Low-regularity Seiberg–Witten moduli spaces on manifolds with boundary

For a compact spinc\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$^{c}$$\end{document} manifold X with boundary b1(∂X)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$b_1(\partial X)=0$$\end{document}, we consider moduli spaces of solutions to the Seiberg–Witten equations in a generalized double Coulomb slice in W1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{1,2}$$\end{document} Sobolev regularity. We prove they are Hilbert manifolds, prove denseness and “semi-infinite-dimensionality” properties of the restriction to ∂X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial X$$\end{document}, and establish a gluing theorem. To achieve these, we prove a general regularity theorem and a strong unique continuation principle for Dirac operators, and smoothness of a restriction map to configurations of higher regularity on the interior, all of which are of independent interest.


Introduction
This article is motivated by an effort to provide a new framework for constructing socalled Floer theories, which are used to construct invariants of low-dimensional manifolds, knots, as well as Lagrangians in symplectic manifolds.Originally, Floer implanted ideas from finite-dimensional Morse theory (cf.[Sch93,Wit82]) to the study of functions on infinite-dimensional spaces (often called functionals on configuration spaces), leading to the resolution of the Arnold conjecture in symplectic topology [Flo87,Flo88].Inspired by Donaldson's proof of the diagonalization theorem for 4-manifolds [Don83], Floer used these ideas to construct homological invariants of 3-manifolds, aiming to establish gluing formulas for Donaldson's invariants.An analogous construction for knots in 3-manifolds has been carried out by Kronheimer and Mrowka and plays the key role in the proof that Khovanov homology detects the unknot [KM11].An analogue for the Seiberg-Witten equations (in place of the anti-self-duality (ASD) equations of Donaldson's theory) is the central ingredient of Manolescu's disproof of the triangulation conjecture in dimensions ≥ 5 [Man13].
While the idea of using Morse theory in infinite-dimensional settings dates back to Atiyah and Bott [AB83], Floer's novelty was dealing with functionals having critical points of infinite index.They key observation was that the unstable manifolds of critical points were of finite index with respect to a certain subbundle of the tangent space, allowing one to define finite relative indices of critical points and eventually leading to a computation of "middle-dimensional" homology groups of the space.As noted by Atiyah [Ati88] (cf.[CJS95]), Floer theory was, from the very beginning, understood as describing the behavior of so-called semi-infinite-dimensional cycles.
As in Morse homology (cf.[Sch93]), one needs to overcome analytical difficulties to even define Floer theories and prove they are well-defined and independent of choices -one needs to establish regularity and compactness of moduli spaces of trajectories between critical points.Moreover, Morse theory is generally not well-suited for equivariant constructions since one in general cannot guarantee regularity of the moduli spaces without breaking the symmetry coming from a group action.To address these problems and to allow general constructions of equivariant (cf.[DS19,KM07,Lin18,Mil19]) and generalized Floer homologies (cf.[Man14, Lin15, AB21]), Lipyanskiy [Lip08] introduced a framework for using such semi-infinite-dimensional cycles as a tool for defining Floer theories and the author has further developed these methods in [Suw20].This article contains results required to rigorously define the relative invariants of 4-manifolds with boundary and maps induced by cobordisms in this construction, using the Seiberg-Witten equations.Moreover, the key result is a gluing theorem which is fundamental for establishing functoriality of the cobordism maps.
Results.Consider a 4-manifold X with boundary a nonempty collection of rational homology spheres (i.e., b 1 (∂X) = 0) and a spin c structure ŝ on X.We study the moduli space of solutions to the Seiberg-Witten equations for pairs (A, Φ) of a spin c connection A on X and a section Φ of the spinor bundle S + over X associated to the spin c structure ŝ: + A Φ = 0. (Definition 3.4).The split Coulomb slice with respect to a reference connection A 0 is given by the equations d * (A − A 0 ) = 0, d * (ι * ∂X (A − A 0 )) = 0, (where ι ∂X : ∂X ֒→ X denotes the inclusion) together with a condition restricting A − A 0 to a subset of codimension b 0 (∂X) − 1 (Definition 3.15).This additional condition depends on a choice of a gauge splitting s (Definition 3.11).The moduli space M • s (X, ŝ) (Definition 3.31) is the quotient of the space of L 2 1 -solutions to the Seiberg-Witten equations in this split Coulomb slice by the (discrete) action of the split gauge group (Definition 3.18); this gauge group preserves the split Coulomb slice as well as the set of solutions to the Seiberg-Witten equations.There is also a residual action of S 1 on M • s (X, ŝ) given by multiplication of the spinor component, Φ → zΦ, by complex numbers in the unit circle z ∈ S 1 ⊂ C.
