# Tangles, relative character varieties, and holonomy perturbed traceless flat
moduli spaces

Guillem Cazassus  Mathematical institute, University of Oxford, Oxford OX2 6GG
<EMAIL_ADDRESS>,  Chris Herald  Department of Mathematics and
Statistics, University of Nevada, Reno, NV 89557<EMAIL_ADDRESS>and  Paul
Kirk  Department of Mathematics, Indiana University, Bloomington, IN 47405
<EMAIL_ADDRESS>
(Date: Jan. 30, 2021)

###### Abstract.

We prove that the restriction map from the subspace of regular points of the
holonomy perturbed SU(2) traceless flat moduli space of a tangle in a
3-manifold to the traceless flat moduli space of its boundary marked surface
is a Lagrangian immersion. A key ingredient in our proof is the use of
composition in the Weinstein category, combined with the fact that SU(2)
holonomy perturbations in a cylinder induce Hamiltonian isotopies. In
addition, we show that $(S^{2},4)$, the 2-sphere with four marked points, is
its own traceless flat SU(2) moduli space.

###### Key words and phrases:

holonomy perturbation, Lagrangian immersion, Floer homology, flat moduli
space, traceless character variety, quilted Floer homology

###### 2010 Mathematics Subject Classification:

Primary 57K18, 57K31, 57R58; Secondary 81T13

CH was supported by Simons Collaboration Grants for Mathematicians, GC was
funded by EPSRC grant reference EP/T012749/1.

##  1\. introduction

We gather together some of the key symplectic properties of character
varieties and traceless character varieties, as well as variants which
correspond to perturbed flat moduli spaces that arise in the gauge theoretic
study of 3-manifolds. Some of these results are well known to the experts, but
the proofs in the literature are framed in contexts that include gauge theory,
Hodge theory, and symplectic reduction. In the present exposition, we provide
a general proof of the fact that, roughly, the character variety of a
3-manifold provides an immersed Lagrangian in the character variety of its
boundary surface, for any compact Lie group $G$, whether the 3-manifold and
its boundaries include tangles, or whether there are trace or other conjugacy
restrictions on some meridional generators, and furthermore we extend the
result to the holonomy perturbed situation. Moreover, we clarify why different
natural definitions of symplectic structures on the pillowcase, 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 Poincaré-Lefschetz duality
theorem, basic algebraic topology, and the notion of composition in the
Weinstein category.

Let $(X,{\mathcal{L}})$ be a tangle in a compact, oriented 3-manifold $X$;
that is, assume that ${\mathcal{L}}$ is a properly embedded, compact
1-manifold. For our initial discussion, we consider $G=SU(2)$. Let $\pi$
denote some holonomy perturbation data supported in a finite disjoint union of
solid tori in the interior of $X\setminus{\mathcal{L}}$.

This data determines two moduli spaces,

${\mathcal{M}}_{\pi}(X,{\mathcal{L}}),$

the moduli space of $\pi$ holonomy-perturbed flat $SU(2)$ connections on
$X\setminus{\mathcal{L}}$ with traceless holonomy on small meridians of
${\mathcal{L}}$, and the (well studied) moduli space:

${\mathcal{M}}(\partial X,\partial{\mathcal{L}})$

of flat $SU(2)$ connections on the punctured surface $\partial
X\setminus\partial{\mathcal{L}}$ with traceless holonomy around the marked
points $\partial{\mathcal{L}}$.

Holonomy perturbed flat connections on a 3-manifold are flat near the
boundary, and restriction to the boundary defines a map

(1.1) $r:{\mathcal{M}}_{\pi}(X,{\mathcal{L}})\to{\mathcal{M}}(\partial
X,\partial{\mathcal{L}})$

The moduli space ${\mathcal{M}}(\partial X,\partial{\mathcal{L}})$ is the
cartesian product of the moduli spaces of its path components. 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 symplectic form called the Atiyah-Bott-Goldman form [AB83, Gol84].
Thus the smooth top stratum of the cartesian product ${\mathcal{M}}(\partial
X,\partial{\mathcal{L}})$, denoted ${\mathcal{M}}(\partial
X,\partial{\mathcal{L}})^{*}$, is endowed with the product symplectic form.

The main result of this article is the following theorem (Theorem 6.9 below),
concerning the regular points (see Section 4.3) of the perturbed moduli space.

######  Theorem A.

Suppose $A\in{\mathcal{M}}_{\pi}(X,{\mathcal{L}})$ is a regular point. Then
$A$ has a neighborhood $U$ so that the restriction
$r|_{U}:U\subset{\mathcal{M}}_{\pi}(X,{\mathcal{L}})\to{\mathcal{M}}(\partial
X,\partial{\mathcal{L}})^{*}$ is a Lagrangian immersion.

This is not a surprising result; indeed, many special cases are known, for
example, when ${\mathcal{L}}$ is empty this result is proven in [Her94]. Our
primary aim is to provide details of the assertion that well-known arguments
in the flat case extend to the holonomy-perturbed flat case when
${\mathcal{L}}$ is nonempty, in support of one claim of the main result of the
article [CHKK20]. In that article it is shown that a certain process
introduced by Kronheimer-Mrowka to ensure admissibility of bundles in
instanton homology [KM11] manifests itself on the symplectic side of the
Atiyah-Floer conjecture [Ati88] (i.e., Lagrangian Floer theory of character
varieties or flat moduli spaces) as a certain Lagrangian immersion of a smooth
closed genus 3 surface into the smooth stratum
$\mathbb{P}^{*}\times\mathbb{P}^{*}$ of the product of two pillowcases (cf.
Equation (1.3)). What is proved in [CHKK20] is that 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 algebraic topology proof of Theorem A in the flat case, for any
compact Lie group $G$. The statement can be found in Corollary 4.5. We
emphasize that the flat case of Theorem A, when ${\mathcal{L}}$ is nonempty,
is known (e.g., to those who attended the appropriate Oxford seminars in the
1980s) and indeed discussed on pages 15-16 of Atiyah’s monograph [Ati90].

Having stated Theorem A and described its relation to the gauge theoretic
literature using the language of flat and perturbed flat moduli spaces, we
note that these spaces can also be identified with certain character
varieties, the definitions of which do not require any of the analytical
machinery of gauge theory; the proofs in this article are most simply
explained without it, so we shall henceforth revert to the character variety
terminology in our exposition. In the simplest situation of a connected 2- or
3-manifold $M$ with base point $x_{0}$, for a fixed compact Lie group $G$,
each flat $G$ connection determines a holonomy representation from
$\pi_{1}(M,x_{0})$ to $G$, and this correspondence induces a bijection between
the moduli space of flat connections and the set of $G$ representations of
$\pi_{1}$ modulo conjugation, known as the character variety. We describe
various extensions of the notion of the character variety corresponding to
traceless and perturbed flat moduli spaces below.

We also take this opportunity to provide an elementary exposition of holonomy
perturbations from the perspective of fundamental groups and character
varieties, in the language of composition of Lagrangian immersions. Algebraic
topology arguments simplify the task of explaining how to understand the
extension from the flat to the holonomy perturbed flat situation
algebraically. Other arguments, which instead appeal to Hodge and elliptic
theory of perturbed flat bundles over 3-manifolds with boundary, can be made
when ${\mathcal{L}}$ is empty, and can be found in detail in [Tau90, Her94].
But when ${\mathcal{L}}$ is nonempty, proper treatment of the traceless
condition requires more subtle analytic tools.

Our focus on holonomy perturbations is motivated by the fact that they are
compatible with the 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
character varieties in Section 6.2.

In addition to proving Theorem A, we prove Theorem B, whose statement roughly
says the four punctured 2-sphere is its own traceless $SU(2)$ moduli space.
This is a variant, for $(S^{2},4)$, of the ubiquitous mathematical statement
that a torus is isomorphic to its Jacobian. We now set some notation in
preparation for the formal statement. First, denote $U(1)\times U(1)$ by
$\mathbb{T}$. The group $SL(2,{\mathbb{Z}})$ acts on $\mathbb{T}$ via

(1.2) $\begin{pmatrix}p&r\\\ q&s\end{pmatrix}\cdot(e^{x{{\bf i}}},e^{y{{\bf
i}}})=(e^{(px+ry){{\bf i}}},e^{(qx+sy){{\bf i}}}).$

The 2-form $dx\wedge dy$ defines the standard symplectic structure on
$\mathbb{T}$ and is invariant under the $SL(2,{\mathbb{Z}})$ action. Note that
the action of the central element $-1\in SL(2,{\mathbb{Z}})$ on $\mathbb{T}$
defines the elliptic involution, denoted by $\iota(e^{{{\bf i}}x},e^{{{\bf
i}}y})=(e^{-{{\bf i}}x},e^{-{{\bf i}}y})$; this involution has four fixed
points $(\pm 1,\pm 1)$. Denote by $\mathbb{T}^{*}\subset\mathbb{T}$ the
complement of the four fixed points, on which $\iota$ acts freely. Then set

(1.3) $\mathbb{P}=\mathbb{T}/\iota\text{ and
}\mathbb{P}^{*}=\mathbb{T}^{*}/\iota.$

The quotient $PSL(2,{\mathbb{Z}})=SL(2,{\mathbb{Z}})/\\{\pm 1\\}$ acts on the
smooth locus $\mathbb{P}^{*}$, a 4-punctured 2-sphere, preserving the
symplectic form $dx\wedge dy$.

$a$ at 150 238 $b$ at 200 120 $c$ at 140 110 $d$ at 60 115 $H_{2}$ at 130 23
$H_{1}$ at 105 105

Figure 1.

Next, we consider the relative character variety (defined below) of the
2-sphere with four marked points $\chi_{SU(2),J}(S^{2},4)$, where the $J$
subscript indicates the four meridians are sent to the conjugacy class $J$ of
traceless elements. This character variety is equipped with its relative
Atiyah-Bott-Goldman 2-form $\omega_{(S^{2},4)}$ (defined below).

From the presentation $\pi_{1}(S^{2}-4)=\langle a,b,c,d~{}|~{}abcd=1\rangle,$
we see that this group is freely generated by $a,b,c$. Hence a representation
is uniquely defined by where it sends the generators $a,b,c$. Define the
function (identifying $SU(2)$ with the group of unit quaternions; see Section
2.1.2):

$\hat{\rho}:\mathbb{P}\to\chi_{SU(2),J}(S^{2},4),~{}[e^{{{\bf i}}x},e^{{{\bf
i}}y}]\mapsto\big{[}a\mapsto{{\bf j}},b\mapsto e^{{{\bf i}}x}{{\bf
j}},c\mapsto e^{{{\bf i}}y}{{\bf j}}\big{]}.$

######  Theorem B.

Half Dehn twists in the two twice-punctured disks indicated in Figure 1
generate a $PSL(2,{\mathbb{Z}})$ action on $\chi(S^{2},4)$ which preserves the
Atiyah-Bott-Goldman symplectic form $\widehat{\omega}_{(S^{2},4)}$. For some
non-zero constant $c$, the map

$\hat{\rho}:(\mathbb{P}^{*},c~{}dx\wedge
dy)\to(\chi(S^{2},4)^{*},\widehat{\omega}_{(S^{2},4)})$

is a $PSL(2,{\mathbb{Z}})$ equivariant symplectomorphism.

