Tangles, relative character varieties, and holonomy perturbed traceless flat moduli spaces
TTANGLES, RELATIVE CHARACTER VARIETIES, AND HOLONOMYPERTURBED TRACELESS FLAT MODULI SPACES
GUILLEM CAZASSUS, CHRIS HERALD, AND PAUL KIRK
Abstract.
We prove that the restriction map from the subspace of regular points of the ho-lonomy perturbed SU(2) traceless flat moduli space of a tangle in a 3-manifold to the tracelessflat moduli space of its boundary marked surface is a Lagrangian immersion. A key ingredientin our proof is the use of composition in the Weinstein category, combined with the fact thatSU(2) holonomy perturbations in a cylinder induce Hamiltonian isotopies. In addition, we showthat ( S , introduction We gather together some of the key symplectic properties of character varieties and tracelesscharacter varieties, as well as variants which correspond to perturbed flat moduli spaces thatarise in the gauge theoretic study of 3-manifolds. Some of these results are well known to theexperts, but the proofs in the literature are framed in contexts that include gauge theory, Hodgetheory, and symplectic reduction. In the present exposition, we provide a general proof of thefact that, roughly, the character variety of a 3-manifold provides an immersed Lagrangian in thecharacter variety of its boundary surface, for any compact Lie group G , whether the 3-manifoldand its boundaries include tangles, or whether there are trace or other conjugacy restrictions onsome meridional generators, and furthermore we extend the result to the holonomy perturbedsituation. Moreover, we clarify why different natural definitions of symplectic structures on thepillowcase, arising as the character variety of the torus or the 2-sphere with four marked points,are equivalent. The results are proved using only the Poincar´e-Lefschetz duality theorem, basicalgebraic topology, and the notion of composition in the Weinstein category.Let ( X, L ) be a tangle in a compact, oriented 3-manifold X ; that is, assume that L is aproperly embedded, compact 1-manifold. For our initial discussion, we consider G = SU (2).Let π denote some holonomy perturbation data supported in a finite disjoint union of solid toriin the interior of X \ L .This data determines two moduli spaces, M π ( X, L ) , the moduli space of π holonomy-perturbed flat SU (2) connections on X \ L with traceless holo-nomy on small meridians of L , and the (well studied) moduli space: M ( ∂X, ∂ L )of flat SU (2) connections on the punctured surface ∂X \ ∂ L with traceless holonomy around themarked points ∂ L . Date : Jan. 30, 2021.2010
Mathematics Subject Classification.
Primary 57K18, 57K31, 57R58; Secondary 81T13.
Key words and phrases. holonomy perturbation, Lagrangian immersion, Floer homology, flat moduli space,traceless character variety, quilted Floer homology.CH was supported by Simons Collaboration Grants for Mathematicians, GC was funded by EPSRC grant referenceEP/T012749/1. a r X i v : . [ m a t h . G T ] F e b GUILLEM CAZASSUS, CHRIS HERALD, AND PAUL KIRK
Holonomy perturbed flat connections on a 3-manifold are flat near the boundary, and restric-tion to the boundary defines a map(1.1) r : M π ( X, L ) → M ( ∂X, ∂ L )The moduli space M ( ∂X, ∂ L ) is the cartesian product of the moduli spaces of its path com-ponents. The flat SU (2) moduli space of an oriented connected surface of genus g with k marked points is known to be a singular variety with smooth top stratum carrying a symplecticform called the Atiyah-Bott-Goldman form [AB83, Gol84]. Thus the smooth top stratum of thecartesian product M ( ∂X, ∂ L ), denoted M ( ∂X, ∂ L ) ∗ , is endowed with the product symplecticform.The main result of this article is the following theorem (Theorem 6.9 below), concerning theregular points (see Section 4.3) of the perturbed moduli space. Theorem A.
Suppose A ∈ M π ( X, L ) is a regular point. Then A has a neighborhood U so thatthe restriction r | U : U ⊂ M π ( X, L ) → M ( ∂X, ∂ L ) ∗ is a Lagrangian immersion. This is not a surprising result; indeed, many special cases are known, for example, when L is empty this result is proven in [Her94]. Our primary aim is to provide details of the assertionthat well-known arguments in the flat case extend to the holonomy-perturbed flat case when L is nonempty, in support of one claim of the main result of the article [CHKK20]. In that articleit is shown that a certain process introduced by Kronheimer-Mrowka to ensure admissibility ofbundles in instanton homology [KM11] manifests itself on the symplectic side of the Atiyah-Floerconjecture [Ati88] (i.e., Lagrangian Floer theory of character varieties or flat moduli spaces) asa certain Lagrangian immersion of a smooth closed genus 3 surface into the smooth stratum P ∗ × P ∗ of the product of two pillowcases (cf. Equation (1.3)). What is proved in [CHKK20] isthat this genus 3 surface satisfies the hypotheses of Theorem A at every point.Since it causes no extra work, we take the opportunity to provide an elementary algebraictopology proof of Theorem A in the flat case , for any compact Lie group G . The statement canbe found in Corollary 4.5. We emphasize that the flat case of Theorem A, when L is nonempty,is known (e.g., to those who attended the appropriate Oxford seminars in the 1980s) and indeeddiscussed on pages 15-16 of Atiyah’s monograph [Ati90].Having stated Theorem A and described its relation to the gauge theoretic literature using thelanguage of flat and perturbed flat moduli spaces, we note that these spaces can also be identifiedwith certain character varieties, the definitions of which do not require any of the analyticalmachinery of gauge theory; the proofs in this article are most simply explained without it, so weshall henceforth revert to the character variety terminology in our exposition. In the simplestsituation of a connected 2- or 3-manifold M with base point x , for a fixed compact Lie group G , each flat G connection determines a holonomy representation from π ( M, x ) to G , and thiscorrespondence induces a bijection between the moduli space of flat connections and the set of G representations of π modulo conjugation, known as the character variety . We describe variousextensions of the notion of the character variety corresponding to traceless and perturbed flatmoduli spaces below.We also take this opportunity to provide an elementary exposition of holonomy perturbationsfrom the perspective of fundamental groups and character varieties, in the language of compo-sition of Lagrangian immersions. Algebraic topology arguments simplify the task of explaininghow to understand the extension from the flat to the holonomy perturbed flat situation alge-braically. Other arguments, which instead appeal to Hodge and elliptic theory of perturbed flatbundles over 3-manifolds with boundary, can be made when L is empty, and can be found indetail in [Tau90, Her94]. But when L is nonempty, proper treatment of the traceless conditionrequires more subtle analytic tools. ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 3
Our focus on holonomy perturbations is motivated by the fact that they are compatible withthe analytical framework of the instanton gauge theory side of the Atiyah-Floer conjecture[Tau90, Flo88]. Holonomy perturbations modify the flatness condition (i.e. the non-linear PDE
Curvature = 0) in a specific way on a collection of solid tori, as described in Lemma 8.1 of Taubes[Tau90] (see also Lemma 16 of [Her94]). We translate this result into the language of charactervarieties in Section 6.2.In addition to proving Theorem A, we prove Theorem B, whose statement roughly says thefour punctured 2-sphere is its own traceless SU (2) moduli space . This is a variant, for ( S , U (1) × U (1) by T . Thegroup SL (2 , Z ) acts on T via(1.2) (cid:18) p rq s (cid:19) · ( e x i , e y i ) = ( e ( px + ry ) i , e ( qx + sy ) i ) . The 2-form dx ∧ dy defines the standard symplectic structure on T and is invariant under the SL (2 , Z ) action. Note that the action of the central element − ∈ SL (2 , Z ) on T defines the elliptic involution , denoted by ι ( e i x , e i y ) = ( e − i x , e − i y ); this involution has four fixed points( ± , ± T ∗ ⊂ T the complement of the four fixed points, on which ι acts freely.Then set(1.3) P = T /ι and P ∗ = T ∗ /ι. The quotient
P SL (2 , Z ) = SL (2 , Z ) / {± } acts on the smooth locus P ∗ , a 4-punctured 2-sphere,preserving the symplectic form dx ∧ dy . a bcd H H Figure 1
Next, we consider the relative character variety (de-fined below) of the 2-sphere with four marked points χ SU (2) ,J ( S , J subscript indicates the fourmeridians are sent to the conjugacy class J of traceless el-ements. This character variety is equipped with its relativeAtiyah-Bott-Goldman 2-form ω ( S , (defined below).From the presentation π ( S −
4) = (cid:104) a, b, c, d | abcd = 1 (cid:105) , we see that this group is freely generated by a, b, c . Hencea representation is uniquely defined by where it sends thegenerators a, b, c . Define the function (identifying SU (2)with the group of unit quaternions; see Section 2.1.2):ˆ ρ : P → χ SU (2) ,J ( S , , [ e i x , e i y ] (cid:55)→ (cid:2) a (cid:55)→ j , b (cid:55)→ e i x j , c (cid:55)→ e i y j (cid:3) . Theorem B.