The maps Π ± come from a decomposition C cC (∂X, s) = H + (∂X, s)⊕ H − (∂X, s) according to the eigenvalues of the operator (⋆d, / D B 0 ), also called a polarization (Definition 3.3).The proof utilizes the Atiyah-Patodi-Singer boundary value problem for an extended linearized Seiberg-Witten operator (Definition 4.1).Then we prove its properties transfer to the restriction to the Coulomb slice.
Our low regularity setting requires us to prove a regularity theorem (Regularity Theorem) for an operator of the form D = D 0 + K : L 2 1 → L 2 , where D 0 is a Dirac operator and K is any compact operator, making it a result of independent interest.We also prove a strong version of the unique continuation principle for D (Strong UCP Theorem).
Secondly, we prove a gluing theorem for a composite cobordism.Assume X splits as X = X 1 ∪ Y X 2 along a rational homology sphere Y .If the auxiliary data of gauge splittings and gauge twistings are compatible in a suitable sense (see Proposition 3.29 and Proposition 5.5), then the moduli space M • s (X, ŝ) can be recovered from the fiber product of M • s 1 (X 1 , ŝ) and M • s 2 (X 2 , ŝ) over the configuration space of Y , in a way compatible with the twisted restriction maps: Gluing Theorem.Assume s Z and (s 1,Z , s 2,Z ) are compatible.Then there is an The proof uses the following fact of independent interest.We show that the restriction map of solutions in M • s (X, ŝ) to a submanifold in the interior of X is smooth as a map into a configuration space of higher regularity (Theorem 5.2).While it is easy to prove that its image lies in the space of smooth configurations (Lemma 5.1), the proof of smoothness of this map is non-trivial.
Applications.The Semi-infinite-dimensionality Theorem together with compact- ness of moduli spaces proved in [KM07] (with minor modifications to account for the double Coulomb slice instead of the Coulomb-Neumann slice used in [KM07]) show that the maps R τ : as defined in [Lip08,Suw20], establishing the existence of relative Seiberg-Witten invariants of X.If the boundary ∂X is connected, these do not depend on the choice of an integral splitting (Lemma 3.30).This also implies that cobordisms ∂W = −Y 1 ∪ Y 2 induce correspondences (defined in [Lip08,Suw20]) between the configuration spaces over Y 1 and Y 2 .
Our methods apply to perturbed equations as well, which we did not include for the sake of simplicity.Varying the metrics and perturbations gives cobordisms between the relevant moduli spaces on X, and changing the reference connections on X gives isomorphic moduli spaces with homotopic restriction maps.This means that the relative invariant of X, up to a suitable cobordism relation, does not depend on the choices of perturbations and metric.
Crucially, the Gluing Theorem says that the correspondence induced by a composite cobordism is (homotopic to) the composition of the respective correspondences, proving the theory comes with a TQFT-like structure.We hope to prove that this theory recovers a non-equivariant version of monopole Floer homology HM * and describe a construction supposed to recover all of the flavors of monopole Floer homology (for rational homology spheres) in an upcoming work.
Applications of Seiberg-Witten-Floer theory suggest including the case b 1 (∂X) > 0 or allowing X to have cylindrical or conical ends (like in [KM97]) with certain asymptotic conditions on the configurations.The methods presented here should suffice to prove regularity and semi-infinite-dimensionality of the corresponding moduli spaces.However, to establish the semi-infinite-dimensional theory in full one needs to carefully select a slice for the gauge action and deal with compactness issues which we hope to address in further work.
Finally, the methods presented here should be applicable (with appropriate adjustments) for defining semi-infinite-dimensional Floer theories in other contexts, e.g., in the Yang-Mills-Floer theory.
Organization.In section 2 we prove the regularity theorem for a Dirac operator with compact perturbation, the Regularity Theorem, and a strong unique continuation principle for a Dirac operator with a low-regularity potential, the Strong UCP Theorem.
In section 3 we introduce the basic notions of Seiberg-Witten theory on 3-and 4manifolds.For a chosen gauge splitting the split Coulomb slice is defined, and for a gauge twisting a twisted restriction map to the boundary is introduced.The choices of splittings and twistings are shown to be equivalent to a choice of an integral splitting, one which does not require any twisting.
Section 4 provides proofs of some of the key properties of the moduli spaces: regularity, semi-infinite-dimensionality and denseness of the restriction map (Semi-infinite-dimensionality Theorem).
Finally, section 5 contains the proof of the Gluing Theorem, showing that moduli on a composite cobordism are a fiber product of the moduli on its components.