The proof, as well as as well as an exposition of the simpler case of $SU(2)$
character variety of the torus, is contained in Section 5.

The outline of the proof of the flat case of Theorem A is the following.
Holonomy identifies flat moduli spaces with character varieties. When
${\mathcal{L}}$ is empty, Theorem A follows from Weil’s identification of the
Zariski tangent spaces of character varieties with cohomology and Poincaré-
Lefschetz duality. Symplectic reduction is used to extend to the case when
${\mathcal{L}}$ is nonempty. We use only Poincaré duality with local
coefficients, which is briefly reviewed, to highlight the fact that the proof
that the image of the differential is maximal isotropic (the subtlest part of
any 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 the perturbation data $\pi$. Building on ideas from [Ati88, Flo88,
Wei82] and many others, Wehrheim and Woodward developed quilts and Floer field
theory [WW15, WW16]; a framework that aims to 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 of the flat moduli spaces of surfaces equipped with the Atiyah-Bott
form. This can be considered as an approach to realizing a bordered Lagrangian
Floer theory of character varieties of surfaces and 3-manifolds, as was done
for Heegaard-Floer theory in [LOT18].

Even in the lucky case that one finds perturbation data $\pi$ for which
${\mathcal{M}}_{\pi}(X,{\mathcal{L}})$ is smooth of the correct dimension and
Lagrangian immerses into the smooth stratum
${\mathcal{M}}(\partial(X,{\mathcal{L}}))^{*}$, an understanding of how this
immersion depends on $\pi$ is necessary in order to extract topological
information. 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, and in particular has no effect on the
topology of ${\mathcal{M}}_{\pi}(X,{\mathcal{L}})$. This is discussed in
Section 6.2; see also [HK18].

However, for general choices of perturbation data, perturbations typically
change the topology of ${\mathcal{M}}_{\pi}(X,{\mathcal{L}})$; indeed, the
primary purpose of using perturbation curves is precisely to smooth
${\mathcal{M}}_{\pi}(X,{\mathcal{L}})$. As we discuss in Section 6.3, Floer
field theory posits that, nevertheless, the resulting immersions should be
independent of $\pi$ in a Floer-theoretic sense (i.e., isomorphic in some
Fukaya category).

###  1.1. Acknowledgements

The authors thank C. Judge, E. Toffoli, and L. Jeffrey for helpful
conversations. Special thanks to A. Kotelskiy, who read early versions of this
article and made important suggestions.

##  2\. Review of Poincaré duality and symplectic forms

###  2.1. Preliminaries and Notation

####  2.1.1. Symplectic linear algebra

Let $A$ denote a finite-dimensional ${\mathbb{R}}$ vector space. A skew
symmetric bilinear form $\omega:A\times A\to{\mathbb{R}}$ is called a
symplectic form if it is nondegenerate, that is, the map
$A\to\operatorname{Hom}(A,{\mathbb{R}})$ given by $a\mapsto\omega(a,-)$ is an
isomorphism. If $A$ admits a symplectic form, then its dimension is even;
denote the dimension by $2n$.

A coisotropic subspace $C\subset A$ is a subspace satisfying

${\rm Annihilator}(C):=\\{a\in A~{}|~{}\omega(c,a)=0\text{ for all }c\in
C\\}\subset C.$

A Lagrangian subspace is a coisotropic subspace of dimension $n$, or,
equivalently a subspace $L\subset A$ satisfying ${\rm Annihilator}(L)=L.$

Symplectic reduction refers the following process. Given any coisotropic
subspace $C\subset A$, the restriction

$\omega|_{C}:C\times C\to{\mathbb{R}}$

may be degenerate, but $\omega|_{C}$ descends to a symplectic form
$\widehat{\omega}$ on $\widehat{C}:=C/{\rm Annihilator}(C)$. Furthermore if
$L\subset A$ is any Lagrangian subspace of $(A,\omega)$ , then

$\hat{L}:=\big{(}L\cap C\big{)}/\big{(}L\cap{\rm
Annihilator}(C)\big{)}\subset\widehat{C}$

is a Lagrangian subspace of $(\widehat{C},\widehat{\omega})$. The subquotient
$(\widehat{C},\widehat{\omega})$ is called the symplectic reduction of
$(A,\omega)$ with respect to $C$ and the Lagrangian subspace
$\hat{L}\subset\widehat{C}$ the symplectic reduction of $L$ with respect to
$C$.

####  2.1.2. The Lie group $G$

Let $G$ be connected compact Lie group. Its Lie algebra $\mathfrak{g}$ admits
a positive definite symmetric $\operatorname{ad}$-invariant bilinear form, so
we fix one and denote it by $\\{\ ,\
\\}:\mathfrak{g}\times\mathfrak{g}\to{\mathbb{R}}.$ Fix a conjugacy class
$J\subset G$. It is well known that $G$ embeds in some ${\mathbb{R}}^{N}$ as
an algebraic variety. Thus, if $F_{g}$ denotes the free group on $g$
generators, $\operatorname{Hom}(F_{g},G)=G^{g}$ is an affine real-set variety,
with tangent space $\mathfrak{g}^{g}$ at the trivial homomorphism.

Our main focus, and the context for Theorem A, concerns the case when
$G=SU(2)$, $\\{v,w\\}=-\tfrac{1}{2}{\rm Tr}(vw)$, and $J\subset SU(2)$ is the
conjugacy class of traceless matrices. To keep notation compact, we identify
$SU(2)$ with the group of unit quaternions $\\{a+b{{\bf i}}+c{{\bf j}}+d{{\bf
k}}~{}|~{}a^{2}+b^{2}+c^{2}+d^{2}=1\\}$ and its Lie algebra $su(2)$ with the
purely imaginary quaternions $\\{b{{\bf i}}+c{{\bf j}}+d{{\bf k}}\\}$. With
this identification, ${\operatorname{Re}}:SU(2)\to[-1,1]$ corresponds to
$\frac{1}{2}\operatorname{Tr}$.

####  2.1.3. Notation used for 2- and 3-manifolds

Throughout this article, the notation

$Y,S,C,V,\text{ and }F$

is fixed as follows.

First, $S$ denotes a possibly disconnected compact oriented surface without
boundary. Denote by $S_{+}$ the path connected based space obtained by first
adding a disjoint base point to $S$, then attaching 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 $C_{i},i=1,\dots m$. We assume that either each $C_{i}$
is oriented, or the chosen conjugacy class $J\subset G$ is invariant under
inversion in $G$, so that the condition that a homomorphism from
$\pi_{1}(S_{+})\to G$ 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 $V\subset S$ is a disjoint
union of oriented annuli, one around each $C_{i}$.

Denote by $F$ the complementary surface, determining the decomposition:

(2.1) $S=F\cup_{\partial F=\partial V}V.$

Orient $V$ and $F$ as subsurfaces of $S$.

Finally, $Y$ denotes a compact, connected, and oriented 3-manifold with
boundary $\partial Y=S$; that is, we assume that an orientation preserving
identification of $\partial Y$ with $S$ is given. Fix a base point in the
interior of $Y$, and extend the inclusion $S=\partial Y\subset Y$ to a based
embedding of $S_{+}$ into $Y$.

####  2.1.4. Tangles

A tangle $(X,{\mathcal{L}})$ consists of a connected compact oriented
3-manifold $X$, and ${\mathcal{L}}\subset X$ a properly embedded compact
1-submanifold with boundary. Thus ${\mathcal{L}}$ consists of a disjoint union
of $n$ intervals and $k$ interior circles.

A tangle $(X,{\mathcal{L}})$ gives rise to a triple $(Y,S,C)$ as above by
taking

$Y=X\setminus N({\mathcal{L}}),$

Where $N({\mathcal{L}})$ denotes a tubular neighborhood of ${\mathcal{L}}$,
and letting $S=\partial Y$. Then let $C\subset S$ denote a union of $m=n+k$
meridians of ${\mathcal{L}}$, one for each component, viewed as a curve in
$S$. As before, $Y,S,C$ determine $V$ and $F$. Orientations of the tangle
components are equivalent to orientations of the components of $C$.

The process $(X,{\mathcal{L}})\Rightarrow(Y,S,C)$ is nearly reversible, by
attaching 2-handles to $Y$ along $C$ and setting ${\mathcal{L}}$ to be union
of the co-cores of the 2-handles. The resulting tangle is obtained from
$(X,{\mathcal{L}})$ by removing $k$ disjoint small balls from the interior of
$X$, each meeting a different closed component of ${\mathcal{L}}$ in a trivial
arc. This results in a tangle with no closed components, but which has the
same character variety as the starting tangle. We use the notation $(Y,S,C)$
in our arguments as it leads to simpler expressions, but state consequences in
terms of the tangle $(X,{\mathcal{L}})$, as they are clearly seen as morphisms
in a (more familiar) $(2,0)+1$ cobordism category.

###  2.2. Poincaré duality and intersection forms

We begin by recalling the statement of Poincaré-Lefschetz duality with local
coefficients for an oriented compact connected $n$-manifold $M$ with boundary
$\partial M$ equipped with a finite cell decomposition (see [CLM20] for a
careful exposition and proofs, and [Mil62] for an elementary derivation using
dual regular cell decompositions). Denote by $\xi\in C_{n}(M,\partial
M;{\mathbb{Z}})$ a cellular chain representing the fundamental class.

Fix a base point in $M$ and some homomorphism $\pi_{1}M\to\Gamma$ to some
group $\Gamma$, and denote by $M_{\Gamma}\to M$ the corresponding $\Gamma$
cover, equipped with the lifted cell structure and its cellular left $\Gamma$
covering action. The cellular ${\mathbb{Z}}$ chain complexes of $M_{\Gamma}$
and $(M_{\Gamma},\partial M_{\Gamma})$ are denoted by $C_{*}^{\Gamma}(M)$ and
$C^{\Gamma}_{*}(M,\partial M)$. These are free finitely generated left
${\mathbb{Z}}\Gamma$ modules: a choice of ${\mathbb{Z}}\Gamma$ basis is given
by arbitrarily choosing lifts of cells of $M$ (respectively of cells which
meet $M\setminus\partial M$). A choice of cellular approximation of the
diagonal determines a chain level cap product:

(2.2)
$\cap\xi:\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C_{*}^{\Gamma}(M,\partial
M),{\mathbb{Z}}\Gamma)\to C^{\Gamma}_{*}(M).$

The Poincaré duality theorem asserts that this map is a chain homotopy
equivalence of free ${\mathbb{Z}}\Gamma$ chain complexes.