Half Dehn twists in the two twice-punctured disks indicated in Figure 1 generate a
P SL (2 , Z ) action on χ ( S , which preserves the Atiyah-Bott-Goldman symplectic form (cid:98) ω ( S , .For some non-zero constant c , the map ˆ ρ : ( P ∗ , c dx ∧ dy ) → ( χ ( S , ∗ , (cid:98) ω ( S , ) is a P SL (2 , Z ) equivariant symplectomorphism. The proof, as well as as well as an exposition of the simpler case of SU (2) character varietyof the torus, is contained in Section 5.The outline of the proof of the flat case of Theorem A is the following. Holonomy identifiesflat moduli spaces with character varieties. When L is empty, Theorem A follows from Weil’sidentification of the Zariski tangent spaces of character varieties with cohomology and Poincar´e-Lefschetz duality. Symplectic reduction is used to extend to the case when L is nonempty.We use only Poincar´e duality with local coefficients, which is briefly reviewed, to highlight the GUILLEM CAZASSUS, CHRIS HERALD, AND PAUL KIRK fact that the proof that the image of the differential is maximal isotropic (the subtlest part ofany proof) does not depend on the deeper result that the nondegenerate 2-form is closed, i.e.,symplectic.To put Theorem A in context, notice that the Lagrangian immersion r U depends on theperturbation data π . Building on ideas from [Ati88, Flo88, Wei82] and many others, Wehrheimand Woodward developed quilts and Floer field theory [WW15, WW16]; a framework that aimsto produce a 2 + 1 or, more generally, (2 ,
0) + 1 TQFT which factors as the composition of the(perturbed) flat moduli space functor, followed by passing to the Lagrangian Floer homology ofthe flat moduli spaces of surfaces equipped with the Atiyah-Bott form. This can be consideredas an approach to realizing a bordered
Lagrangian Floer theory of character varieties of surfacesand 3-manifolds, as was done for Heegaard-Floer theory in [LOT18].Even in the lucky case that one finds perturbation data π for which M π ( X, L ) is smoothof the correct dimension and Lagrangian immerses into the smooth stratum M ( ∂ ( X, L )) ∗ , anunderstanding of how this immersion depends on π is necessary in order to extract topologicalinformation. If each perturbation curve is parallel to an embedded curve in the boundary surface,for example, then varying the perturbation parameters changes r by a Hamiltonian isotopy, andin particular has no effect on the topology of M π ( X, L ). This is discussed in Section 6.2; seealso [HK18].However, for general choices of perturbation data, perturbations typically change the topologyof M π ( X, L ); indeed, the primary purpose of using perturbation curves is precisely to smooth M π ( X, L ). As we discuss in Section 6.3, Floer field theory posits that, nevertheless, the resultingimmersions should be independent of π in a Floer-theoretic sense (i.e., isomorphic in some Fukayacategory).1.1. Acknowledgements.
The authors thank C. Judge, E. Toffoli, and L. Jeffrey for helpfulconversations. Special thanks to A. Kotelskiy, who read early versions of this article and madeimportant suggestions.2.
Review of Poincar´e duality and symplectic forms
Preliminaries and Notation.
Symplectic linear algebra.
Let A denote a finite-dimensional R vector space. A skewsymmetric bilinear form ω : A × A → R is called a symplectic form if it is nondegenerate, thatis, the map A → Hom( A, R ) given by a (cid:55)→ ω ( a, − ) is an isomorphism. If A admits a symplecticform, then its dimension is even; denote the dimension by 2 n .A coisotropic subspace C ⊂ A is a subspace satisfyingAnnihilator( C ) := { a ∈ A | ω ( c, a ) = 0 for all c ∈ C } ⊂ C. A Lagrangian subspace is a coisotropic subspace of dimension n , or, equivalently a subspace L ⊂ A satisfying Annihilator( L ) = L. Symplectic reduction refers the following process. Given any coisotropic subspace C ⊂ A , therestriction ω | C : C × C → R may be degenerate, but ω | C descends to a symplectic form (cid:98) ω on (cid:98) C := C/ Annihilator( C ). Fur-thermore if L ⊂ A is any Lagrangian subspace of ( A, ω ) , thenˆ L := (cid:0) L ∩ C (cid:1) / (cid:0) L ∩ Annihilator( C ) (cid:1) ⊂ (cid:98) C is a Lagrangian subspace of ( (cid:98) C, (cid:98) ω ). The subquotient ( (cid:98) C, (cid:98) ω ) is called the symplectic reduction of ( A, ω ) with respect to C and the Lagrangian subspace ˆ L ⊂ (cid:98) C the symplectic reduction of L withrespect to C . ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 5
The Lie group G . Let G be connected compact Lie group. Its Lie algebra g admits apositive definite symmetric ad-invariant bilinear form, so we fix one and denote it by { , } : g × g → R . Fix a conjugacy class J ⊂ G . It is well known that G embeds in some R N as analgebraic variety. Thus, if F g denotes the free group on g generators, Hom( F g , G ) = G g is anaffine real-set variety, with tangent space g g at the trivial homomorphism.Our main focus, and the context for Theorem A, concerns the case when G = SU (2), { v, w } = − Tr( vw ), and J ⊂ SU (2) is the conjugacy class of traceless matrices. To keep notationcompact, we identify SU (2) with the group of unit quaternions { a + b i + c j + d k | a + b + c + d =1 } and its Lie algebra su (2) with the purely imaginary quaternions { b i + c j + d k } . With thisidentification, Re : SU (2) → [ − ,
1] corresponds to Tr.2.1.3.
Notation used for 2- and 3-manifolds.
Throughout this article, the notation
Y, S, C, V, and F is fixed as follows.First, S denotes a possibly disconnected compact oriented surface without boundary. Denoteby S + the path connected based space obtained by first adding a disjoint base point to S , thenattaching a 1-cell from this new base point to each path component of S .Next, C denotes a finite disjoint union of m circles in the surface S . Number its components C i , i = 1 , . . . m . We assume that either each C i is oriented, or the chosen conjugacy class J ⊂ G is invariant under inversion in G , so that the condition that a homomorphism from π ( S + ) → G takes each loop C i into J makes sense.The pair ( S, C ) determines a decomposition of S into two surfaces as follows.Denote by V a tubular neighborhood of C . Then V ⊂ S is a disjoint union of oriented annuli,one around each C i .Denote by F the complementary surface, determining the decomposition:(2.1) S = F ∪ ∂F = ∂V V. Orient V and F as subsurfaces of S .Finally, Y denotes a compact, connected, and oriented 3-manifold with boundary ∂Y = S ;that is, we assume that an orientation preserving identification of ∂Y with S is given. Fix abase point in the interior of Y , and extend the inclusion S = ∂Y ⊂ Y to a based embedding of S + into Y .2.1.4. Tangles.