Notations.
In this article we use the notation L p k for Sobolev spaces of regularity k, in accordance with the literature in gauge theory and Floer theory in low dimensional topology.These are often denoted by W k,p or H k,p or, when p = 2, simply by H k .
All manifolds are assumed to be smooth, submanifolds to be smoothly embedded, and manifolds with boundary to have smooth boundary.

Acknowledgments. I would like to express gratitude to my graduate advisor Tom
Mrowka for his encouragement, support and sharing many insights on geometry, topology and analysis.I would like to thank Maciej Borodzik and Michał Miśkiewicz for helpful comments on a draft of this paper, and an anonymous referee for pointing out an innacuracy in the statement of Regularity Theorem.
Most results of this paper have been part of the author's Ph.D. thesis [Suw20], although the proofs have been revised and some results have been generalized.In particular, [Suw20] does not consider 4-manifolds with more than two boundary components, or the general split Coulomb slice on them.

Analytical preparation
We prove two results which are fundamental for the surjectivity proofs in section 4.
The first one, the Regularity Theorem, is a regularity theorem for operators of the form , where D 0 is a first-order elliptic operator and K is a compact operator.In our applications K is a multiplication by an L 2 1 -configuration on a 4-manifold and thus it does not factor as a map L 2 1 → L 2 1 .If it did, elliptic regularity would immediately imply the regularity result.The novelty here is the general form of the perturbation K which is only assumed to be compact as a map L 2 1 → L 2 .The second result, the Strong UCP Theorem, is a strong unique continuation principle in a similar low regularity setting.This can be understood as a strengthening of the unique continuation results of [KM07, Section 7].

A regularity theorem for Dirac operators
Let X be a smooth Riemannian manifold with asymptotically cylindrical ends (without boundary).Let D 0 be an elliptic operator D 0 : C ∞ (X; E) → C ∞ (X; F ) of order 1 which is asymptotically cylindrical on the ends of X.Let K : L 2 1 (X; E) → L 2 (X; F ) be a compact operator which has a compact formal adjoint and which extends to a continuous operator K : L 2 (X; E) → L 2 −l (X; F ) for some l.Consider the operator The aim of this subsection is to prove the following regularity theorem: In the course of the proof we will make use of the following simple lemma (cf.Lipyanskiy [Lip08, Lemma 44]): Lemma 2.1.Suppose the sequence of Hilbert space operators {A i : V → W } is uniformly bounded and weakly convergent to A in the sense that for any v ∈ V we have We also need the following version of the Gårding inequality, proven in [Shu92, Appendix 1, Lemma 1.4]: Proposition 2.2 (cylindrical Gårding inequality).For every s, t ∈ R there exists . With these in hand, we are ready to prove the Regularity Theorem.Proof (Proof of Regularity Theorem).By considering where D * 0 is the formal adjoint of D 0 we may assume, without loss of generality, that E = F and D 0 is formally self-adjoint.Since X has bounded geometry and D 0 is uniformly elliptic, Proposition 4.1 in [Shu92, Section 1] implies that the minimal and maximal operators D min 0 and D max 0 of D 0 : L 2 (X; E) → L 2 (X; E) coincide, and their domains are equal to L 2 1 (X; E).Thus D 0 is essentially self-adjoint.Consider the equation for large c ∈ R. Certainly, ψ = v ∈ L 2 (X; E) solves the equation.We will prove that for large c equation (1) has a unique solution ψ in L 2 1 (X; E), which is also unique in L 2 (X; E), so that v = ψ ∈ L 2 1 (X; E), as wished.
We proceed to proving existence and uniqueness of solutions to (1) in L 2 1 (X; E) for large |c|.Using Lemma 2.1 we conclude that T = (D 0 +ic) −1 K : L 2 1 (X; E) → L 2 1 (X; E) converges strongly to 0. Thus, for large |c|, the operator Existence and uniqueness of ψ ∈ L 2 1 (X; E) solving (1) follows.To conclude the first part of the proof we need to establish uniqueness of solutions to (1) in L 2 (X; E).We have that . However, we already established that for large |c| the latter operator is invertible and it follows that Finally, it remains to consider the more general case v, h ∈ L 2 loc .For any compact A ⊂ X we can take a compactly supported bump function ρ : X → [0, 1] with ρ| A = 1 and define v ′ = ρv ∈ L 2 (X; E).Then Dv ′ = h ′ for some h ′ ∈ L 2 (X; E) and therefore v ′ ∈ L 2 1 (X; E) as proven above, so v| A ∈ L 2 1 (A; E| A ). Repeating the argument for every compact A ⊂ X shows that v ∈ L 2 1,loc (X; E).