Given any left ${\mathbb{Z}}\Gamma$ module $W$,
$\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C_{*}^{\Gamma}(M,\partial
M),W)=\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C_{*}^{\Gamma}(M,\partial
M),{\mathbb{Z}}\Gamma)\otimes_{{\mathbb{Z}}\Gamma}W$ and hence capping with
$\xi$ induces a chain homotopy equivalence

$\cap\xi:\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C_{*}^{\Gamma}(M,\partial
M),W)\to C^{\Gamma}_{*}(M)\otimes_{{\mathbb{Z}}\Gamma}W.$

Now suppose that $W$ is a finite dimensional ${\mathbb{R}}$ vector space
equipped with a positive definite inner product $\\{\ ,\ \\}:W\times
W\to{\mathbb{R}}$ and $\Gamma\to O(W)$ a representation, determining the left
${\mathbb{Z}}\Gamma$ module structure on $W$. There is an algebraic chain
isomorphism of ${\mathbb{R}}$ chain complexes:

$C^{\Gamma}_{*}(M)\otimes_{{\mathbb{Z}}\Gamma}W\cong\operatorname{Hom}_{{\mathbb{R}}}(\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C^{\Gamma}_{*}(M),W),{\mathbb{R}}),~{}c\otimes
w\mapsto\big{(}h\mapsto\\{h(c),w\\}\big{)}.$

Composing with the chain homotopy equivalence $\cap\xi$, and using the
universal coefficient theorem for ${\mathbb{R}}$, we obtain an isomorphism

$H^{*}(M,\partial M;W)\cong
H^{*}(\operatorname{Hom}_{{\mathbb{R}}}(\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C^{\Gamma}_{*}(M),W),{\mathbb{R}}))=\operatorname{Hom}_{{\mathbb{R}}}(H^{*}(M;W),{\mathbb{R}})$

whose adjoint is the (by construction nondegenerate) cohomology intersection
pairing over $W$

(2.3) $H^{*}(M,\partial M;W)\times H^{*}(M;W)\to{\mathbb{R}}.$

The cohomology intersection pairing can also be expressed in terms of cup
products:

(2.4) $(x,y)\mapsto\\{x\cup y\\}\cap\xi,$

where $\\{~{}\\}:H^{n}(M;\partial M;W\otimes_{{\mathbb{Z}}\Gamma}W)\to
H^{n}(M;\partial M;{\mathbb{R}})$ is induced by the coefficient homomorphism
$W\otimes_{{\mathbb{Z}}\Gamma}W\to{\mathbb{R}}$ determined by the bilinear
form $\\{~{},~{}\\}$.

When the boundary of $M$ is empty, this pairing, which we denote by

(2.5) $\omega_{M}:H^{*}(M;W)\times H^{*}(M;W)\to{\mathbb{R}}$

is therefore a nondegenerate inner product on the ${\mathbb{R}}$ vector space
$H^{*}(M;W)$.

If $\partial M$ is nonempty, precomposing the injective adjoint

$H^{*}(M;W)\to\operatorname{Hom}_{\mathbb{R}}(H^{*}(M,\partial
M;W);{\mathbb{R}})$

with the restriction map $H^{*}(M,\partial M;W)\to H^{*}(M;W)$ yields a map

$H^{*}(M,\partial M;W)\to\operatorname{Hom}_{\mathbb{R}}(H^{*}(M,\partial
M;W);{\mathbb{R}})$

with kernel equal to the kernel of $H^{*}(M,\partial M;W)\to H^{*}(M;W)$. An
equivalent statement is that the pairing

(2.6) $\omega_{(M,\partial M)}:H^{*}(M,\partial M;W)\times H^{*}(M,\partial
M;W)\to{\mathbb{R}}$

has radical equal to $\ker H^{*}(M,\partial M;W)\to H^{*}(M;W)$.

Taking gradings into account, when $\dim M=2n$ restriction defines a pairing

(2.7) $\omega_{(M,\partial M)}:H^{n}(M,\partial M;W)\times H^{n}(M,\partial
M;W)\to{\mathbb{R}}$

with radical equal to $\ker H^{n}(M,\partial M;W)\to H^{n}(M;W)$.

When $\dim M=4\ell+2$, for example when $M$ is a surface, the pairings
$\omega_{M}$, and $\omega_{(M,\partial M)}$ are skew-symmetric:

$\omega_{M}(x,y)=-\omega_{M}(y,x)~{}\text{ and }~{}\omega_{(M,\partial
M)}(x,y)=-\omega_{(M,\partial M)}(y,x).$

##  3\. Two and three dimensional manifolds and symplectic linear algebra

###  3.1.  A symplectic form on the first cohomology of surface

Recall that $S_{+}$ denotes the path 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 $\Gamma=\pi_{1}(S_{+})$. Its
universal cover $\widetilde{S}_{+}\to S_{+}$ is a regular $\Gamma$ cover, and
hence so is its restriction over $S$

$\widetilde{S}\to S.$

A representation $\rho:\Gamma\to G$ is fixed.

Then $\rho$ determines, via the adjoint action of $G$, the representation
$\operatorname{ad}\rho:\Gamma\to O(\mathfrak{g})$, and hence cohomology groups
$H^{*}(S;\mathfrak{g}),H^{*}(S,\partial S;\mathfrak{g}),$ and $H^{*}(\partial
S;\mathfrak{g})$, with, for example,

$H^{*}(S;\mathfrak{g}):=H^{*}(\operatorname{Hom}_{{\mathbb{Z}}\Gamma}(C^{\Gamma}_{*}(S),\mathfrak{g})).$

If we wish to emphasize $\rho$, we write
$H^{*}(S;\mathfrak{g}_{\operatorname{ad}\rho})$.

Equation (2.5) shows that since the boundary of $S$ is empty,

(3.1) $\omega_{S}:H^{1}(S;\mathfrak{g})\times
H^{1}(S;\mathfrak{g})\to{\mathbb{R}}$

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

$\omega_{(F,\partial F)}:H^{1}(F,\partial F;\mathfrak{g})\times
H^{1}(F,\partial F;\mathfrak{g})\to{\mathbb{R}}$

is in general a degenerate skew symmetric form with radical equal to

$\ker H^{1}(F,\partial F;\mathfrak{g})\to H^{1}(F;\mathfrak{g}).$

A degenerate skew symmetric form induces a nondegenerate form on the quotient
by its radical. The exact sequence of the pair $(F,\partial F)$ shows that
$\omega_{(F,\partial F)}$ descends to a nondegenerate skew symmetric form

(3.2)
$\widehat{\omega}_{F}:\widehat{H}^{1}(F;\mathfrak{g})\times\widehat{H}^{1}(F;\mathfrak{g})\to{\mathbb{R}},$

where

$\widehat{H}^{1}(F;\mathfrak{g})={\rm Image}\ H^{1}(F,\partial
F;\mathfrak{g})\to H^{1}(F;\mathfrak{g})=\ker H^{1}(F;\mathfrak{g})\to
H^{1}(\partial F;\mathfrak{g}).$

Summarizing:

######  Proposition 3.1.

Let $S$ be a compact oriented surface without boundary and
$\rho:\pi_{1}S_{+}\to G$ a representation. Then
$(H^{1}(S;\mathfrak{g}_{\operatorname{ad}\rho}),\omega_{S})$ is a symplectic
vector space. If $F\subset S$ is the complement of a nonempty disjoint union
of annuli $V$, then
$(\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho}),\widehat{\omega}_{F})$
is a symplectic vector space.

In the following diagram of inclusions, the two vertical and two horizontal
rows are exact, and the isomorphisms excisions:

(3.3) ${H^{1}(S,F)}$${H^{1}(V,\partial
V)}$${H^{1}(S,V)}$${H^{1}(S)}$${H^{1}(V)}$${H^{2}(S,V)}$${H^{1}(F,\partial
F)}$${H^{1}(F)}$${H^{1}(\partial F)}$${H^{2}(F,\partial
F)}$$\scriptstyle{q_{1}}$$\scriptstyle{\cong}$$\scriptstyle{q_{2}}$$\scriptstyle{\alpha}$$\scriptstyle{p_{0}}$$\scriptstyle{\cong}$$\scriptstyle{\beta}$$\scriptstyle{p_{1}}$$\scriptstyle{\gamma}$$\scriptstyle{p_{2}}$$\scriptstyle{\cong}$$\scriptstyle{a}$$\scriptstyle{b}$$\scriptstyle{c}$

######  Proposition 3.2.

The kernel of $\beta$ is a cosiotropic subspace of $H^{1}(S;\mathfrak{g})$
with annihilator $\ker p_{1}$, and hence $p_{1}$ induces a symplectomorphism

$\ker\beta/\ker p_{1}\xrightarrow{\cong}\widehat{H}^{1}(F;\mathfrak{g}).$

###### Proof.

The surface $V$ is a disjoint union of annuli. The composition $\\{0\\}\times
S^{1}\subset\partial(I\times S^{1})\subset I\times S^{1}$ a homotopy
equivalence, and hence the restriction $H^{1}(I\times S^{1})\to
H^{1}(\partial(I\times S^{1}))$ is injective with any coefficients. Hence
$p_{2}$ is injective, and $q_{2}=0$.

If $s\in H^{1}(S)$ satisfies $\omega_{S}(s,\alpha(y))=0$ for all $y\in
H^{1}(S,V)$, then $\omega_{F,\partial F}(p_{1}(s),a\circ p_{0}(y))=0$ for all
$y\in H^{1}(S,V)$. Hence $\omega_{F,\partial F}(p_{1}(s),a(z))=0$ for all
$z\in H^{1}(F,\partial F)$. Since the pairing $H^{1}(F,\partial F)\times
H^{1}(F)\to{\mathbb{R}}$ is nondegenerate (Equation (2.3)), this implies that
$p_{1}(s)=0$. In other words, the annihilator of $\text{image
}\alpha=\ker\beta$ is contained in $\ker p_{1}$.

Since $q_{2}=0$, $\ker p_{1}\subset\ker\beta$. Therefore $\ker\beta$ contains
its annihilator and hence is coisotropic.

It remains to show that $\ker p_{1}=\text{image }q_{1}$ is contained in the
annihilator of $\ker\beta$. Given $x\in H^{1}(S,V)$ and $y\in H^{1}(S,F)$,
$\omega_{S}(\alpha(x),q_{1}(y))=0$ since the cup product

$H^{1}(S,V)\times H^{1}(S,F)\to H^{2}(S)$

factors through $H^{2}(S,F\cup V)=0$ (see Equation (2.4)). $\square$

######  Corollary 3.3.

If $L\subset H^{1}(S;\mathfrak{g}_{\operatorname{ad}\rho})$ is any Lagrangian
subspace, then the image

$p_{1}(L\cap\ker\beta)\subset\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho})$

is a Lagrangian subspace.

###  3.2. Restriction from a 3-manifold with boundary

Recall that $Y$ is a compact, connected, oriented 3-manifold with boundary
$S=\partial Y$, extended to an embedding $S_{+}\subset Y$.

Assume that the representation $\rho:\pi_{1}(S_{+})\to G$ is a restriction of
a representation (of the same name) $\rho:\pi_{1}(Y)\to G$.

######  Lemma 3.4.

The image of the restriction map,

$L_{Y}:={\rm Image}\ r:H^{1}(Y;\mathfrak{g}_{\operatorname{ad}\rho})\to
H^{1}(S;\mathfrak{g}_{\operatorname{ad}\rho})$

is a Lagrangian subspace of
$(H^{1}(S;\mathfrak{g}_{\operatorname{ad}\rho}),\omega_{S})$.

###### 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 Poincaré-Lefschetz duality. The diagram commutes up to sign
[Spa95]. We suppress the $\mathfrak{g}_{\operatorname{ad}\rho}$ coefficients.