A tangle ( X, L ) consists of a connected compact oriented 3-manifold X , and L ⊂ X a properly embedded compact 1-submanifold with boundary. Thus L consists of adisjoint union of n intervals and k interior circles.A tangle ( X, L ) gives rise to a triple ( Y, S, C ) as above by taking Y = X \ N ( L ) , Where N ( L ) denotes a tubular neighborhood of L , and letting S = ∂Y . Then let C ⊂ S denotea union of m = n + k meridians of L , one for each component, viewed as a curve in S . Asbefore, Y, S, C determine V and F . Orientations of the tangle components are equivalent toorientations of the components of C .The process ( X, L ) ⇒ ( Y, S, C ) is nearly reversible, by attaching 2-handles to Y along C and setting L to be union of the co-cores of the 2-handles. The resulting tangle is obtainedfrom ( X, L ) by removing k disjoint small balls from the interior of X , each meeting a differentclosed component of L in a trivial arc. This results in a tangle with no closed components, butwhich has the same character variety as the starting tangle. We use the notation ( Y, S, C ) in
GUILLEM CAZASSUS, CHRIS HERALD, AND PAUL KIRK our arguments as it leads to simpler expressions, but state consequences in terms of the tangle( X, L ), as they are clearly seen as morphisms in a (more familiar) (2 ,
0) + 1 cobordism category.2.2.
Poincar´e duality and intersection forms.
We begin by recalling the statement ofPoincar´e-Lefschetz duality with local coefficients for an oriented compact connected n -manifold M with boundary ∂M equipped with a finite cell decomposition (see [CLM20] for a carefulexposition and proofs, and [Mil62] for an elementary derivation using dual regular cell decom-positions). Denote by ξ ∈ C n ( M, ∂M ; Z ) a cellular chain representing the fundamental class.Fix a base point in M and some homomorphism π M → Γ to some group Γ, and denote by M Γ → M the corresponding Γ cover, equipped with the lifted cell structure and its cellular leftΓ covering action. The cellular Z chain complexes of M Γ and ( M Γ , ∂M Γ ) are denoted by C Γ ∗ ( M )and C Γ ∗ ( M, ∂M ). These are free finitely generated left Z Γ modules: a choice of Z Γ basis is givenby arbitrarily choosing lifts of cells of M (respectively of cells which meet M \ ∂M ). A choiceof cellular approximation of the diagonal determines a chain level cap product:(2.2) ∩ ξ : Hom Z Γ ( C Γ ∗ ( M, ∂M ) , Z Γ) → C Γ ∗ ( M ) . The Poincar´e duality theorem asserts that this map is a chain homotopy equivalence of free Z Γchain complexes.Given any left Z Γ module W , Hom Z Γ ( C Γ ∗ ( M, ∂M ) , W ) = Hom Z Γ ( C Γ ∗ ( M, ∂M ) , Z Γ) ⊗ Z Γ W and hence capping with ξ induces a chain homotopy equivalence ∩ ξ : Hom Z Γ ( C Γ ∗ ( M, ∂M ) , W ) → C Γ ∗ ( M ) ⊗ Z Γ W. Now suppose that W is a finite dimensional R vector space equipped with a positive definiteinner product { , } : W × W → R and Γ → O ( W ) a representation, determining the left Z Γmodule structure on W . There is an algebraic chain isomorphism of R chain complexes: C Γ ∗ ( M ) ⊗ Z Γ W ∼ = Hom R (Hom Z Γ ( C Γ ∗ ( M ) , W ) , R ) , c ⊗ w (cid:55)→ (cid:0) h (cid:55)→ { h ( c ) , w } (cid:1) . Composing with the chain homotopy equivalence ∩ ξ , and using the universal coefficient the-orem for R , we obtain an isomorphism H ∗ ( M, ∂M ; W ) ∼ = H ∗ (Hom R (Hom Z Γ ( C Γ ∗ ( M ) , W ) , R )) = Hom R ( H ∗ ( M ; W ) , R )whose adjoint is the (by construction nondegenerate) cohomology intersection pairing over W (2.3) H ∗ ( M, ∂M ; W ) × H ∗ ( M ; W ) → R . The cohomology intersection pairing can also be expressed in terms of cup products:(2.4) ( x, y ) (cid:55)→ { x ∪ y } ∩ ξ, where { } : H n ( M ; ∂M ; W ⊗ Z Γ W ) → H n ( M ; ∂M ; R ) is induced by the coefficient homomor-phism W ⊗ Z Γ W → R determined by the bilinear form { , } .When the boundary of M is empty, this pairing, which we denote by(2.5) ω M : H ∗ ( M ; W ) × H ∗ ( M ; W ) → R is therefore a nondegenerate inner product on the R vector space H ∗ ( M ; W ).If ∂M is nonempty, precomposing the injective adjoint H ∗ ( M ; W ) → Hom R ( H ∗ ( M, ∂M ; W ); R )with the restriction map H ∗ ( M, ∂M ; W ) → H ∗ ( M ; W ) yields a map H ∗ ( M, ∂M ; W ) → Hom R ( H ∗ ( M, ∂M ; W ); R )with kernel equal to the kernel of H ∗ ( M, ∂M ; W ) → H ∗ ( M ; W ). An equivalent statement isthat the pairing(2.6) ω ( M,∂M ) : H ∗ ( M, ∂M ; W ) × H ∗ ( M, ∂M ; W ) → R ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 7 has radical equal to ker H ∗ ( M, ∂M ; W ) → H ∗ ( M ; W ).Taking gradings into account, when dim M = 2 n restriction defines a pairing(2.7) ω ( M,∂M ) : H n ( M, ∂M ; W ) × H n ( M, ∂M ; W ) → R with radical equal to ker H n ( M, ∂M ; W ) → H n ( M ; W ).When dim M = 4 (cid:96) + 2, for example when M is a surface, the pairings ω M , and ω ( M,∂M ) areskew-symmetric: ω M ( x, y ) = − ω M ( y, x ) and ω ( M,∂M ) ( x, y ) = − ω ( M,∂M ) ( y, x ) . Two and three dimensional manifolds and symplectic linear algebra
A symplectic form on the first cohomology of surface.
Recall that S + denotes thepath connected CW complex obtained by adding a disjoint base point to the oriented surface S and a 1-cell connecting each path component of S to this base point. Let Γ = π ( S + ). Itsuniversal cover (cid:101) S + → S + is a regular Γ cover, and hence so is its restriction over S (cid:101) S → S. A representation ρ : Γ → G is fixed.Then ρ determines, via the adjoint action of G , the representation ad ρ : Γ → O ( g ), and hencecohomology groups H ∗ ( S ; g ) , H ∗ ( S, ∂S ; g ) , and H ∗ ( ∂S ; g ), with, for example, H ∗ ( S ; g ) := H ∗ (Hom Z Γ ( C Γ ∗ ( S ) , g )) . If we wish to emphasize ρ , we write H ∗ ( S ; g ad ρ ).Equation (2.5) shows that since the boundary of S is empty,(3.1) ω S : H ( S ; g ) × H ( S ; g ) → R is a nondegenerate skew-symmetric form.On the other hand, if C is nonempty, so that the boundary of F is nonempty, Equation (2.7)shows that ω ( F,∂F ) : H ( F, ∂F ; g ) × H ( F, ∂F ; g ) → R is in general a degenerate skew symmetric form with radical equal toker H ( F, ∂F ; g ) → H ( F ; g ) . A degenerate skew symmetric form induces a nondegenerate form on the quotient by itsradical. The exact sequence of the pair (
F, ∂F ) shows that ω ( F,∂F ) descends to a nondegenerate skew symmetric form(3.2) (cid:98) ω F : (cid:98) H ( F ; g ) × (cid:98) H ( F ; g ) → R , where (cid:98) H ( F ; g ) = Image H ( F, ∂F ; g ) → H ( F ; g ) = ker H ( F ; g ) → H ( ∂F ; g ) . Summarizing:
Proposition 3.1.