A strong UCP for Dirac operators
Let M be a connected Riemannian manifold and S a real (resp.complex) Dirac bundle over it (e.g., the real (resp.complex) spinor bundle associated to a spin c -structure on M ) with connection A. Denote by / D A : Γ(S) → Γ(S) the corresponding Dirac operator.Let V ∈ L n (M ; R) be a potential.Here we prove the following unique continuation theorem for spinors and potentials of low regularity.
Strong UCP Theorem.The differential inequality

The version of this theorem for
n+2 , and ω ∈ L p 1,loc (Ω * (M )) such that |dω| + |d * ω| ≤ V |ω| with V ∈ L n loc (M ).Then if ω vanishes on an open set, it vanishes identically.It thus suffices to reduce our problem to Wolff's result, following the idea of [Man94].Since the proof in [Man94] does not explain the reduction rigorously, we describe the procedure below.Proof.The problem is local, thus we need only to consider the case of M being R n with some metric g.If A 0 is the flat connection on S, / D A − / D A 0 is a smooth operator of order 0, so (again using locality) we can assume that A is flat.Moreover, by contractibility of R n , we can decompose S into irreducible components, and irreducible components must be isomorphic to the real (resp.complex) spinor bundle S (resp.S C ) associated to the unique spin structure.Thus we have reduced the problem to the case where S = S ⊗ R k (resp.S = S ⊗ C k ) for some k.
The real (resp.complex) spinor bundle S (resp.S C ) embeds into Cℓ n (R n ) (resp.Cℓ n (R n )).Furthermore, [LM89, Theorem 5.12] implies that the Dirac operator on the The proof of Theorem 2.3 goes through for differential forms with coefficients in R k (resp.C k ), establishing the unique continuation property for This finishes the proof.
In the article we will use the following special case: Corollary 2.4 (UCP for Dirac operators in 4d).In the setting of the Strong UCP Theorem, assume n = 4 and V ∈ L 2 1 (M ; Aut(S)).Then any solution which is zero on some open set is identically zero.
3 Seiberg-Witten moduli spaces in split Coulomb slice In this section we introduce the moduli spaces of the Seiberg-Witten equations on a Riemannian 4-manifold with boundary a collection of rational homology spheres, together with the restriction maps to the boundary.For 3-manifolds, we introduce the Coulomb slice and its polarization, a decomposition of the tangent space into a sum of two infinite-dimensional subspaces.For 4-manifolds, we introduce the double Coulomb slice and what we call the split Coulomb slice together with the split gauge group.Since the restriction maps are generally not invariant with respect to the split gauge group, we need to introduce appropriate twisted restriction maps as well.
The split gauge fixing is a key novel element that generalizes the gauge slice introduced by Khandhawit [Kha] and Lipyanskiy [Lip08].It simplifies the proof of the Gluing Theorem letting us to reduce it to the case of untwisted restriction maps.

Coulomb slice on 3-manifolds
We begin by introducing polarizations on the Seiberg-Witten configuration space in Coulomb gauge on an oriented rational homology sphere Y and collections of such.
Let g be a Riemannian metric and s be a spin c structure on Y .Denote by S Y the associated spinor bundle and choose a smooth spin c connection B 0 .The Seiberg-Witten configuration space on Y is the space consisting of pairs (B, Ψ) of a spin c connection and a spinor on Y .In Seiberg-Witten theory one investigates the Chern-Simons-Dirac functional on the configuration space.The gauge group G(Y ) = L 2 3/2 (Y ; S 1 ) acts on C(Y, s) via u(A, Φ) = (A − u −1 du, uΦ) where u ∈ G(Y ), leaving L invariant.If one used spaces of higher regularity, one could work with the quotient of the configuration space by the action of the gauge group.However, in the low regularity setting the action of G(Y ) on the spinors is not continuous.Because of that (and in applications concerning the Seiberg-Witten stable homotopy type, cf.[KLS18]), it is preferable to take the Coulomb slice as the model for the quotient by the identity component of the gauge group.Indeed, for Y a rational homology sphere the Hodge decomposition gives the L 2 -orthogonal decomposition where Lemma 3.1 (gauge fixing in 3d).On Y , there is a continuous choice of based and contractible gauge transformations putting forms in the Coulomb slice, i.e., a homomorphism For each a, there is exactly one such u a . Proof.
We take u a = e G d Π d a which has the desired properties.Uniqueness follows from the fact that df = 0 and Y f = 0 imply f = 0.
Moreover, for Y a rational homology sphere we have which implies dũ = 0 and thus ũ is constant.It follows that there is are bijections justifying the restriction to the Coulomb slice: The following subspaces are crucial to the analysis of Atiyah-Patodi-Singer boundary value problem for the Seiberg-Witten equations on 4-manifolds with boundary Y .Definition 3.3 (polarization on the Coulomb slice).We define to be the closure of the span of positive (resp.nonpositive) eigenvalues of We denote by One of our goals is to prove that the moduli spaces of solutions to the Seiberg-Witten equations on X are, in a precise sense, comparable to the negative subspace Moreover, note that there are a natural identifications between configuration spaces for Y and oppositely oriented −Y .For a spin c structure s with its spinor bundle S Y there is the conjugate spin c structure s determined by the conjugate bundle S Y , and the anti-linear isomorphism S Y ≃ S Y induces natural affine isomorphisms