${\operatorname{Hom}(H_{1}(Y);{\mathbb{R}})}$${\operatorname{Hom}(H_{1}(S);{\mathbb{R}})}$${\operatorname{Hom}(H_{2}(Y,S);{\mathbb{R}})}$${H^{1}(Y)}$${H^{1}(S)}$${H^{2}(Y,S)}$${H_{2}(Y,S)}$${H_{1}(S)}$${H_{1}(Y)}$$\scriptstyle{\delta^{*}}$$\scriptstyle{r}$$\scriptstyle{\nu}$$\scriptstyle{\delta}$

The image of $r$ equals the kernel of $\nu$, which is isomorphic to the kernel
of $\delta^{*}$. The kernel of $\delta^{*}$ is isomorphic to the cokernel of
its dual $\delta$, which in turn is isomorphic to the cokernel of $r$. Hence
the image and cokernel of $r$ are isomorphic, and so
$\dim(\text{image}(r))=\tfrac{1}{2}\dim H^{1}(S)$.

Commutativity of the following diagram is a consequence of naturality of cup
product and Poincaré duality:

${H^{1}(Y;\mathfrak{g})\times
H^{1}(Y;\mathfrak{g})}$${H^{2}(Y;{\mathbb{R}})}$${H_{1}(Y,S;{\mathbb{R}})}$${H^{1}(S;\mathfrak{g})\times
H^{1}(S;\mathfrak{g})}$${H^{2}(S;{\mathbb{R}})}$${H_{0}(S;{\mathbb{R}})}$${{\mathbb{R}}}$${H_{0}(Y;{\mathbb{R}})}$$\scriptstyle{\cup_{\\{,\\}}}$$\scriptstyle{r\times
r}$$\scriptstyle{\cap{[Y,S]}}$$\scriptstyle{\cup_{\\{,\\}}}$$\scriptstyle{\cap{[S]}}$$\scriptstyle{\epsilon}$$\scriptstyle{\epsilon}$

Exactness of the vertical sequence shows that the dotted arrow is zero, which
implies that the image of $r$ is isotropic, and therefore Lagrangian.
$\square$

Recall that the boundary $S=\partial Y$ is equipped with a embedded collection
$C\subset S$ of circles, with tubular neighborhood $V$ and complementary
subsurface $F$, producing the decomposition $S=F\cup V$ as in Equation (2.1).
Consider the ladder of exact sequences, with all maps induced by inclusions.
The $\mathfrak{g}_{\operatorname{ad}\rho}$ coefficients are supressed. The
bottom two rows coincide with those of Diagram (3.3).

(3.4)
${\cdots}$${H^{1}(Y,V)}$${H^{1}(Y)}$${H^{1}(V)}$${\cdots}$${\cdots}$${H^{1}(S,V)}$${H^{1}(S)}$${H^{1}(V)}$${\cdots}$${\cdots}$${H^{1}(F,\partial
F)}$${H^{1}(F)}$${H^{1}(\partial
F)}$${\cdots}$$\scriptstyle{A}$$\scriptstyle{r_{0}}$$\scriptstyle{B}$$\scriptstyle{r}$$\scriptstyle{\alpha}$$\scriptstyle{p_{0}}$$\scriptstyle{\cong}$$\scriptstyle{\beta}$$\scriptstyle{p_{1}}$$\scriptstyle{p_{2}}$$\scriptstyle{a}$$\scriptstyle{b}$

A diagram chase shows that ${\rm Image}\ r\cap\ker\beta={\rm Image}\
\alpha\circ r_{0}$. Hence

$p_{1}({\rm Image}\ r\cap\ker\beta)={\rm Image}\ a\circ p_{0}\circ
r_{0}:H^{1}(Y,V)\to H^{1}(F)\subset\ker b=\widehat{H}^{1}(F)$

Lemma 3.4 and Corollary 3.3 imply the following.

######  Corollary 3.5.

Suppose that $\partial Y=S=V\cup F$ with $V$ a disjoint union of annuli.

Then

$L_{Y,V}:={\rm Image}\ H^{1}(Y,V;\mathfrak{g}_{\operatorname{ad}\rho})\to
H^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho})\subset\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho})$

is a Lagrangian subspace. Moreover, $L_{Y,V}$ is the symplectic reduction of
$L_{Y}$ with respect to
$\ker\beta:H^{1}(S;\mathfrak{g}_{\operatorname{ad}\rho})\to
H^{1}(V;\mathfrak{g}_{\operatorname{ad}\rho})$.

##  4\. Character varieties, relative character varieties, and their tangent
spaces

###  4.1. Character varieties

######  Definition 4.1.

Given a finitely presented group $\Gamma$, its $G$ character variety
$\chi_{G}(\Gamma)$ is the real semi-algebraic set defined to be the orbit
space of the $G$-conjugation action on the affine ${\mathbb{R}}$-algebraic set
$\operatorname{Hom}(\Gamma,G)$. A set of $g$ generators of $\Gamma$ embeds
$\operatorname{Hom}(\Gamma,G)$ in $G^{g}$ equivariantly, and, since $G$ is a
compact Lie group, $\operatorname{Hom}(\Gamma,G)$ is an affine
${\mathbb{R}}$-algebraic set, with orbit space $\chi_{G}(\Gamma)$ a semi-
algebraic set. We call $\chi_{G}(\Gamma)$ and (orbit spaces of conjugation
invariant) Zariski closed subsets character varieties. If $Z$ is a path
connected space, write $\chi_{G}(Z)$ instead of $\chi_{G}(\pi_{1}(Z))$. The
character variety of a non-path connected space, by definition, is the
cartesian product of the character varieties of its path components.

As observed by Weil [Wei64], the formal tangent space at  the conjugacy class
of any representation $\rho:\Gamma\to G$ to the character variety
$\chi_{G}(\Gamma)$ is naturally identified with first cohomology:

(4.1)
$T_{\rho}\chi_{G}(\Gamma)=H^{1}(\Gamma;\mathfrak{g}_{\operatorname{ad}\rho}).$

We take Equation (4.1) as the definition of the formal tangent space at $\rho$
for any $\rho\in\chi_{G}(\Gamma)$. Recall that for any space $X$ with
fundamental group $\Gamma$,
$H^{1}(\Gamma;\mathfrak{g}_{\operatorname{ad}\rho})$ and
$H^{1}(X;\mathfrak{g}_{\operatorname{ad}\rho})$ are canonically isomorphic.

Weil’s argument is based on the calculation that if a path of representations
is expressed in the form $\rho_{s}=\exp(\alpha_{s})\rho_{0}$ for some path
$\alpha_{s}:\Gamma\to\mathfrak{g}$, then
$\tfrac{d}{ds}|_{s=0}~{}\alpha_{s}:\Gamma\to\mathfrak{g}$ is a 1-cocycle
[Bro94].

###  4.2. Relative character varieties

As above, assume that $Y$ is a compact connected 3-manifold with boundary
$S=\partial Y$, $C\subset S$ is a union of $m$ circles $C_{i}$, $V$ is the
tubular neighborhood of $C$ in $S$, with complementary surface $F$.

Assume further that either $J$ is invariant via the inversion map of $G$ (as
is the case for the conjugacy class of traceless matrices in $SU(2)$) or else
assume that every circle $C_{i}$ is equipped with an orientation.

Then define the relative character variety to be

(4.2) $\chi_{G,J}(Y,C)\subset\chi_{G}(Y)$

to be the subvariety consisting of conjugacy classes of representations
$\pi_{1}(Y)\to G$ which send (any based representative of the homotopy class
of) each circle in $C$ into $J$. Define the formal tangent space of
$\chi_{G,J}(Y,C)$ at $\rho$ to be

(4.3) $T_{\rho}\chi_{G,J}(Y,C)=\ker
H^{1}(Y;\mathfrak{g}_{\operatorname{ad}\rho})\to
H^{1}(C;\mathfrak{g}_{\operatorname{ad}\rho})$

Given an oriented surface $F$, define

$\chi_{G,J}(F,\partial F)=\chi_{G,J}(F\times[0,1],\partial
F\times\\{\tfrac{1}{2}\\}).$

Its formal tangent space is

(4.4) $T_{\rho}\chi_{G,J}(F,\partial F)=\ker
H^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho})\to H^{1}(\partial
F;\mathfrak{g}_{\operatorname{ad}\rho})=\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho})$

###  4.3. Regular points

The term “formal tangent space” may be replaced by its usual elementary
differential topology notion in neighborhoods of regular points of
$\chi_{G,J}(Y,C)$ and $\chi_{G,J}(F,\partial F)$, as we next explain. In
brief, as elsewhere in gauge theory, a regular point is one which has a
neighborhood diffeomorphic to Euclidean space of the correct index-theoretic
dimension. We provide a stripped-down explanation, suitable for our purposes,
of what this means, for the benefit of the reader.

First, given a connected compact surface $F$ (with possibly empty boundary) we
call $\rho\in\chi_{G,J}(F,\partial F)$ a regular point provided $\rho$ has a
neighborhood $U\subset\chi_{G,J}(F,\partial F)$ so that
$\dim\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho^{\prime}})$ is
independent of $\rho^{\prime}\in U$.

Next, given a disjoint union of connected compact surfaces
$F=\sqcup_{i}F_{i}$, call

$\rho\in\chi_{G,J}(F,\partial F)=\prod_{i}\chi_{G,J}(F_{i},\partial F_{i})$

a regular point provided each of its components is a regular point.

Finally, for a pair $(Y,C)$ (with $C$ possibly empty) $\rho\in\chi_{G,J}(Y,C)$
is called a regular point provided $\rho$ admits a neighborhood
$U\subset\chi_{G,J}(Y,C)$ so that for all $\rho^{\prime}\in U$:

  * •

The restriction map $\chi_{G,J}(Y,C)\to\chi_{G,J}(F,\partial F)$ takes
$\rho^{\prime}$ to a regular point, and

  * •

$\dim T_{\rho^{\prime}}\chi_{G,J}(Y,C)=\tfrac{1}{2}\dim
T_{\rho}\chi_{G,J}(F,\partial F)=\tfrac{1}{2}\sum_{i}\dim
T_{\rho}\chi_{G,J}(F_{i},\partial F_{i})$

Hence, if $\rho\in\chi_{G,J}(Y,C)$ is a regular point, the map
$\chi_{G,J}(Y,C)\to\chi_{G,J}(F,\partial F)$, which takes a representation of
a 3-manifold group to its restriction to the boundary surface, is, near
$\rho$, a smooth map of a smooth $n$-disk into ${\mathbb{R}}^{2n}$ for some
$n$.

Notice that $\chi_{G,J}(Y,C)$ is the preimage of the point $(J,\cdots,J)$
under the restriction map

$\chi_{G}(Y)\to\chi_{G}(C)=\prod_{i=1}^{m}\chi_{G}(C_{i})=(G/_{\rm
conjugation})^{m}.$

and that $\chi_{G,J}(F,\partial F)$ is the preimage of the point
$(J,\cdots,J)$ under the restriction map

$\chi_{G}(F)\to\chi_{G}(\partial F).$

Since $C\subset V$ is a deformation retract, the exact sequence of the pair
shows that

$T_{\rho}\chi_{G,J}(Y,C)\cong{\rm Image}\
H^{1}(Y,V;\mathfrak{g}_{\operatorname{ad}\rho})\to
H^{1}(Y;\mathfrak{g}_{\operatorname{ad}\rho}).$

The image of the differential of the restriction map
$\chi_{G,J}(Y,C)\to\chi_{G,J}(F,\partial F)$ at $\rho\in\chi_{G,J}(Y,C)$ is
therefore identified with $L_{Y,V}$, the image of the composition

$H^{1}(Y,V;\mathfrak{g}_{\operatorname{ad}\rho})\to H^{1}(F,\partial
F;\mathfrak{g}_{\operatorname{ad}\rho})\to\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho}),$

which by Corollary 3.5 is a Lagrangian subspace of
$(H^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho}),\widehat{\omega}_{F})$.
Summarizing:

######  Corollary 4.2.

If $\rho\in\chi_{G,J}(Y,C)$ is a regular point, then there exists a
neighborhood of $\rho$ so that the differential of the restriction
$\chi_{G,J}(Y,C)\to\chi_{G,J}(F,\partial F)$ at any point $\rho^{\prime}$ in
this neighborhood has image a Lagrangian subspace of
$(\widehat{H}^{1}(F;\mathfrak{g}_{\operatorname{ad}\rho^{\prime}}),\widehat{\omega}_{F})$.