Let S be a compact oriented surface without boundary and ρ : π S + → G arepresentation. Then ( H ( S ; g ad ρ ) , ω S ) is a symplectic vector space. If F ⊂ S is the complementof a nonempty disjoint union of annuli V , then ( (cid:98) H ( F ; g ad ρ ) , (cid:98) ω F ) is a symplectic vector space. GUILLEM CAZASSUS, CHRIS HERALD, AND PAUL KIRK
In the following diagram of inclusions, the two vertical and two horizontal rows are exact,and the isomorphisms excisions:(3.3) H ( S, F ) H ( V, ∂V ) H ( S, V ) H ( S ) H ( V ) H ( S, V ) H ( F, ∂F ) H ( F ) H ( ∂F ) H ( F, ∂F ) q ∼ = q αp ∼ = βp γp ∼ = a b c Proposition 3.2.
The kernel of β is a cosiotropic subspace of H ( S ; g ) with annihilator ker p ,and hence p induces a symplectomorphism ker β/ ker p ∼ = −→ (cid:98) H ( F ; g ) . Proof.
The surface V is a disjoint union of annuli. The composition { }× S ⊂ ∂ ( I × S ) ⊂ I × S a homotopy equivalence, and hence the restriction H ( I × S ) → H ( ∂ ( I × S )) is injective withany coefficients. Hence p is injective, and q = 0.If s ∈ H ( S ) satisfies ω S ( s, α ( y )) = 0 for all y ∈ H ( S, V ), then ω F,∂F ( p ( s ) , a ◦ p ( y )) = 0for all y ∈ H ( S, V ). Hence ω F,∂F ( p ( s ) , a ( z )) = 0 for all z ∈ H ( F, ∂F ). Since the pairing H ( F, ∂F ) × H ( F ) → R is nondegenerate (Equation (2.3)), this implies that p ( s ) = 0. Inother words, the annihilator of image α = ker β is contained in ker p .Since q = 0, ker p ⊂ ker β . Therefore ker β contains its annihilator and hence is coisotropic.It remains to show that ker p = image q is contained in the annihilator of ker β . Given x ∈ H ( S, V ) and y ∈ H ( S, F ), ω S ( α ( x ) , q ( y )) = 0 since the cup product H ( S, V ) × H ( S, F ) → H ( S )factors through H ( S, F ∪ V ) = 0 (see Equation (2.4)). (cid:3) Corollary 3.3. If L ⊂ H ( S ; g ad ρ ) is any Lagrangian subspace, then the image p ( L ∩ ker β ) ⊂ (cid:98) H ( F ; g ad ρ ) is a Lagrangian subspace. Restriction from a 3-manifold with boundary.
Recall that Y is a compact, connected,oriented 3-manifold with boundary S = ∂Y , extended to an embedding S + ⊂ Y .Assume that the representation ρ : π ( S + ) → G is a restriction of a representation (of thesame name) ρ : π ( Y ) → G . Lemma 3.4.
The image of the restriction map, L Y := Image r : H ( Y ; g ad ρ ) → H ( S ; g ad ρ ) is a Lagrangian subspace of ( H ( S ; g ad ρ ) , ω S ) .Proof. In the following diagram, the middle row is part of the exact sequence of the pair (
Y, S ).The vertical arrows all isomorphisms, with the downward pointing isomorphisms Poincar´e-Lefschetz duality. The diagram commutes up to sign [Spa95]. We suppress the g ad ρ coefficients. ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 9
Hom( H ( Y ); R ) Hom( H ( S ); R ) Hom( H ( Y, S ); R ) H ( Y ) H ( S ) H ( Y, S ) H ( Y, S ) H ( S ) H ( Y ) δ ∗ r νδ The image of r equals the kernel of ν , which is isomorphic to the kernel of δ ∗ . The kernel of δ ∗ is isomorphic to the cokernel of its dual δ , which in turn is isomorphic to the cokernel of r .Hence the image and cokernel of r are isomorphic, and so dim(image( r )) = dim H ( S ).Commutativity of the following diagram is a consequence of naturality of cup product andPoincar´e duality: H ( Y ; g ) × H ( Y ; g ) H ( Y ; R ) H ( Y, S ; R ) H ( S ; g ) × H ( S ; g ) H ( S ; R ) H ( S ; R ) R H ( Y ; R ) ∪ { , } r × r ∩ [ Y,S ] ∪ { , } ∩ [ S ] (cid:15)(cid:15) Exactness of the vertical sequence shows that the dotted arrow is zero, which implies that theimage of r is isotropic, and therefore Lagrangian. (cid:3) Recall that the boundary S = ∂Y is equipped with a embedded collection C ⊂ S of circles,with tubular neighborhood V and complementary subsurface F , producing the decomposition S = F ∪ V as in Equation (2.1). Consider the ladder of exact sequences, with all maps inducedby inclusions. The g ad ρ coefficients are supressed. The bottom two rows coincide with those ofDiagram (3.3).(3.4) · · · H ( Y, V ) H ( Y ) H ( V ) · · ·· · · H ( S, V ) H ( S ) H ( V ) · · ·· · · H ( F, ∂F ) H ( F ) H ( ∂F ) · · · Ar Brαp ∼ = βp p a b A diagram chase shows that Image r ∩ ker β = Image α ◦ r . Hence p (Image r ∩ ker β ) = Image a ◦ p ◦ r : H ( Y, V ) → H ( F ) ⊂ ker b = (cid:98) H ( F )Lemma 3.4 and Corollary 3.3 imply the following. Corollary 3.5.
Suppose that ∂Y = S = V ∪ F with V a disjoint union of annuli.Then L Y,V := Image H ( Y, V ; g ad ρ ) → H ( F ; g ad ρ ) ⊂ (cid:98) H ( F ; g ad ρ ) is a Lagrangian subspace. Moreover, L Y,V is the symplectic reduction of L Y with respect to ker β : H ( S ; g ad ρ ) → H ( V ; g ad ρ ) . Character varieties, relative character varieties, and their tangent spaces
Character varieties.Definition 4.1.
Given a finitely presented group Γ, its G character variety χ G (Γ) is the realsemi-algebraic set defined to be the orbit space of the G -conjugation action on the affine R -algebraic set Hom(Γ , G ). A set of g generators of Γ embeds Hom(Γ , G ) in G g equivariantly, and,since G is a compact Lie group, Hom(Γ , G ) is an affine R -algebraic set, with orbit space χ G (Γ)a semi-algebraic set. We call χ G (Γ) and (orbit spaces of conjugation invariant) Zariski closedsubsets character varieties. If Z is a path connected space, write χ G ( Z ) instead of χ G ( π ( Z )).The character variety of a non-path connected space, by definition , is the cartesian product ofthe character varieties of its path components.As observed by Weil [Wei64], the formal tangent space at the conjugacy class of any repre-sentation ρ : Γ → G to the character variety χ G (Γ) is naturally identified with first cohomology:(4.1) T ρ χ G (Γ) = H (Γ; g ad ρ ) . We take Equation (4.1) as the definition of the formal tangent space at ρ for any ρ ∈ χ G (Γ).Recall that for any space X with fundamental group Γ, H (Γ; g ad ρ ) and H ( X ; g ad ρ ) are canon-ically isomorphic.Weil’s argument is based on the calculation that if a path of representations is expressed inthe form ρ s = exp( α s ) ρ for some path α s : Γ → g , then dds | s =0 α s : Γ → g is a 1-cocycle [Bro94].4.2. Relative character varieties.