Split Coulomb slice on 4-manifolds
We turn our attention to the Seiberg-Witten equations and gauge fixings for configurations on a connected oriented 4-manifold X with nonempty boundary ∂X = ∅ satisfying b 1 (∂X) = 0, oriented using the outward normal.As explained by Khandawit [Kha], the most convenient slice for these is a kind of a double Coulomb slice (which was already used by Lipyanskiy [Lip08]), which imposes both coclosedness of the connection 1-form on both X and ∂X, as well as an auxiliary condition near ∂X.We drop this auxiliary condition from the definition of the double Coulomb slice and instead introduce the split Coulomb slice which generalizes the constructions of Khandhawit and Lipyanskiy.This allows one to choose a gauge fixing which do not require twisting or ones that are more geometric in nature, depending on one's needs.Indeed, twisting is necessary in Khandhawit's and Lipyanskiy's gauge fixing, in which the restriction map may not commute with the residual gauge group action.
We begin by introducing the Seiberg-Witten equations on 4-manifolds.
Definition 3.4 (Seiberg-Witten equations).The Seiberg-Witten map is defined by where A t denotes the connection induced by A on det(S + X ) and F + A t denotes the selfdual part of its curvature, according to the splitting Λ 2 (X) = Λ + (X) ⊕ Λ − (X) by the eigenspaces of the Hodge star ⋆.
The Seiberg-Witten equations are the equations given by SW(A, Φ) = 0, that is, Note that continuity and smoothness of the map SW follow from Theorem A.1 and the fact that continuous multilinear maps on Banach spaces are smooth.These equations are equivariant with respect to the action of the gauge group G(X) = L 2 2 (X; S 1 ).As is easily seen, the solution set is invariant under this action.
Note that this action is not continuous since the multiplication In order to prove that the moduli spaces of solutions are manifolds we need to investigate the differential of SW.
Similarly, at (e, A, Φ) the differential of the gauge group action is: As in dimension 3, we can fix gauge using the Coulomb condition, i.e., require that the 1-form is coclosed.Adding the same condition on the boundary ∂X ensures that the restriction to the boundary lies in the previously defined Coulomb slice (cf.Definition 3.2): Definition 3.6 (double Coulomb slice).We define the double Coulomb slice: The gauge group preserving it is called the harmonic gauge group: While the action of the full gauge group G(X) is not continuous, the action of G h (X) is, which will be proven in Lemma 3.9.
To define the split Coulomb slice we need to first understand the harmonic gauge group and its relation to harmonic functions and forms on X.Notice that we have a well-defined homomorphism which, restricted to harmonic gauge transformations, induces where is the space of harmonic 1-forms with Dirichlet boundary conditions.Note that δ (both on G(X) and on G h (X)) is an inclusion modulo S 1 , i.e., ker δ = S 1 , the group of constant gauge transformations.
On the other hand, the exponential map restricted to the space of doubly harmonic functions since the conditions ∆f = 0 and ∆(f Denote the image of this exponential map by G h,e (X) = exp(iH(X)).As the next proposition explains, G h,e (X) is the identity component of G h (X).Thus, our goal will be to find a gauge fixing that dispenses with the action of this identity component, saving only the action by S 1 , the constant elements.
Proposition 3.7 (sequence of harmonic gauge groups).The following sequence is exact: The identity component G h,e (X) is isomorphic to S 1 × R b 0 (∂X)−1 and the group of components π 0 G h (X) is naturally isomorphic to H 1 (X; Z).Remark 3.8.Recall that H 1 (X; Z) ≃ Hom(π 1 (X), Z) has no torsion.
Proof.Crucial to the understanding of the gauge group is the homomorphism (2).Hodge theory provides an identification H 1 D (X) ≃ H 1 (X, ∂X; R).Thus, we will consider δ as a map δ : G h (X) → H 1 (X, ∂X; iR) with kernel S 1 .This will be used to establish a map of horizontal short exact sequences where the vertical sequences are also exact, as will be shown in the course of the proof.Firstly, we prove that π 0 G h (X) ≃ H 1 (X; Z).Notice that for any closed loop γ ⊂ X the period of δ(u), i.e., the integral γ δ(u), is an integer multiple of 2πi; and it is zero whenever γ is contractible.(In fact it is the obstruction to lifting u| γ : γ → S 1 to a map u| γ : γ → R.) This way any u ∈ G h (X) determines an element [u] ∈ H 1 (X; 2πiZ) and we get a homomorphism G h (X) → H 1 (X; 2πiZ).Since the periods (having values in 2πiZ) do not change under homotopy, this descends to a map π 0 G h (X) → H 1 (X; 2πiZ) and from its construction it follows that it coincides with the composition which has image in H 1 (X; 2πiZ).It remains to notice that any element in x ∈ H 1 (X; 2πiZ) can be lifted to an element x = H 1 (X, ∂X; iR) ≃ iH 1 D (X) and then integrated along curves to obtain an element u x ∈ G h (X) mapping to x (see (6)).After dividing 1-forms by 2πi we obtain a natural isomorphism π 0 G h (X) ≃ H 1 (X; Z), as wished.
Moreover, this shows that the kernel K of the map G h (X) → H 1 (X; 2πiZ) maps via δ to the kernel of H 1 (X, ∂X; iR) → H 1 (X; iR).We thus get the map With this in mind, we turn our focus to G h,e (X) = exp(H(X)).Since any harmonic function f on X is determined by its restriction f | ∂X to ∂X, and harmonic functions on ∂X are locally constant, we have an isomorphism and we will denote any element g in H 0 (∂X; R) by f | ∂X for the unique f ∈ H(X) such that f | ∂X = g.We obtain a canonically defined surjection with kernel generated by restrictions of H 0 (X; 2πiZ).After dividing by 2πi we get an isomorphism G h,e (X) ≃ H 0 (∂X; R)/H 0 (X; Z) ≃ S 1 × (H 0 (∂X; R)/H 0 (X; R)), as wished.
Finally, recall that the composition is given by f | ∂X → df and therefore, by Hodge theory, represents the boundary map in the exact sequence Thus δ(G h,e (X)) = im(H 0 (∂X; iR)) = δ(K).Since ker δ| G h,e (X) = S 1 = ker δ| K and G h,e (X) ⊆ K, we obtain that G h,e (X) = K, i.e., the sequence (3) is exact, as wished.
Lemma 3.9 (G h acts continuously).The action of G h (X) on C(X, ŝ) is smooth.
Proof.By Proposition 3.7 it suffices prove that the action of G h,e (X) is smooth.Further, by Theorem A.1 it suffices to prove that G h,e (X) ⊂ L 2 3 (X; S 1 ) and that this injection is continuous.Since f ∈ iH(X) and H(X) is finite-dimensional, there are constants Let us compare different splittings.If s, s ′ are two different splittings, then for any . Therefore any two gauge splittings differ by a homomorphism π 0 G h (X) → G h,e (X).
Our goal is to reduce the gauge group action to the action of S 1 and the action of a chosen lift of π 0 G h (X) to G h (X).Precisely, we will consider splittings s : π 0 G h (X) → G h (X) of (3).In order to choose the gauge fixing we need to understand that a gauge splitting induces another splitting on the level of homology.
We clarify the relationship between gauge splittings and homological splittings.Proposition 3.12 (gauge splittings from homological splittings).Let σ : H 1 (X; R) → H 1 (X, ∂X; R) be a homological splitting, i.e., a splitting of (5).Then up to action of S 1 there exists a unique gauge splitting s such that σ = s H . Proof.For existence, choose x 0 ∈ X and consider the map The uniqueness up to action of S 1 follows from the exactness of the middle vertical sequence in (4).
In order to find the appropriate gauge fixing we need the following analogue of Proposition 3.7 for 1-forms.
Proof.This follows from the proof of [Kha, Proposition 2.2].(Note that our definition of Ω 1 CC (X) differs from Khandhawit's, which we denote by Ω 1 s ⊥ (X) (cf.Definition 3.21).) In particular, we can decompose 1-forms as where Denoting by Π CC the projection onto Ω 1 CC (X) along d(Ω 0 ∂ (X)) we obtain the following analog of Lemma 3.1.