It is known that for surfaces, with the exception of a few low genus cases,
the regular points coincide with the irreducible representations.

###  4.4. Symplectic structure

The proof of Corollary 4.2 does not rely of the following fundamental 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 $\chi_{G}(S)$ and
$\chi_{G,J}(F,\partial F)$, the 2-forms $\omega_{S}$ and
$\widehat{\omega}_{F}$ 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 $\rho\in\chi_{G,J}(Y,C)$ is a regular point. Then there exists a
neighborhood $U$ of $\rho$ in $\chi_{G,J}(Y,C)$ so that the restriction of $r$
to $U$,

$r|_{U}:U\to\chi_{G,J}(F,\partial F),$

is a Lagrangian embedding.

In particular, if $\chi_{G,J}(Y,C)$ contains only regular points, then the
restriction map is a Lagrangian immersion.

Given a tangle $(X,{\mathcal{L}})$, We write $\chi_{G,J}(X,{\mathcal{L}})$
rather than $\chi_{G,J}(Y,C)$ where $Y,S,C,F$ and $V$ are determined by
$(X,{\mathcal{L}})$ as in Section 2.1.4. Also write
$\chi_{G,J}(\partial(X,{\mathcal{L}}))$ rather than $\chi_{G,J}(F,\partial
F)$. Then Theorem 4.4 can be restated in the new notation as follows.

######  Corollary 4.5.

Suppose that $\rho\in\chi_{G,J}(X,{\mathcal{L}})$ is a regular point. Then
there exists a neighborhood $U$ of $\rho$ in $\chi_{G,J}(X,{\mathcal{L}})$ so
that the restriction of $r$ to $U$

$r|_{U}:U\to\chi_{G,J}(\partial(X,{\mathcal{L}})),$

is a Lagrangian embedding.

In particular, if $\chi_{G,J}(X,{\mathcal{L}})$ contains only regular points,
then the restriction map is a Lagrangian immersion.

In what follows, we simply write $\chi(A)$ for $\chi_{G}(A)$ and $\chi(A,B)$
for $\chi_{G,J}(A,B)$. In addition, we denote by $\chi(S)^{*}\subset\chi(S)$
and $\chi(F,\partial F)^{*}\subset\chi(F,\partial F)$ the smooth top strata,
as real algebraic varieties, equipped with the symplectic forms $\omega_{S}$,
$\widehat{\omega}_{F}$.

##  5\. Three pillowcases

In this section, take $G=SU(2)$, and let $J\subset SU(2)$ be the conjugacy
class of unit quaternions with zero real part, so $J=su(2)\cap SU(2)$, a
2-sphere.

###  5.1. The quotient of the torus by the elliptic involution

Let $\mathbb{T}$ denote $U(1)\times U(1)$ with its symplectic form $dx\wedge
dy$ and symplectic $SL(2,{\mathbb{Z}})$ action given in Equation 1.2.
Multiplication by $-1\in SL(2,{\mathbb{Z}})$ is central and hence the quotient

$\mathbb{P}=\mathbb{T}/\\{\pm 1\\}$

inherits a $PSL(2,{\mathbb{Z}})=SL(2,{\mathbb{Z}})/\\{\pm 1\\}$ action. The
quotient map $\mathbb{T}\to\mathbb{P}$ is the 2-fold branched cover of the
2-sphere with branch points the four points $\\{(\pm 1,\pm 1)\\}$. The
complement, $\mathbb{P}^{*}$ of the four branch points is a smooth surface,
with symplectic form $dx\wedge dy$ and symplectic $PSL(2,{\mathbb{Z}})$
action, and the restriction

$\mathbb{T}^{*}\to\mathbb{P^{*}}$

is a smooth symplectic 2-fold covering map.

###  5.2. The $SU(2)$ character variety of the genus 1 surface

Consider the genus one closed oriented surface $T$, equipped with generators
$\mu,\lambda\in\pi_{1}T$ represented by a pair of oriented loops intersecting
geometrically and algebraically once. The Dehn twists $D_{\mu},D_{\lambda}$
about $\mu$ and $\lambda$ induce the automorphisms

$D_{\mu}:\big{(}\mu\mapsto\mu,\lambda\mapsto\mu\lambda\big{)}\text{ and
}D_{\lambda}:\big{(}\mu\mapsto\mu\lambda^{-1},\lambda\mapsto\lambda\big{)}.$

of $\pi_{1}(T,t_{0})\cong{\mathbb{Z}}\mu\oplus{\mathbb{Z}}\lambda$ (where
$t_{0}$ lies outside the support of these 2 Dehn twists). These automorphisms
induce, by precomposition, homeomorphisms

$D_{\mu}^{*},D_{\lambda}^{*}:\chi(T)\to\chi(T).$

Let $\mathcal{H}(T)=\operatorname{Hom}(\pi_{1}(T),SU(2))$. Denote by

$p:\mathcal{H}(T)\to\chi(T)$

the (surjective) orbit map of the conjugation action.

Define

(5.1) $\rho:\mathbb{T}\to\mathcal{H}(T),~{}\rho(e^{x{{\bf i}}},e^{y{{\bf
i}}})=\big{(}\mu\mapsto e^{x{{\bf i}}},~{}\lambda\mapsto e^{y{{\bf
i}}}\big{)}.$

######  Proposition 5.1.

The Dehn twists $D_{\mu}$ and $D_{\lambda}$ define a symplectic
$PSL(2,{\mathbb{Z}})$ action on $(\chi(T),\omega_{T})$. The map $\rho$ of
Equation (5.1) descends to a $PSL(2,{\mathbb{Z}})$ equivariant homeomorphism

$\rho:\mathbb{P}\to\chi(T)$

which restricts to a $PSL(2,{\mathbb{Z}})$ equivariant symplectomorphism

$(\mathbb{P}^{*},c~{}dx\wedge dy)\to(\chi(T)^{*},\omega_{T})$

on the top strata of the $SU(2)$ character varieties.

###### Proof.

It is well known that $\rho:\mathbb{P}\to\chi(T)$ is a well defined
homeomorphism, as well as an analytic diffeomorphism of the smooth strata
$\mathbb{P}^{*}\to\chi(T)^{*}$. This follows simply from the observations that
the fundamental group of $T$ is abelian and the Weyl group of $SU(2)$ is
${\mathbb{Z}}/2$.

We show that $\rho:(\mathbb{P}^{*},dx\wedge dy)\to(\chi(T)^{*},\omega_{T})$ is
a symplectomorphism. Since this is a local statement, we work in
${\mathbb{R}}^{2}$ for simplicity. Define $m=\rho\circ
e:{\mathbb{R}}^{2}\to\chi(T)$, where $e(x,y)=(e^{x{{\bf i}}},e^{y{{\bf i}}})$.

The differential of $m$ at $(x,y)\in{\mathbb{R}}^{2}$,
$dm:T_{(x,y)}{\mathbb{R}}^{2}\to T_{m(x,y)}(SU(2)\times SU(2))$ is given by

$dm(\tfrac{\partial}{\partial x})=\big{(}\mu\mapsto e^{x{{\bf i}}}{{\bf
i}},\lambda\mapsto 0\big{)},dm(\tfrac{\partial}{\partial y})=\big{(}\mu\mapsto
0,\lambda\mapsto e^{y{{\bf i}}}{{\bf i}}\big{)}.$

Following Weil, left translation in $SU(2)\times SU(2)$ identifies these with
the $su(2)$-valued 1-cochains

$z_{x}=(\mu\mapsto{{\bf i}},\lambda\mapsto 0),~{}z_{y}=(\mu\mapsto
0,\lambda\mapsto{{\bf i}})\in C^{1}(T;su(2)_{\operatorname{ad}m(x,y)}).$

The subspaces $L={{\bf i}}{\mathbb{R}}$ and $V={{\bf j}}{\mathbb{R}}+{{\bf
k}}{\mathbb{R}}$ are invariant and complementary with respect to
$\operatorname{ad}m(x,y)$, and therefore

$H^{1}(T;su(2)_{\operatorname{ad}m(x,y)})=H^{1}(T;L_{\operatorname{ad}m(x,y)})\oplus
H^{1}(T;V_{\operatorname{ad}m(x,y)}).$

Note that the action $\operatorname{ad}m(x,y)$ on $L$ is trivial, since
$L={{\bf i}}{\mathbb{R}}$ and $m(x,y)$ has values in the abelian subgroup
$\\{e^{{{\bf i}}u}\\}$. Hence

$H^{1}(T;L_{\operatorname{ad}m(x,y)})\cong
H^{1}(T;{\mathbb{R}})\cong{\mathbb{R}}^{2}.$

The branched cover $m:{\mathbb{R}}^{2}\to\chi(T)$ is a local diffeomorphism
near any
$(x,y)\in({\mathbb{R}}^{2})^{*}={\mathbb{R}}^{2}\setminus(\pi{\mathbb{Z}})^{2}$,
and hence it follows that for such $(x,y)$,

$H^{1}(T;{\mathbb{R}})\otimes{\mathbb{R}}{{\bf
i}}=H^{1}(T;L_{\operatorname{ad}m(x,y)})=\operatorname{Span}\\{z_{x},z_{y}\\}\text{
and }H^{1}(T;V_{\operatorname{ad}m(x,y)})=0$

(these calculations can also be easily checked directly), so that

$T_{m(x,y)}(\chi(T))=H^{1}(T;{\mathbb{R}})\otimes{\mathbb{R}}{{\bf i}}.$

The cup product

$H^{1}(T;L_{\operatorname{ad}m(x,y)})\times
H^{1}(T;L_{\operatorname{ad}m(x,y)})\to H^{2}(T;{\mathbb{R}})$

is thereby identified with

$H^{1}(T;{\mathbb{R}})\otimes{\mathbb{R}}{{\bf i}}\times
H^{1}(T;{\mathbb{R}})\otimes{\mathbb{R}}{{\bf i}}\to H^{2}(T;{\mathbb{R}})$
$(z_{1}\otimes{{\bf i}},z_{2}\otimes{{\bf i}})=(z_{1}\cup
z_{2})\cdot(-{\operatorname{Re}}({{\bf i}}{{\bf i}}))=z_{1}\cup z_{2}.$