As above, assume that Y is a compact connected 3-manifold with boundary S = ∂Y , C ⊂ S is a union of m circles C i , V is the tubular neighborhoodof C in S , with complementary surface F .Assume further that either J is invariant via the inversion map of G (as is the case for theconjugacy class of traceless matrices in SU (2)) or else assume that every circle C i is equippedwith an orientation.Then define the relative character variety to be(4.2) χ G,J ( Y, C ) ⊂ χ G ( Y )to be the subvariety consisting of conjugacy classes of representations π ( Y ) → G which send(any based representative of the homotopy class of) each circle in C into J . Define the formaltangent space of χ G,J ( Y, C ) at ρ to be(4.3) T ρ χ G,J ( Y, C ) = ker H ( Y ; g ad ρ ) → H ( C ; g ad ρ )Given an oriented surface F , define χ G,J ( F, ∂F ) = χ G,J ( F × [0 , , ∂F × { } ) . Its formal tangent space is(4.4) T ρ χ G,J ( F, ∂F ) = ker H ( F ; g ad ρ ) → H ( ∂F ; g ad ρ ) = (cid:98) H ( F ; g ad ρ )4.3. Regular points.
The term “formal tangent space” may be replaced by its usual elementarydifferential topology notion in neighborhoods of regular points of χ G,J ( Y, C ) and χ G,J ( F, ∂F ),as we next explain. In brief, as elsewhere in gauge theory, a regular point is one which has aneighborhood diffeomorphic to Euclidean space of the correct index-theoretic dimension. Weprovide a stripped-down explanation, suitable for our purposes, of what this means, for thebenefit of the reader.First, given a connected compact surface F (with possibly empty boundary) we call ρ ∈ χ G,J ( F, ∂F ) a regular point provided ρ has a neighborhood U ⊂ χ G,J ( F, ∂F ) so thatdim (cid:98) H ( F ; g ad ρ (cid:48) ) is independent of ρ (cid:48) ∈ U . ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 11
Next, given a disjoint union of connected compact surfaces F = (cid:116) i F i , call ρ ∈ χ G,J ( F, ∂F ) = (cid:89) i χ G,J ( F i , ∂F i )a regular point provided each of its components is a regular point.Finally, for a pair ( Y, C ) (with C possibly empty) ρ ∈ χ G,J ( Y, C ) is called a regular pointprovided ρ admits a neighborhood U ⊂ χ G,J ( Y, C ) so that for all ρ (cid:48) ∈ U : • The restriction map χ G,J ( Y, C ) → χ G,J ( F, ∂F ) takes ρ (cid:48) to a regular point, and • dim T ρ (cid:48) χ G,J ( Y, C ) = dim T ρ χ G,J ( F, ∂F ) = (cid:88) i dim T ρ χ G,J ( F i , ∂F i )Hence, if ρ ∈ χ G,J ( Y, C ) is a regular point, the map χ G,J ( Y, C ) → χ G,J ( F, ∂F ), which takesa representation of a 3-manifold group to its restriction to the boundary surface, is, near ρ , asmooth map of a smooth n -disk into R n for some n .Notice that χ G,J ( Y, C ) is the preimage of the point ( J, · · · , J ) under the restriction map χ G ( Y ) → χ G ( C ) = m (cid:89) i =1 χ G ( C i ) = ( G/ conjugation ) m . and that χ G,J ( F, ∂F ) is the preimage of the point ( J, · · · , J ) under the restriction map χ G ( F ) → χ G ( ∂F ) . Since C ⊂ V is a deformation retract, the exact sequence of the pair shows that T ρ χ G,J ( Y, C ) ∼ = Image H ( Y, V ; g ad ρ ) → H ( Y ; g ad ρ ) . The image of the differential of the restriction map χ G,J ( Y, C ) → χ G,J ( F, ∂F ) at ρ ∈ χ G,J ( Y, C ) is therefore identified with L Y,V , the image of the composition H ( Y, V ; g ad ρ ) → H ( F, ∂F ; g ad ρ ) → (cid:98) H ( F ; g ad ρ ) , which by Corollary 3.5 is a Lagrangian subspace of ( H ( F ; g ad ρ ) , (cid:98) ω F ). Summarizing: Corollary 4.2. If ρ ∈ χ G,J ( Y, C ) is a regular point, then there exists a neighborhood of ρ so thatthe differential of the restriction χ G,J ( Y, C ) → χ G,J ( F, ∂F ) at any point ρ (cid:48) in this neighborhoodhas image a Lagrangian subspace of ( (cid:98) H ( F ; g ad ρ (cid:48) ) , (cid:98) ω F ) . It is known that for surfaces, with the exception of a few low genus cases, the regular pointscoincide with the irreducible representations.4.4.
Symplectic structure.
The proof of Corollary 4.2 does not rely of the following fun-damental result of Atiyah-Bott [AB83] and its extensions due to Goldman [Gol84], Karshon[Kar92], Biswas-Guruprasad [BG93], King-Sengupta [KS96], Guruprasad-Huebschmann-Jeffrey-Weinstein [GHJW97].
Theorem 4.3.
On the top stratum of regular points of χ G ( S ) and χ G,J ( F, ∂F ) , the 2-forms ω S and (cid:98) ω F are closed, that is, are symplectic forms. In light of this result, Corollary 4.2 can be restated as follows.
Theorem 4.4.
Suppose that ρ ∈ χ G,J ( Y, C ) is a regular point. Then there exists a neighborhood U of ρ in χ G,J ( Y, C ) so that the restriction of r to U , r | U : U → χ G,J ( F, ∂F ) , is a Lagrangian embedding.In particular, if χ G,J ( Y, C ) contains only regular points, then the restriction map is a La-grangian immersion. Given a tangle ( X, L ), We write χ G,J ( X, L ) rather than χ G,J ( Y, C ) where
Y, S, C, F and V are determined by ( X, L ) as in Section 2.1.4. Also write χ G,J ( ∂ ( X, L )) rather than χ G,J ( F, ∂F ).Then Theorem 4.4 can be restated in the new notation as follows.
Corollary 4.5.
Suppose that ρ ∈ χ G,J ( X, L ) is a regular point. Then there exists a neighborhood U of ρ in χ G,J ( X, L ) so that the restriction of r to Ur | U : U → χ G,J ( ∂ ( X, L )) , is a Lagrangian embedding.In particular, if χ G,J ( X, L ) contains only regular points, then the restriction map is a La-grangian immersion. In what follows, we simply write χ ( A ) for χ G ( A ) and χ ( A, B ) for χ G,J ( A, B ). In addition,we denote by χ ( S ) ∗ ⊂ χ ( S ) and χ ( F, ∂F ) ∗ ⊂ χ ( F, ∂F ) the smooth top strata , as real algebraicvarieties, equipped with the symplectic forms ω S , (cid:98) ω F .5. Three pillowcases
In this section, take G = SU (2), and let J ⊂ SU (2) be the conjugacy class of unit quaternionswith zero real part, so J = su (2) ∩ SU (2), a 2-sphere.5.1. The quotient of the torus by the elliptic involution.
Let T denote U (1) × U (1) withits symplectic form dx ∧ dy and symplectic SL (2 , Z ) action given in Equation 1.2. Multiplicationby − ∈ SL (2 , Z ) is central and hence the quotient P = T / {± } inherits a P SL (2 , Z ) = SL (2 , Z ) / {± } action. The quotient map T → P is the 2-fold branchedcover of the 2-sphere with branch points the four points { ( ± , ± } . The complement, P ∗ of thefour branch points is a smooth surface, with symplectic form dx ∧ dy and symplectic P SL (2 , Z )action, and the restriction T ∗ → P ∗ is a smooth symplectic 2-fold covering map.5.2. The SU (2) character variety of the genus 1 surface. Consider the genus one closedoriented surface T , equipped with generators µ, λ ∈ π T represented by a pair of oriented loopsintersecting geometrically and algebraically once. The Dehn twists D µ , D λ about µ and λ inducethe automorphisms D µ : (cid:0) µ (cid:55)→ µ, λ (cid:55)→ µλ (cid:1) and D λ : (cid:0) µ (cid:55)→ µλ − , λ (cid:55)→ λ (cid:1) . of π ( T, t ) ∼ = Z µ ⊕ Z λ (where t lies outside the support of these 2 Dehn twists). Theseautomorphisms induce, by precomposition, homeomorphisms D ∗ µ , D ∗ λ : χ ( T ) → χ ( T ) . Let H ( T ) = Hom( π ( T ) , SU (2)). Denote by p : H ( T ) → χ ( T )the (surjective) orbit map of the conjugation action.Define(5.1) ρ : T → H ( T ) , ρ ( e x i , e y i ) = (cid:0) µ (cid:55)→ e x i , λ (cid:55)→ e y i (cid:1) . Proposition 5.1.
