Invent. Math. 85 (1986), no. 2, 263–302.
- 8[Ha] R. Hain, Hodge Theory of the Turaev Cobracket and the Kashiwara–Vergne Problem. ar Xiv:1807.09209.