In particular, $z_{x}=\mu^{*}\otimes{{\bf i}}$ and
$z_{y}=\lambda^{*}\otimes{{\bf i}}$, where $\mu^{*},\lambda^{*}\in
H^{1}(T;{\mathbb{Z}})=\operatorname{Hom}(H_{1}(T;{\mathbb{Z}}),{\mathbb{Z}})$
is the basis dual to $\mu,\lambda$. This basis is symplectic with respect to
the (usual, untwisted) intersection form. Hence

$\big{(}(p\circ m)^{*}(\omega_{T})\big{)}|_{(x,y)}(\tfrac{\partial}{\partial
x},\tfrac{\partial}{\partial
y})=\omega_{T}(z_{x},z_{y})=(\mu^{*}\cup\lambda^{*})\cap[T]=1=dx\wedge
dy(\tfrac{\partial}{\partial x},\tfrac{\partial}{\partial y})=1.$

Naturality of cup products shows that $D_{\mu}^{*}$, $D_{\lambda}^{*}$
preserve the symplectic form $\omega_{T}$. Next,

$D^{*}_{\mu}(m(x,y))=m(x,y)\circ D_{\mu}=\big{(}\mu\mapsto
m(x,y)(\mu)=e^{x{{\bf i}}},\lambda\mapsto m(x,y)(\mu\lambda)=e^{(x+y){{\bf
i}}}\big{)}$

and similarly

$D^{*}_{\lambda}(m(x,y))=m(x,y)\circ D_{\lambda}=\big{(}\mu\mapsto
e^{(x-y){{\bf i}}},\lambda\mapsto e^{y{{\bf i}}}\big{)}$

It follows that the subgroup of Homeo$(\chi(T))$ generated by $D^{*}_{\mu}$
and $D^{*}_{\lambda}$ pulls back, via the homeomorphism
$\rho:\mathbb{P}\to\chi(T)$, to the subgroup of $PSL(2,{\mathbb{Z}})$
generated by

$\rho^{*}(D^{*}_{\mu})=\begin{pmatrix}1&1\\\
0&1\end{pmatrix},~{}\rho^{*}(D^{*}_{\lambda})=\begin{pmatrix}1&0\\\
-1&1\end{pmatrix}.$

These two matrices generate $PSL(2,{\mathbb{Z}})$ ([Ser80]), finishing the
proof. $\square$

####  5.2.1. The solid torus and the restriction to its boundary

Let $X$ denote the solid torus with boundary $T$. Equip $T$ with based loops
$\mu,\lambda$ generating $\pi_{1}(T)$, so that $\mu$ is trivial in
$\pi_{1}(X)$ and $\lambda$ generates $\pi_{1}(X)$. Then

$\chi(X)=\chi({\mathbb{Z}}\lambda)=SU(2)/_{\rm conjugation}.$

An explicit slice of the conjugation action
$\operatorname{Hom}({\mathbb{Z}}\lambda,SU(2))\to\chi({\mathbb{Z}}\lambda)$ is
given by the map

(5.2)
$[0,\pi]\to\operatorname{Hom}({\mathbb{Z}}\lambda,SU(2)),~{}s\mapsto(\lambda\mapsto
e^{{{\bf i}}s})$

with composition
$[0,\pi]\to\chi(X)\xrightarrow{{\operatorname{Re}}_{\lambda}}[-1,1]$ equal to
the analytic isomorphism $\cos(s)$. Simple cohomology calculations show $\dim
H^{1}({\mathbb{Z}};su(2))$ equals 1 when $0<s<\pi$ and equals $3$ when $s=0$
or $\pi$. This shows that the interior of the interval forms the smooth top
stratum of $\chi({\mathbb{Z}})$, and the endpoints are singular.

Since $\mu=1\in\pi_{1}(X)$, the restriction-to-the boundary map

(5.3) $\chi(X)\to\chi(T)$

is is easily computed, in $\mathbb{P}$, to be the smooth (necessarily
Lagrangian) embedded arc given by:

(5.4) $[0,\pi]\ni s\mapsto[e^{s{{\bf i}}},1]\in\mathbb{P}$

with endpoints at $[-1,1]$ and $[1,1]$.

###  5.3. The traceless $SU(2)$ character variety of the 4-punctured 2-sphere

Let $4D^{2}\subset S^{2}$ be four disjoint open disks, and set

$F=S^{2}\setminus 4D^{2},~{}\partial F=4S^{1}.$

Then $\pi_{1}(F)$ has presentation

$\pi_{1}(F)=\langle a,b,c,d~{}|~{}abcd=1\rangle$

and is free on $a,b,c$. Set

$\mathcal{H}(S^{2},4)=\\{\rho\in\operatorname{Hom}(\pi_{1}(F),SU(2))~{}|~{}\rho(a),\rho(b),\rho(c),\rho(d)\in
J\\}.$

Let $p:\mathcal{H}(S^{2},4)\to\chi(S^{2},4)$ denote the orbit map of the
conjugation action.

Define

(5.5) ${\hat{\rho}}:\mathbb{T}\to\mathcal{H}(S^{2},4),~{}(e^{x{{\bf
i}}},e^{y{{\bf i}}})\mapsto\big{(}a\mapsto{{\bf j}},~{}b\mapsto e^{x{{\bf
i}}}{{\bf j}},~{}c\mapsto e^{y{{\bf i}}}{{\bf j}}\big{)}.$

A half Dehn twist of $(D^{2},p,q)$, where $p,q$ are a fixed pair of interior
points, is a homeomorphism of the disk which fixes the boundary and permutes
$p$ and $q$, and which generates the infinite cyclic mapping class group rel
boundary of a disk with two marked interior points. The generator 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 $\hat{\rho}:\mathbb{P}\to\chi(S^{2},4)$ is a well defined homeomorphism,
as well as an analytic diffeomorphism of the smooth strata
$\mathbb{P}^{*}\to\chi(S^{2},4)^{*}$, is simple; its proof can be found in
[Lin92, HK98].

We show that $\hat{\rho}:(\mathbb{P}^{*},c~{}dx\wedge
dy)\to(\chi(S^{2},4)^{*},\widehat{\omega}_{(S^{2},4)})$ is a
symplectomorphism, for some constant $c$. Since this is a local statement, we
work in ${\mathbb{R}}^{2}$ for simplicity. Define $\hat{m}=\hat{\rho}\circ
e:{\mathbb{R}}^{2}\to\chi(S^{2},4)$, where $e(x,y)=(e^{x{{\bf i}}},e^{y{{\bf
i}}})$.

The differential of $\hat{m}$ at $(x,y)\in{\mathbb{R}}^{2}$ is given by

$d\hat{m}(\tfrac{\partial}{\partial x})=\big{(}a\mapsto 0,b\mapsto e^{x{{\bf
i}}}{{\bf i}}{{\bf j}},c\mapsto 0\big{)},d\hat{m}(\tfrac{\partial}{\partial
y})=\big{(}a\mapsto 0,b\mapsto 0,c\mapsto e^{y{{\bf i}}}{{\bf i}}{{\bf
j}}\big{)}.$

Translation in $SU(2)^{3}$ identifies these with the $su(2)$-valued 1-cochains

$z_{x}=\big{(}a\mapsto 0,b\mapsto-{{\bf i}},c\mapsto
0\big{)},z_{y}=\big{(}a\mapsto 0,b\mapsto 0,c\mapsto-{{\bf i}}\big{)}.$

Since $\hat{m}$ is a local diffeomorphism away from $(\pi{\mathbb{Z}})^{2}$
[HHK14], and $\tfrac{\partial}{\partial x},\tfrac{\partial}{\partial y}$ span
$T_{(x,y)}{\mathbb{R}}^{2}$, the cohomology classes $[z_{x}],~{}[z_{y}]$ span

$\widehat{H}^{1}(S^{2}-4D^{2};su(2)_{\operatorname{ad}\hat{m}(x,y)})=T_{[\hat{m}(x,y)]}(\chi(S^{2},4)).$

For any $(x,y)\in{\mathbb{R}}^{2}\setminus(\pi{\mathbb{Z}})^{2}$, the adjoint
action $\operatorname{ad}\hat{m}(x,y):\pi_{1}(S^{2}-4D^{2})\to GL(su(2))$
reduces as the direct sum

$\operatorname{ad}\hat{m}(x,y)=\operatorname{ad}\hat{m}(x,y)_{1}\oplus\operatorname{ad}\hat{m}(x,y)_{2}:\pi_{1}(S^{2}-4D^{2})\to
GL({\mathbb{R}}{{\bf i}})\times GL({\mathbb{C}}{{\bf j}})$

and hence

$\widehat{H}^{1}(S^{2}-4D^{2};su(2)_{\operatorname{ad}\hat{m}(x,y)})=\widehat{H}^{1}(S^{2}-4D^{2};{\mathbb{R}}{{\bf
i}}_{{\operatorname{ad}\hat{m}(x,y)}_{1}})\oplus\widehat{H}^{1}(S^{2}-4D^{2};{\mathbb{C}}{{\bf
j}}_{{\operatorname{ad}\hat{m}(x,y)}_{2}}).$

Since ${\mathbb{R}}{{\bf i}}$ and ${\mathbb{C}}{{\bf j}}$ are orthogonal, the
symplectic form $\widehat{\omega}_{S^{2},4}$ splits orthogonally

$\widehat{\omega}_{S^{2},4}=\widehat{\omega}_{S^{2},4}^{1}\oplus\widehat{\omega}_{S^{2},4}^{2}.$

The cocyles $z_{x}$ and $z_{y}$ lie in the first summand, and hence they span
the first summand and the second summand is zero (these two facts can also be
easily calculated directly). Hence
$\widehat{\omega}_{S^{2},4}=\widehat{\omega}_{S^{2},4}^{1}.$

The representation on the first summand independent of $(x,y)$: indeed $a,b,$
and $c$ (and hence also $d$) act by $-1$ for all $x,y$. The cocycles
$z_{x},z_{y}$ are independent of $x,y$, and hence

$\hat{m}^{*}(\widehat{\omega}_{S^{2},4})|_{(x,y)}(\tfrac{\partial}{\partial
x},\tfrac{\partial}{\partial
y})=\widehat{\omega}^{1}_{S^{2},4}(z_{x},z_{y})=cdx\wedge
dy(\tfrac{\partial}{\partial x},\tfrac{\partial}{\partial y})$

for a non-zero constant $c$ (since $z_{x},z_{y}$ span).

The half-Dehn twists along the disks $H_{1}$ and $H_{2}$ illustrated in Figure
1 induce automorphisms of $\pi_{1}(S^{2}\setminus 4,s_{0})$, for a base point
chosen outside the supports of $H_{1}$ and $H_{2}$, as indicated in the
figure. These automorphisms are given by

$H_{1}:a\mapsto a,b\mapsto b,c\mapsto d=\bar{c}\bar{b}\bar{a}$ $H_{2}:a\mapsto
a,b\mapsto cd\bar{c}=\bar{b}\bar{a}\bar{c},c\mapsto c$

and induce homeomorphisms $H_{1}^{*},H_{2}^{*}:\chi(S^{2},4)\to\chi(S^{2},4)$.
Naturality of cup products shows that $H_{1}^{*}$, $H_{2}^{*}$ preserve the
symplectic form $\widehat{\omega}_{(S^{2},4)}$.