The Dehn twists D µ and D λ define a symplectic P SL (2 , Z ) action on ( χ ( T ) , ω T ) . The map ρ of Equation (5.1) descends to a P SL (2 , Z ) equivariant homeomorphism ρ : P → χ ( T ) ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 13 which restricts to a
P SL (2 , Z ) equivariant symplectomorphism ( P ∗ , c dx ∧ dy ) → ( χ ( T ) ∗ , ω T ) on the top strata of the SU (2) character varieties.Proof. It is well known that ρ : P → χ ( T ) is a well defined homeomorphism, as well as an analyticdiffeomorphism of the smooth strata P ∗ → χ ( T ) ∗ . This follows simply from the observationsthat the fundamental group of T is abelian and the Weyl group of SU (2) is Z / ρ : ( P ∗ , dx ∧ dy ) → ( χ ( T ) ∗ , ω T ) is a symplectomorphism. Since this is a localstatement, we work in R for simplicity. Define m = ρ ◦ e : R → χ ( T ), where e ( x, y ) = ( e x i , e y i ).The differential of m at ( x, y ) ∈ R , dm : T ( x,y ) R → T m ( x,y ) ( SU (2) × SU (2)) is given by dm ( ∂∂x ) = (cid:0) µ (cid:55)→ e x i i , λ (cid:55)→ (cid:1) , dm ( ∂∂y ) = (cid:0) µ (cid:55)→ , λ (cid:55)→ e y i i (cid:1) . Following Weil, left translation in SU (2) × SU (2) identifies these with the su (2)-valued 1-cochains z x = ( µ (cid:55)→ i , λ (cid:55)→ , z y = ( µ (cid:55)→ , λ (cid:55)→ i ) ∈ C ( T ; su (2) ad m ( x,y ) ) . The subspaces L = i R and V = j R + k R are invariant and complementary with respect toad m ( x, y ), and therefore H ( T ; su (2) ad m ( x,y ) ) = H ( T ; L ad m ( x,y ) ) ⊕ H ( T ; V ad m ( x,y ) ) . Note that the action ad m ( x, y ) on L is trivial, since L = i R and m ( x, y ) has values in the abeliansubgroup { e i u } . Hence H ( T ; L ad m ( x,y ) ) ∼ = H ( T ; R ) ∼ = R . The branched cover m : R → χ ( T ) is a local diffeomorphism near any ( x, y ) ∈ ( R ) ∗ = R \ ( π Z ) , and hence it follows that for such ( x, y ), H ( T ; R ) ⊗ R i = H ( T ; L ad m ( x,y ) ) = Span { z x , z y } and H ( T ; V ad m ( x,y ) ) = 0(these calculations can also be easily checked directly), so that T m ( x,y ) ( χ ( T )) = H ( T ; R ) ⊗ R i . The cup product H ( T ; L ad m ( x,y ) ) × H ( T ; L ad m ( x,y ) ) → H ( T ; R )is thereby identified with H ( T ; R ) ⊗ R i × H ( T ; R ) ⊗ R i → H ( T ; R )( z ⊗ i , z ⊗ i ) = ( z ∪ z ) · ( − Re( ii )) = z ∪ z . In particular, z x = µ ∗ ⊗ i and z y = λ ∗ ⊗ i , where µ ∗ , λ ∗ ∈ H ( T ; Z ) = Hom( H ( T ; Z ) , Z ) is thebasis dual to µ, λ . This basis is symplectic with respect to the (usual, untwisted) intersectionform. Hence (cid:0) ( p ◦ m ) ∗ ( ω T ) (cid:1) | ( x,y ) ( ∂∂x , ∂∂y ) = ω T ( z x , z y ) = ( µ ∗ ∪ λ ∗ ) ∩ [ T ] = 1 = dx ∧ dy ( ∂∂x , ∂∂y ) = 1 . Naturality of cup products shows that D ∗ µ , D ∗ λ preserve the symplectic form ω T . Next, D ∗ µ ( m ( x, y )) = m ( x, y ) ◦ D µ = (cid:0) µ (cid:55)→ m ( x, y )( µ ) = e x i , λ (cid:55)→ m ( x, y )( µλ ) = e ( x + y ) i (cid:1) and similarly D ∗ λ ( m ( x, y )) = m ( x, y ) ◦ D λ = (cid:0) µ (cid:55)→ e ( x − y ) i , λ (cid:55)→ e y i (cid:1) It follows that the subgroup of Homeo( χ ( T )) generated by D ∗ µ and D ∗ λ pulls back, via thehomeomorphism ρ : P → χ ( T ), to the subgroup of P SL (2 , Z ) generated by ρ ∗ ( D ∗ µ ) = (cid:18) (cid:19) , ρ ∗ ( D ∗ λ ) = (cid:18) − (cid:19) . These two matrices generate
P SL (2 , Z ) ([Ser80]), finishing the proof. (cid:3) The solid torus and the restriction to its boundary.
Let X denote the solid torus withboundary T . Equip T with based loops µ, λ generating π ( T ), so that µ is trivial in π ( X ) and λ generates π ( X ). Then χ ( X ) = χ ( Z λ ) = SU (2) / conjugation . An explicit slice of the conjugation action Hom( Z λ, SU (2)) → χ ( Z λ ) is given by the map(5.2) [0 , π ] → Hom( Z λ, SU (2)) , s (cid:55)→ ( λ (cid:55)→ e i s )with composition [0 , π ] → χ ( X ) Re λ −−→ [ − ,
1] equal to the analytic isomorphism cos( s ). Simplecohomology calculations show dim H ( Z ; su (2)) equals 1 when 0 < s < π and equals 3 when s = 0 or π . This shows that the interior of the interval forms the smooth top stratum of χ ( Z ),and the endpoints are singular.Since µ = 1 ∈ π ( X ), the restriction-to-the boundary map(5.3) χ ( X ) → χ ( T )is is easily computed, in P , to be the smooth (necessarily Lagrangian) embedded arc given by:(5.4) [0 , π ] (cid:51) s (cid:55)→ [ e s i , ∈ P with endpoints at [ − ,
1] and [1 , The traceless SU (2) character variety of the 4-punctured 2-sphere. Let 4 D ⊂ S be four disjoint open disks, and set F = S \ D , ∂F = 4 S . Then π ( F ) has presentation π ( F ) = (cid:104) a, b, c, d | abcd = 1 (cid:105) and is free on a, b, c . Set H ( S ,
4) = { ρ ∈ Hom( π ( F ) , SU (2)) | ρ ( a ) , ρ ( b ) , ρ ( c ) , ρ ( d ) ∈ J } . Let p : H ( S , → χ ( S ,
4) denote the orbit map of the conjugation action.Define(5.5) ˆ ρ : T → H ( S , , ( e x i , e y i ) (cid:55)→ (cid:0) a (cid:55)→ j , b (cid:55)→ e x i j , c (cid:55)→ e y i j (cid:1) . A half Dehn twist of ( D , p, q ), where p, q are a fixed pair of interior points, is a homeomor-phism of the disk which fixes the boundary and permutes p and q , and which generates theinfinite cyclic mapping class group rel boundary of a disk with two marked interior points. Thegenerator which veers to the right is called a positive half Dehn twist.5.3.1. Proof of Theorem B.
We prove the Theorem B of the introduction.
Proof.