Lemma 3.14 (Coulomb gauge fixing in 4d).
There is a unique homomorphism We can further decompose A homological splitting s H provides a decomposition which is an analogue of (3) for H 1 D (X).With these in hand, we are ready to define the split Coulomb slice.Definition 3.15 (split Coulomb slice).Let s be a gauge splitting and s H its associated homological splitting.The split Coulomb slice is In particular, we have that and parallel to Lemma 3.14 and Lemma 3.1 we can use the projection Π s onto the first factor along the second one to obtain: Lemma 3.16 (split gauge fixing in 4d).There is a unique homomorphism Proof.The projection (1 − Π s ) on Ω 1 (X) has image in d(Ω 0 (X)) and d is injective on Ω 0 0 (X).Therefore there is a unique homomorphism L 2 1 (iΩ 1 (X)) → L 2 2 (iΩ 0 0 (X)) sending a to the unique f s a ∈ Ω 0 0 (X) such that a−df s a ∈ Ω 1 (X).Then we take u s a = exp(f s a ).
Remark 3.17 (continuous gauge fixing within double Coulomb slice).If we only consider a ∈ Ω 1 CC (X), then the above map has image in G h (X), which is finite-dimensional.Since the latter gauge group acts continuously on the configuration space by Lemma 3.9, we conclude that putting (A, Φ) ∈ C CC (X, ŝ) into split Coulomb slice C s (X, ŝ) can be done continuously with respect to (A, Φ).
The gauge group acting on this split Coulomb slice is the product of S 1 and the split gauge group: Definition 3.18 (split harmonic gauge group).Let s be a gauge splitting.The split gauge group is defined to be Lemma 3.19 (split gauge group preserves the split Coulomb slice).
Proof.One direction follows directly from the definition of s