Next:

$\displaystyle H_{1}^{*}(m(x,y))$ $\displaystyle=\big{(}a\mapsto
m(x,y)(a)={{\bf j}},b\mapsto m(x,y)(b)=e^{x{{\bf i}}},c\mapsto
m(x,y)(\bar{c}\bar{b}\bar{a})=e^{(y-x){{\bf i}}}{{\bf j}}\big{)}$
$\displaystyle=m(x,y-x)$

and similarly

$H_{2}^{*}(m(x,y))=\big{(}a\mapsto{{\bf j}},b\mapsto
m(x,y)(\bar{b}\bar{a}\bar{c})=e^{(x+y){{\bf i}}}{{\bf j}},c\mapsto e^{y{{\bf
i}}})=m(x+y,y).$

It follows that the subgroup of Homeo$(\chi(S^{2},4))$ generated by
$H_{1}^{*}$ and $H_{2}^{*}$ pulls back, via the homeomorphism
$\hat{\rho}:\mathbb{P}\to\chi(T)$, to the subgroup of $PSL(2,{\mathbb{Z}})$
generated by

$\hat{\rho}^{*}(H_{1}^{*})=\begin{pmatrix}1&-1\\\
0&1\end{pmatrix},~{}\hat{\rho}^{*}(H_{2}^{*})=\begin{pmatrix}1&0\\\
1&1\end{pmatrix}.$

These two matrices generate $PSL(2,{\mathbb{Z}})$ ([Ser80]), finishing the
proof. $\square$

Mapping classes of $(S^{2},4)$ permute the four punctures. The subgroup of the
mapping class group of $(S^{2},4)$ which fixes the point labelled $a$ in
Figure 1 can be shown to act on $\chi(S^{2},4)$; this subgroup is isomorphic
to $PSL(2,{\mathbb{Z}})$ (see, e.g., [FM12]), generated by these half twists.

##  6\. Perturbations

The holonomy perturbation process is easily understood, as well as motivated,
in the language of Weinstein composition of Lagrangian immersions.

###  6.1. Composition

Given any two (set) maps

$\alpha:A\to M\text{ and }\beta=\beta_{M}\times\beta_{N}:B\to M\times N,$

define the composition $(A\times_{M}B,\beta^{\alpha}_{N})$ by

(6.1) $A\times_{M}B:=\\{(a,b)\in A\times
B~{}|~{}\alpha(a)=\beta_{M}(b)\\}=(\alpha\times\beta_{M})^{-1}(\Delta_{M})$

and

(6.2) $\beta^{\alpha}_{N}:A\times_{M}B\to
N,~{}\beta^{\alpha}_{N}(a,b):=\beta_{N}(b).$

####  6.1.1. Composition in character varieties

Recall that if $\rho:\Gamma\to G$ is a homomorphism,

$Stab(\rho)=\\{g\in G~{}|~{}g\rho(\gamma)g^{-1}=\rho(\gamma)\text{ for all
}\gamma\in\Gamma\\}.$

The proof of the following simple lemma can be found in [HHK14, Lemma 4.2].

######  Lemma 6.1.

Suppose that $\Gamma_{0},\Gamma_{1}$, $H$ are groups, $h_{0}:H\to\Gamma_{0}$,
$h_{1}:H\to\Gamma_{1}$ homomorphisms. Set $\Gamma=\Gamma_{0}*_{H}\Gamma_{1}$,
the pushout along $h_{0},h_{1}$. Then there is a surjection

$\chi(\Gamma)\to\chi(\Gamma_{0})\times_{\chi(H)}\chi(\Gamma_{1})$

with fiber over $([\rho_{0}],[\rho_{1}])$ the double coset

$Stab(\rho_{0})\backslash Stab(\rho_{0}|_{H})/Stab(\rho_{1}).$

The fibers $Stab(\rho_{0})\backslash Stab(\rho_{0}|_{H})/Stab(\rho_{1})$ are
called gluing parameters.

######  Proposition 6.2.

If $Z=Z_{0}\cup_{\Sigma}Z_{1}$ is a decomposition of a compact 3-manifold
along a closed separating surface $\Sigma$, with
$\pi_{1}(\Sigma)\to\pi_{1}(Y_{0})$ surjective, then
$\chi(Z)\to\chi(Z_{0})\times_{\chi(\Sigma)}\chi(Z_{1})$ is a homeomorphism, in
fact, an algebraic isomorphism.

###### Proof.

Choose
$([\rho_{0}],[\rho_{1}])\in\chi(Z_{0})\times_{\chi(\Sigma)}\chi(Z_{1})$. Since
$\pi_{1}(\Sigma)\to\pi_{1}(Y_{0})$ is surjective, $\rho_{0}$ and
$\rho_{0}|_{\pi_{1}(\Sigma)}$ have the same image, and hence equal
stabilizers. Therefore $Stab(\rho_{0})\backslash
Stab(\rho_{0}|_{\Sigma})/Stab(\rho_{1})$ is a single point, and the proof
follows from Lemma 6.1. $\square$

######  Corollary 6.3.

If $Z=Z_{0}\cup_{\Sigma}Z_{1}$ with $Z_{0}$ a handlebody, then

$\chi(Z)=\chi(Z_{0})\times_{\chi(\Sigma)}\chi(Z_{1}).$

####  6.1.2. Composition in the Weinstein category

The Weinstein category [Wei82] is, roughly speaking, a category with objects
symplectic manifolds and morphisms Lagrangian immersions. Composition is not
always defined, however. The following criterion ensures that a composition of
Lagrangian immersions is defined.

######  Lemma 6.4.

[GS82, §4.1], [BW18, lemma 2.0.5] Suppose that $M,N$ are symplectic manifolds,

$\alpha:A\to M\text{ and }\beta=\beta_{M}\times\beta_{N}:B\to M^{-}\times N$

are Lagrangian immersions (with $M^{-}$ obtained from $M$ by reversing the
sign of the symplectic form). If $\alpha\times\beta_{M}$ is transverse to the
diagonal $\Delta_{M}\in M\times M$, then $A\times_{M}B$ is a smooth manifold
and $\beta^{\alpha}_{N}:A\times_{M}B\to N$ is a Lagrangian immersion.

When the transversality assumption in Lemma 6.4 holds, one says the
composition

$(A\times_{M}B,\beta^{\alpha}_{N})$

of $(A,\alpha)$ and $(B,\beta)$ is defined and immersed.

In cases where the transversality assumption in Lemma 6.4 does not hold, a
differential topological approach to remedying the situation would be to
deform either or both of the immersions $\alpha,\beta$. In order to retain the
symplectic properties, one would typically deform them by Hamiltonian flows.
In the context in this article, where character varieties correspond to flat
moduli spaces in the gauge theoretic framework, we seek deformations in Lemma
6.4 that correspond to holonomy perturbations in the gauge theory context; we
describe these in the next section.

###  6.2. SU(2) and holonomy perturbations

We return to the pair $(Y,C)$, determining $S=V\cup F$ as above, so
$S=\partial Y$ and $V=nbd(C)$.

Suppose that $e:D^{2}\times S^{1}\hookrightarrow Int(Y)$ is an embedding of a
solid torus. Denote its image by $Y_{0}$, the closure of the complement by
$Y_{1}$, and the separating torus by $T$, so that $Y=Y_{0}\cup_{T}Y_{1}.$
Corollary 6.3 shows that

$\chi(Y,C)=\chi(Y_{0})\times_{\chi(T)}\chi(Y_{1},C),$

so that $\chi(Y,C)\to\chi(F,\partial F)$ is exhibited as the composition of

$\alpha:\chi(Y_{0})\to\chi(T)\text{ and
}\beta:\chi(Y_{1},C)\to\chi(T)\times\chi(F,\partial F).$

For a deformation $\alpha_{\pi}:\chi(Y_{0})\to\chi(T)$ of $\alpha$, or more
generally a family of functions $\alpha_{\pi},~{}\pi\in\mathcal{U}$ with
$\mathcal{U}$ a manifold (for example, a small open interval), we can view the
deformed composition

(6.3)
$\chi_{\pi}(Y,C):=\chi(Y_{0})\circ_{\pi}\chi(Y_{1},C),~{}\beta^{\alpha_{\pi}}:\chi(Y_{0})\circ_{\pi}\chi(Y_{1},C)\to\chi(F,\partial
F)$

as a perturbed character variety with perturbation data $\pi$.

Restrict to the case when $G=SU(2)$ and $J$ is the conjugacy class of
imaginary unit quaternions. Then $\chi(Y_{0})$ is simply an arc
$[e^{is}],0\leq s\leq\pi$, and the immersion to $\chi(\partial Y_{0})$ is
$\alpha:[e^{is}]\mapsto[e^{is},1]$ in the pillowcase, described in (5.3) and
(5.4). This map descends from the smooth, ${\mathbb{Z}}_{2}$ equivariant map
$e^{is}\to(e^{is},1)$ from $S^{1}$ to $T$.

Let $f:{\mathbb{R}}\to{\mathbb{R}}$ be a smooth, odd, $2\pi$ periodic
function. Then $f$ determines a ${\mathbb{Z}}_{2}$ equivariant Hamiltonian
deformation $e^{is}\mapsto(e^{is},e^{if(s)})$, inducing the deformation of
$\alpha$ given by

(6.4) $\alpha_{f}:[0,\pi]\to\chi(S^{1}\times S^{1}),~{}\alpha_{f}(s)=[e^{{{\bf
i}}s},e^{{{\bf i}}f(s)}].$

The definition extends easily to the setting of a finite disjoint collection
of embeddings of solid tori $e=\\{e_{i}\\}_{i=1}^{k}$ of disjointly embedded
solid tori in $Y$ and a corresponding collection of smooth, odd, periodic
functions $f_{1},\dots,f_{k}$ as above. Denote by $\pi$ this set of
perturbation data $\\{(e_{i},f_{i})\\}_{i=1}^{k}$. Letting $Y_{0}$ denote the
union of the solid tori, Equation (6.3) defines a way to deform the character
variety.

Notice that it suffices to think of $e$ as framed link in $Y$, since isotopic
embeddings yield equal perturbed character varieties.

######  Definition 6.5.

Let $\mathcal{V}$ be the vector space of smooth, odd, $2\pi$ periodic
functions ${\mathbb{R}}\to{\mathbb{R}}$. Fix $(Y,C)$ as above, where $C$ may
or may not be nonempty). Given perturbation data

$\pi=(e,f)=(\sqcup_{i=1}^{k}e_{i}:D^{2}\times S^{1}\subset
Y,f=(f_{i})\in\mathcal{V}^{k}),$

define $\chi_{\pi}(Y,C)$ to be the resulting perturbed (traceless) character
variety.

In light of Equation (6.4), $\chi_{\pi}(Y,C)$ 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 $\pi=(e,f)$ be a choice of perturbation data, and define
$Y_{0}\subset Y$ to be the disjoint union of solid tori
$\sqcup_{i}e_{i}(D^{2}\times S^{1})$. Finally define
$\lambda_{i},\mu_{i}\in\pi_{1}(Y\setminus Y_{0})$ to be the loops
$e_{i}(S^{1}\times\\{1\\})$ and $e_{i}(\\{1\\}\times\partial D^{2})$,
connected to the base point in some way.