That ˆ ρ : P → χ ( S ,
4) is a well defined homeomorphism, as well as an analytic diffeomor-phism of the smooth strata P ∗ → χ ( S , ∗ , is simple; its proof can be found in [Lin92, HK98].We show that ˆ ρ : ( P ∗ , c dx ∧ dy ) → ( χ ( S , ∗ , (cid:98) ω ( S , ) is a symplectomorphism, for someconstant c . Since this is a local statement, we work in R for simplicity. Define ˆ m = ˆ ρ ◦ e : R → χ ( S , e ( x, y ) = ( e x i , e y i ).The differential of ˆ m at ( x, y ) ∈ R is given by d ˆ m ( ∂∂x ) = (cid:0) a (cid:55)→ , b (cid:55)→ e x i ij , c (cid:55)→ (cid:1) , d ˆ m ( ∂∂y ) = (cid:0) a (cid:55)→ , b (cid:55)→ , c (cid:55)→ e y i ij (cid:1) . Translation in SU (2) identifies these with the su (2)-valued 1-cochains z x = (cid:0) a (cid:55)→ , b (cid:55)→ − i , c (cid:55)→ (cid:1) , z y = (cid:0) a (cid:55)→ , b (cid:55)→ , c (cid:55)→ − i (cid:1) . ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 15
Since ˆ m is a local diffeomorphism away from ( π Z ) [HHK14], and ∂∂x , ∂∂y span T ( x,y ) R , thecohomology classes [ z x ] , [ z y ] span (cid:98) H ( S − D ; su (2) ad ˆ m ( x,y ) ) = T [ ˆ m ( x,y )] ( χ ( S , . For any ( x, y ) ∈ R \ ( π Z ) , the adjoint action ad ˆ m ( x, y ) : π ( S − D ) → GL ( su (2)) reducesas the direct sumad ˆ m ( x, y ) = ad ˆ m ( x, y ) ⊕ ad ˆ m ( x, y ) : π ( S − D ) → GL ( R i ) × GL ( C j )and hence (cid:98) H ( S − D ; su (2) ad ˆ m ( x,y ) ) = (cid:98) H ( S − D ; R i ad ˆ m ( x,y ) ) ⊕ (cid:98) H ( S − D ; C j ad ˆ m ( x,y ) ) . Since R i and C j are orthogonal, the symplectic form (cid:98) ω S , splits orthogonally (cid:98) ω S , = (cid:98) ω S , ⊕ (cid:98) ω S , . The cocyles z x and z y lie in the first summand, and hence they span the first summand andthe second summand is zero (these two facts can also be easily calculated directly). Hence (cid:98) ω S , = (cid:98) ω S , . The representation on the first summand independent of ( x, y ): indeed a, b, and c (and hencealso d ) act by − x, y . The cocycles z x , z y are independent of x, y , and henceˆ m ∗ ( (cid:98) ω S , ) | ( x,y ) ( ∂∂x , ∂∂y ) = (cid:98) ω S , ( z x , z y ) = cdx ∧ dy ( ∂∂x , ∂∂y )for a non-zero constant c (since z x , z y span).The half-Dehn twists along the disks H and H illustrated in Figure 1 induce automorphismsof π ( S \ , s ), for a base point chosen outside the supports of H and H , as indicated in thefigure. These automorphisms are given by H : a (cid:55)→ a, b (cid:55)→ b, c (cid:55)→ d = ¯ c ¯ b ¯ aH : a (cid:55)→ a, b (cid:55)→ cd ¯ c = ¯ b ¯ a ¯ c, c (cid:55)→ c and induce homeomorphisms H ∗ , H ∗ : χ ( S , → χ ( S , H ∗ , H ∗ preserve the symplectic form (cid:98) ω ( S , .Next: H ∗ ( m ( x, y )) = (cid:0) a (cid:55)→ m ( x, y )( a ) = j , b (cid:55)→ m ( x, y )( b ) = e x i , c (cid:55)→ m ( x, y )(¯ c ¯ b ¯ a ) = e ( y − x ) i j (cid:1) = m ( x, y − x )and similarly H ∗ ( m ( x, y )) = (cid:0) a (cid:55)→ j , b (cid:55)→ m ( x, y )(¯ b ¯ a ¯ c ) = e ( x + y ) i j , c (cid:55)→ e y i ) = m ( x + y, y ) . It follows that the subgroup of Homeo( χ ( S , H ∗ and H ∗ pulls back, via thehomeomorphism ˆ ρ : P → χ ( T ), to the subgroup of P SL (2 , Z ) generated byˆ ρ ∗ ( H ∗ ) = (cid:18) −
10 1 (cid:19) , ˆ ρ ∗ ( H ∗ ) = (cid:18) (cid:19) . These two matrices generate
P SL (2 , Z ) ([Ser80]), finishing the proof. (cid:3) Mapping classes of ( S ,
4) permute the four punctures. The subgroup of the mapping classgroup of ( S ,
4) which fixes the point labelled a in Figure 1 can be shown to act on χ ( S , P SL (2 , Z ) (see, e.g., [FM12]), generated by these half twists.6. Perturbations
The holonomy perturbation process is easily understood, as well as motivated, in the languageof Weinstein composition of Lagrangian immersions.
Composition.
Given any two (set) maps α : A → M and β = β M × β N : B → M × N, define the composition ( A × M B, β αN ) by(6.1) A × M B := { ( a, b ) ∈ A × B | α ( a ) = β M ( b ) } = ( α × β M ) − (∆ M )and(6.2) β αN : A × M B → N, β αN ( a, b ) := β N ( b ) . Composition in character varieties.
Recall that if ρ : Γ → G is a homomorphism, Stab ( ρ ) = { g ∈ G | gρ ( γ ) g − = ρ ( γ ) for all γ ∈ Γ } . The proof of the following simple lemma can be found in [HHK14, Lemma 4.2].
Lemma 6.1.
Suppose that Γ , Γ , H are groups, h : H → Γ , h : H → Γ homomorphisms.Set Γ = Γ ∗ H Γ , the pushout along h , h . Then there is a surjection χ (Γ) → χ (Γ ) × χ ( H ) χ (Γ ) with fiber over ([ ρ ] , [ ρ ]) the double coset Stab ( ρ ) \ Stab ( ρ | H ) /Stab ( ρ ) . The fibers
Stab ( ρ ) \ Stab ( ρ | H ) /Stab ( ρ ) are called gluing parameters . Proposition 6.2. If Z = Z ∪ Σ Z is a decomposition of a compact 3-manifold along a closedseparating surface Σ , with π (Σ) → π ( Y ) surjective, then χ ( Z ) → χ ( Z ) × χ (Σ) χ ( Z ) is ahomeomorphism, in fact, an algebraic isomorphism.Proof. Choose ([ ρ ] , [ ρ ]) ∈ χ ( Z ) × χ (Σ) χ ( Z ). Since π (Σ) → π ( Y ) is surjective, ρ and ρ | π (Σ) have the same image, and hence equal stabilizers. Therefore Stab ( ρ ) \ Stab ( ρ | Σ ) /Stab ( ρ ) is asingle point, and the proof follows from Lemma 6.1. (cid:3) Corollary 6.3. If Z = Z ∪ Σ Z with Z a handlebody, then χ ( Z ) = χ ( Z ) × χ (Σ) χ ( Z ) . Composition in the Weinstein category.
The Weinstein category [Wei82] is, roughly speak-ing, a category with objects symplectic manifolds and morphisms Lagrangian immersions. Com-position is not always defined, however. The following criterion ensures that a composition ofLagrangian immersions is defined.