The other direction follows by chasing arrows in the diagram (4).
The circle • in the superscript indicates that the only constant gauge transformation contained in G h,• s (X) is the identity.This way we do not forget the S 1 -action when taking the quotient by the split gauge group.
We want to compare different split slices together with the split gauge group actions.Choose two splittings s, s ′ .These determine a map Proposition 3.20 (equivalence of slices).The map F ν is well-defined, a diffeomorphism, equivariant with respect to the action of π Proof.Firstly, we need to show that the image of F ν actually lies in C s (X, ŝ).Equivalently, we want to show that ), as wished.The map F ν is smooth because the map ν is smooth and the action of the finitedimensional G h (X) on C(X, ŝ) is smooth.
It is invertible because F 1/ν is its inverse.Indeed, since im ν ⊂ G h,e (X), we have that δν ∈ d(H(X)), so Π im s H (δν) ≡ 0. Therefore Finally, we discuss the gauge slice used by Lipyanskiy [Lip08] and Khandhawit [Kha,KLS18].They require that a ∈ Ω 1 CC (X) and that for each component Y i ⊂ ∂X we have Y i ι * (⋆a) = 0. Using Stokes' theorem one can show that for a ∈ Ω 1 CC (X) this integral condition is equivalent to the condition that X df ∧ ⋆a = 0 for any f ∈ H(X).This fits into our setup perfectly, since there is exactly one homological splitting s Definition 3.21 (orthogonal splitting).We call s H ⊥ the orthogonal homological splitting.We say that a splitting s is a orthogonal splitting if s H = s H ⊥ .

Restriction to the boundary and twisting
Unless ∂X is connected, we are not guaranteed that the restriction to the boundary is invariant under the action of the split gauge group G h,• s (X).If it happens to be invariant for some s, we call such s an integral splitting.For a general s, we introduce and prove the existence of twistings of the restriction map, making it invariant under the action of G h,• s (X) action even for non-integral s.As mentioned before, integral splittings are utilized in the proof of the Gluing Theorem, while non-integral splittings may be more convenient in other contexts (e.g., in constructions of [Lip08,Kha,KLS18]).
We start by defining the restriction maps for an embedding ι Y : Y ֒→ X of an oriented 3-manifold Y .Denote by s the restriction to Y of the spin c structure ŝ on X.We get canonical identifications S ± X | Y ≃ S Y .Assuming Y is a geodesic codimension-1 submanifold of X, the spin c connection A 0 induces a spin c connection B 0 on Y by simple restriction: and u ∈ G(X).We define the restrictions: Integral splittings are the ones for which restriction maps are invariant under the split gauge group.