Then

$\chi_{\pi}(Y,C)=\\{\rho\in\chi(Y\setminus Y_{0},C)~{}|~{}\text{ if
$\rho(\lambda_{i})=e^{{{\bf i}}s}$, then $\rho(\mu_{i})=e^{{{\bf
i}}f(s)}$}\\}.$

######  Theorem 6.7 ([Tau90, Her94]).

Given a set of perturbation data $\pi=(e,f)$ as in Definition 6.5, there is a
holonomy perturbation $h_{(e,f)}$ of the flatness equation on $SU(2)$
connections for which the perturbed flat moduli space is identified with the
perturbed character variety as described above.

######  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 a similar composition interpretation of the perturbed
character variety, with $Y_{0}$ the handlebody. But in this setting, an
explicit description of the perturbed character variety and the counterpart to
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 $A\in\chi_{\pi}(Y,C)$ is a regular point. Then $A$ admits a
neighborhood $U$ so that $r|_{U}:U\to\chi(F,\partial F)^{*}$ is a Lagrangian
embedding.

###### Proof.

Apply Lemma 6.4. $\square$

###  6.3. Dependence on perturbations

In [Her94], the second author showed that (when ${\mathcal{L}}$ is empty)
different holonomy perturbations in general yield Legendrian cobordant
immersions. Unfortunately this usually does not guarantee that they have
isomorphic Floer homology. We now 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 to the immersed case as
developed by Bottman-Wehrheim [BW18].

The reader will find a discussion of holonomy perturbations in cylinders
$(S,p)\times[0,1]$, with $p$ a finite set of points in a surface $S$, in
[HK18]. In particular, Theorem 6.3 of that article states the following. Take
perturbation data $\pi$ with framed perturbation curve obtained by pushing a
simple closed curve in $S\setminus p$ into the interior of $S\times I$. Then
define the 1-parameter family of holonomy perturbations
$s\pi,s\in[0,\epsilon)$.

The restriction

$\chi_{s\pi}((S,p)\times I)\to\chi(S_{0},p)\times\chi(S_{1},p)$

can be identified with the family of graphs of a Hamiltonian isotopy of
$\chi(S)$ known as the Goldman twist flow associated to the simple closed
curve [Gol84]. This implies that the holonomy perturbation process can be
viewed as a combination of a decomposition induced by cutting a 3 manifold
along a separating torus $T$, followed by a perturbation as in Equation (6.3),
with $\alpha_{s\pi}:\chi(S^{1}\times D^{2})\to\chi(T)$ the composition of the
unperturbed inclusion $\alpha_{0}$ followed by a small time 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

$\iota_{0}\colon L_{0}\looparrowright M,~{}\iota_{1}\colon
L_{1}\looparrowright M$

_secretly Hamiltonian isotopic_ if they can be expressed as compositions with
some $\beta:\Lambda\looparrowright X\times M$:

$\displaystyle i_{0}:L_{0}\looparrowright M$
$\displaystyle=\beta^{j_{0}}_{M}\colon
L_{0}^{\prime}\times_{X}\Lambda\looparrowright M,$ $\displaystyle
i_{1}:L_{1}\looparrowright M$ $\displaystyle=\beta^{j_{1}}_{M}\colon
L_{1}^{\prime}\times_{X}\Lambda\looparrowright M.$

in such a way that $j_{0},j_{1}:L_{0}^{\prime},L_{1}^{\prime}\looparrowright
X$ are Hamiltonian isotopic in $X$. It follows from the discussion in Section
6.2 that different choices of holonomy perturbations induce secretly
Hamiltonian isotopic immersed Lagrangians.

Being secretly Hamiltonian isotopic does not necessarily imply being
Hamiltonian isotopic. Indeed, $L_{0}$ and $L_{1}$ need not even be
diffeomorphic. Moreover, in the absence of extra hypotheses (embedded
composition, monotonicity, exactness, etc.), the conclusion of the Wehrheim-
Woodward composition theorem [WW10] need not hold. In particular, given a
third Lagrangian $L_{2}$, $HF(L_{0},L_{2})$ and $HF(L_{1},L_{2})$ need not be
isomorphic, even if they are both well-defined.

However, provided:

  * •

all Lagrangian immersions and symplectic manifolds satisfy suitable
assumptions so to be able to define Lagrangian (quilted) Floer homology,

  * •

all Lagrangian immersions come equipped with suitable bounding cochains, in a
way consistent with composition.

Then it would follow from the Bottman-Wehrheim conjecture [BW18, Sec. 4.4]
(see also a similar statement in [Fuk17]) that the secretly Hamiltonian
isotopic Lagrangian immersions $L_{0}$ and $L_{1}$, when paired with any test
Lagrangian $L_{2}\subset M$, produce isomorphic Floer homology groups
$HF(L_{0},L_{2})\simeq HF(L_{1},L_{2})$ (since these would respectively
correspond to the quilted Floer homology groups
$HF(L_{0}^{\prime},\Lambda,L_{2})$ and $HF(L_{1}^{\prime},\Lambda,L_{2})$,
which are isomorphic).

In that sense secretly Hamiltonian isotopic Lagrangian immersions $L_{0}$ and
$L_{1}$ can be thought as being equivalent. In particular, the problem of
dependence of Floer theory on the choice of holonomy perturbation is seen as a
special case of the general problem of dependence of quilted Floer homology on
composition.

## References

  * [AB83] M. F. Atiyah and R. Bott.  The Yang-Mills equations over Riemann surfaces.  Philos. Trans. Roy. Soc. London Ser. A, 308(1505):523–615, 1983\.
  * [Ati88] M. Atiyah.  New invariants of $3$\- and $4$-dimensional manifolds.  In The mathematical heritage of Hermann Weyl (Durham, NC, 1987), volume 48 of Proc. Sympos. Pure Math., pages 285–299. Amer. Math. Soc., Providence, RI, 1988.
  * [Ati90] M. Atiyah.  The geometry and physics of knots.  Lezioni Lincee. [Lincei Lectures]. Cambridge University Press, Cambridge, 1990.
  * [BG93] I. Biswas and K. Guruprasad.  Principal bundles on open surfaces and invariant functions on Lie groups.  Internat. J. Math., 4(4):535–544, 1993.
  * [Bro94] Kenneth S. Brown.  Cohomology of groups, volume 87 of Graduate Texts in Mathematics.  Springer-Verlag, New York, 1994.  Corrected reprint of the 1982 original.
  * [BW18] N. Bottman and K. Wehrheim.  Gromov compactness for squiggly strip shrinking in pseudoholomorphic quilts.  Selecta Math. (N.S.), 24(4):3381–3443, 2018.  ArXiv: 1503.03486.
  * [CHKK20] G. Cazassus, C. M. Herald, P. Kirk, and A. Kotelskiy.  The correspondence induced on the pillowcase by the earring tangle.  2020\.  ArXiv preprint 2010.04320.
  * [CLM20] D Crowley, W Lück, and T. Macko.  Surgery theory: Foundations.  2020\.
http://thales.doa.fmph.uniba.sk/macko/surgery-book.html.

  * [Flo88] A. Floer.  An instanton-invariant for $3$-manifolds.  Comm. Math. Phys., 118(2):215–240, 1988.
  * [FM12] Benson Farb and Dan Margalit.  A primer on mapping class groups, volume 49 of Princeton Mathematical Series.  Princeton University Press, Princeton, NJ, 2012.
  * [Fuk17] K. Fukaya.  Unobstructed immersed Lagrangian correspondence and filtered A infinity functor.  2017\.  ArXiv preprint 1706.02131.
  * [GHJW97] K. Guruprasad, J. Huebschmann, L. Jeffrey, and A. Weinstein.  Group systems, groupoids, and moduli spaces of parabolic bundles.  Duke Math. J., 89(2):377–412, 1997.
  * [Gol84] W. M. Goldman.  The symplectic nature of fundamental groups of surfaces.  Adv. in Math., 54(2):200–225, 1984.
  * [GS82] Victor Guillemin and Shlomo Sternberg.  Moments and reductions.  In Differential geometric methods in mathematical physics (Clausthal, 1980), volume 905 of Lecture Notes in Math., pages 52–65. Springer, Berlin-New York, 1982.
  * [Her94] C. M. Herald.  Legendrian cobordism and Chern-Simons theory on $3$-manifolds with boundary.  Comm. Anal. Geom., 2(3):337–413, 1994.
  * [HHK14] M. Hedden, C. M. Herald, and P. Kirk.  The pillowcase and perturbations of traceless representations of knot groups.  Geom. Topol., 18(1):211–287, 2014.  ArXiv: 1301.0164.
  * [HK98] M. Heusener and J. Kroll.  Deforming abelian ${\rm SU}(2)$-representations of knot groups.  Comment. Math. Helv., 73(3):480–498, 1998.
  * [HK18] C. M. Herald and P. Kirk.  Holonomy perturbations and regularity for traceless $\rm SU(2)$ character varieties of tangles.  Quantum Topol., 9(2):349–418, 2018.  ArXiv: 1511.00308.
  * [Kar92] Yael Karshon.  An algebraic proof for the symplectic structure of moduli space.  Proc. Amer. Math. Soc., 116(3):591–605, 1992.
  * [KM11] P. B. Kronheimer and T. S. Mrowka.  Khovanov homology is an unknot-detector.  Publ. Math. Inst. Hautes Études Sci., (113):97–208, 2011.  ArXiv: 1005.4346.
  * [KS96] Christopher King and Ambar Sengupta.  A symplectic structure for connections on surfaces with boundary.  Comm. Math. Phys., 175(3):657–671, 1996.
  * [Lin92] X.-S. Lin.  A knot invariant via representation spaces.  J. Differential Geom., 35(2):337–357, 1992.
  * [LOT18] R. Lipshitz, P. S. Ozsváth, and D. P. Thurston.  Bordered Heegaard Floer homology.  Mem. Amer. Math. Soc., 254(1216):viii+279, 2018.  ArXiv: 0810.0687v5.
  * [Mil62] John Milnor.  A duality theorem for Reidemeister torsion.  Ann. of Math. (2), 76:137–147, 1962.
  * [Ser80] Jean-Pierre Serre.  Trees.  Springer-Verlag, Berlin-New York, 1980.  Translated from the French by John Stillwell.
  * [Spa95] Edwin H. Spanier.  Algebraic topology.  Springer-Verlag, New York, [1995].  Corrected reprint of the 1966 original.
  * [Tau90] C. H. Taubes.  Casson’s invariant and gauge theory.  J. Differential Geom., 31(2):547–599, 1990.
  * [Wei64] André Weil.  Remarks on the cohomology of groups.  Ann. of Math. (2), 80:149–157, 1964.
  * [Wei82] A. Weinstein.  The symplectic “category”.  In Differential geometric methods in mathematical physics (Clausthal, 1980), volume 905 of Lecture Notes in Math., pages 45–51. Springer, Berlin-New York, 1982.
  * [Wei10] Alan Weinstein.  Symplectic categories.  Port. Math., 67(2):261–278, 2010.
  * [WW10] K. Wehrheim and C. Woodward.  Quilted Floer cohomology.  Geom. Topol., 14(2):833–902, 2010.  ArXiv: 0905.1370.
  * [WW15] K. Wehrheim and C. Woodward.  Floer field theory for tangles.  2015\.  ArXiv preprint 1503.07615.
  * [WW16] K. Wehrheim and C. Woodward.  Floer field theory for coprime rank and degree.  2016\.  ArXiv preprint 1601.04924.