Lemma 6.4. [GS82, § , [BW18, lemma 2.0.5] Suppose that
M, N are symplectic manifolds, α : A → M and β = β M × β N : B → M − × N are Lagrangian immersions (with M − obtained from M by reversing the sign of the symplecticform). If α × β M is transverse to the diagonal ∆ M ∈ M × M , then A × M B is a smooth manifoldand β αN : A × M B → N is a Lagrangian immersion. When the transversality assumption in Lemma 6.4 holds, one says the composition ( A × M B, β αN ) of ( A, α ) and ( B, β ) is defined and immersed .In cases where the transversality assumption in Lemma 6.4 does not hold, a differential topo-logical approach to remedying the situation would be to deform either or both of the immersions α, β . In order to retain the symplectic properties, one would typically deform them by Hamil-tonian flows. In the context in this article, where character varieties correspond to flat modulispaces in the gauge theoretic framework, we seek deformations in Lemma 6.4 that correspondto holonomy perturbations in the gauge theory context; we describe these in the next section. ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 17
SU(2) and holonomy perturbations.
We return to the pair (
Y, C ), determining S = V ∪ F as above, so S = ∂Y and V = nbd ( C ).Suppose that e : D × S (cid:44) → Int ( Y ) is an embedding of a solid torus. Denote its image by Y , the closure of the complement by Y , and the separating torus by T , so that Y = Y ∪ T Y . Corollary 6.3 shows that χ ( Y, C ) = χ ( Y ) × χ ( T ) χ ( Y , C ) , so that χ ( Y, C ) → χ ( F, ∂F ) is exhibited as the composition of α : χ ( Y ) → χ ( T ) and β : χ ( Y , C ) → χ ( T ) × χ ( F, ∂F ) . For a deformation α π : χ ( Y ) → χ ( T ) of α , or more generally a family of functions α π , π ∈ U with U a manifold (for example, a small open interval), we can view the deformed composition(6.3) χ π ( Y, C ) := χ ( Y ) ◦ π χ ( Y , C ) , β α π : χ ( Y ) ◦ π χ ( Y , C ) → χ ( F, ∂F )as a perturbed character variety with perturbation data π .Restrict to the case when G = SU (2) and J is the conjugacy class of imaginary unitquaternions. Then χ ( Y ) is simply an arc [ e is ] , ≤ s ≤ π , and the immersion to χ ( ∂Y ) is α : [ e is ] (cid:55)→ [ e is ,
1] in the pillowcase, described in (5.3) and (5.4). This map descends from thesmooth, Z equivariant map e is → ( e is ,
1) from S to T .Let f : R → R be a smooth, odd, 2 π periodic function. Then f determines a Z equivariantHamiltonian deformation e is (cid:55)→ ( e is , e if ( s ) ), inducing the deformation of α given by(6.4) α f : [0 , π ] → χ ( S × S ) , α f ( s ) = [ e i s , e i f ( s ) ] . The definition extends easily to the setting of a finite disjoint collection of embeddings ofsolid tori e = { e i } ki =1 of disjointly embedded solid tori in Y and a corresponding collection ofsmooth, odd, periodic functions f , . . . , f k as above. Denote by π this set of perturbation data { ( e i , f i ) } ki =1 . Letting Y denote the union of the solid tori, Equation (6.3) defines a way todeform the character variety.Notice that it suffices to think of e as framed link in Y , since isotopic embeddings yield equalperturbed character varieties. Definition 6.5.
Let V be the vector space of smooth, odd, 2 π periodic functions R → R . Fix( Y, C ) as above, where C may or may not be nonempty). Given perturbation data π = ( e, f ) = ( (cid:116) ki =1 e i : D × S ⊂ Y, f = ( f i ) ∈ V k ) , define χ π ( Y, C ) to be the resulting perturbed (traceless) character variety.In light of Equation (6.4), χ π ( Y, C ) has the following explicit description.
Proposition 6.6.
Let ( Y, C ) be a compact oriented 3-manifold with a collection of curves C in its boundary. Let π = ( e, f ) be a choice of perturbation data, and define Y ⊂ Y to be thedisjoint union of solid tori (cid:116) i e i ( D × S ) . Finally define λ i , µ i ∈ π ( Y \ Y ) to be the loops e i ( S × { } ) and e i ( { } × ∂D ) , connected to the base point in some way.Then χ π ( Y, C ) = { ρ ∈ χ ( Y \ Y , C ) | if ρ ( λ i ) = e i s , then ρ ( µ i ) = e i f ( s ) } . Theorem 6.7 ([Tau90, Her94]) . Given a set of perturbation data π = ( e, f ) as in Definition 6.5,there is a holonomy perturbation h ( e,f ) of the flatness equation on SU (2) connections for whichthe perturbed flat moduli space is identified with the perturbed character variety as describedabove. Remark 6.8.
More flexible holonomy perturbations can be defined using a solid handlebody,rather than disjoint solid tori (see, for example, [Flo88]). In light of Corollary 6.3, there is asimilar composition interpretation of the perturbed character variety, with Y the handlebody. But in this setting, an explicit description of the perturbed character variety and the counterpartto the restriction map in Equation (6.4), are not known to the authors.We can now prove the following theorem, which is equivalent to Theorem A.
Theorem 6.9.
Suppose A ∈ χ π ( Y, C ) is a regular point. Then A admits a neighborhood U sothat r | U : U → χ ( F, ∂F ) ∗ is a Lagrangian embedding.Proof. Apply Lemma 6.4. (cid:3)
Dependence on perturbations.
In [Her94], the second author showed that (when L is empty) different holonomy perturbations in general yield Legendrian cobordant immersions.Unfortunately this usually does not guarantee that they have isomorphic Floer homology. Wenow outline how one can address this point using Wehrheim and Woodward’s quilt theory[WW10] (a rigorous formulation of the Weinstein category [Wei10]), as well as its extension tothe immersed case as developed by Bottman-Wehrheim [BW18].The reader will find a discussion of holonomy perturbations in cylinders ( S, p ) × [0 , p a finite set of points in a surface S , in [HK18]. In particular, Theorem 6.3 of that article statesthe following. Take perturbation data π with framed perturbation curve obtained by pushinga simple closed curve in S \ p into the interior of S × I . Then define the 1-parameter family ofholonomy perturbations sπ, s ∈ [0 , (cid:15) ).The restriction χ sπ (( S, p ) × I ) → χ ( S , p ) × χ ( S , p )can be identified with the family of graphs of a Hamiltonian isotopy of χ ( S ) known as the Goldman twist flow associated to the simple closed curve [Gol84]. This implies that the holonomyperturbation process can be viewed as a combination of a decomposition induced by cutting a3 manifold along a separating torus T , followed by a perturbation as in Equation (6.3), with α sπ : χ ( S × D ) → χ ( T ) the composition of the unperturbed inclusion α followed by a smalltime flow of the Hamiltonian Goldman twist flow associated to a curve in this torus.We formalize this in the following way. Call two Lagrangian immersions ι : L (cid:35) M, ι : L (cid:35) M secretly Hamiltonian isotopic if they can be expressed as compositions with some β : Λ (cid:35) X × M : i : L (cid:35) M = β j M : L (cid:48) × X Λ (cid:35) M,i : L (cid:35) M = β j M : L (cid:48) × X Λ (cid:35) M. in such a way that j , j : L (cid:48) , L (cid:48) (cid:35) X are Hamiltonian isotopic in X . It follows from the discus-sion in Section 6.2 that different choices of holonomy perturbations induce secretly Hamiltonianisotopic immersed Lagrangians.Being secretly Hamiltonian isotopic does not necessarily imply being Hamiltonian isotopic.Indeed, L and L need not even be diffeomorphic. Moreover, in the absence of extra hypothe-ses (embedded composition, monotonicity, exactness, etc.), the conclusion of the Wehrheim-Woodward composition theorem [WW10] need not hold. In particular, given a third Lagrangian L , HF ( L , L ) and HF ( L , L ) need not be isomorphic, even if they are both well-defined.However, provided: • all Lagrangian immersions and symplectic manifolds satisfy suitable assumptions so tobe able to define Lagrangian (quilted) Floer homology, • all Lagrangian immersions come equipped with suitable bounding cochains, in a wayconsistent with composition. ANGLES, RELATIVE CHARACTER VARIETIES, AND PERTURBED FLAT MODULI SPACES 19
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