Definition 3.22 (integral splitting). We call a gauge splitting s integral if for each
The integrality of s is closely connected to the integrality of s H . Proposition 3.23 (homological classification of integral splittings).If s is integral, then s H (H 1 (X; Z)) ⊂ H 1 (X, ∂X; Z), i.e., s H is integral as well.
Given any integral homological splitting σ there exists a unique integral splitting s such that σ = s H . Proof.Assume s is integral.Choose y 0 ∈ ∂X and consider the map I y 0 defined in (6).We know s and I y 0 • s H • [δ] differ by action of S 1 , but since both are equal to 1 at y 0 , thus s = I y 0 • s H • [δ].This implies that for any y ∈ ∂X and any embedded curve γ : [0, 1] → X with γ(0) = y 0 and γ(1) = y we have that exp y y 0 s H ([δ(u)]) = 1 and thus To find a consistent way of twisting the boundary values of 1-forms we consider ways to "undo" the action of G h,• s (X) on the boundary "in a linear fashion".
Definition 3.24 (gauge twisting).We call a homomorphism τ : H 1 (X; iR) → G h (∂X) ≃ (S 1 ) π 0 (∂X) a gauge twisting for s if the composition agrees with the action of the split gauge group on the boundary, R Continuous homomorphisms from a vector space to S 1 correspond to linear functionals on the vector space.Thus, every such twisting τ comes from a linear map dτ : H 1 (X; iR) → H 0 (∂X; iR) and τ = exp •(dτ ).We utilize it to prove the existence of gauge twistings for s, and one could use it to classify all possible gauge twistings for s.Actually, every such homomorphism τ is a gauge twisting for some s, but we do not use this fact in this article.
Lemma 3.25 (existence of gauge twistings).For a given gauge splitting s there exists a gauge twisting τ .
With τ in hand, there is a way of defining a twisting on the whole Coulomb slice, enabling us to finally define the twisted restriction maps.Definition 3.26 (twisted restriction map).We define the Coulomb slice twisting τ CC : L 2 1 iΩ 1 CC X → G h (∂X) associated to τ to be the composition We define the twisted restriction map Remark 3.27.What is of importance for defining the twisted restriction maps is the map τ CC : iΩ 1 s (X) → (S 1 ) π 0 (∂X) .The extension of τ CC to the whole of iΩ 1 CC (X) is artificial: it does not undo the action of G h,e (X) on the boundary as one might expect.
With more work, including a choice of a based gauge group ) and a more general twisting, one could work with the full iΩ 1 CC (X) and then quotient by the action of G h o (X).However, this would introduce unnecessary complications.
These twisted restriction maps are indeed invariant under G h,• s (X).
Lemma 3.28 (twisted restriction map is invariant under split gauge group).
Let τ be a gauge twisting for s.For any but that is equivalent to ).This follows directly from Definition 3.24 of the twistings and Definition 3.26 of the twisted restriction map, since We conclude these sections by showing that choosing τ is essentially equivalent to choosing an integral splitting.In general, one can restrict themselves to considering integral splittings without any twisting at all.Proposition 3.29 (twistings are integral splittings).Let τ be a twisting for s.Then there is an integral splitting s Z and an equivariant diffeomorphism Proof.Every function f ∈ H(X) is determined by its restriction to the boundary f | ∂X , which is locally constant.We thus have the exact sequence and from these two it follows that is exact.Since the group to the left is discrete it follows that there exists a unique lift τ of τ to G h,e (X): G h,e (X) for any [u] ∈ π 0 G(X) ≃ H 1 (X; 2πiZ), and using the construction of F ν of Proposition 3.20.This gives an equivariant diffeomorphism from C s (X, ŝ) to C s Z (X, ŝ).
The equality R • F s,τ = R τ follows from the construction.
Even though the spaces C s Z (X, ŝ) and C s ′ Z (X, ŝ) are equivariantly diffeomorphic by Proposition 3.20, the corresponding restriction maps differ by a twist.Thus, a priori we cannot get rid of the choice of a splitting.However, this is not relevant to most of the applications because for connected boundary there is no choice to make.Lemma 3.30 (uniqueness of integral splittings).If ∂X = ∅ is connected, there exists exactly one integral splitting s Z .
Proof.In this case, the restriction map G h,e (X) → G h (∂X) is an isomorphism (cf.(10)).Therefore for each element π 0 G(X) there exists exactly one representative u ∈ G h (X) such that u| ∂X = 1.

Seiberg-Witten moduli in split Coulomb slice
We conclude this section by defining the Seiberg-Witten moduli spaces, the main object of study of this article.We also prove they only depend on the choice of s Z associated to s and τ .
Thanks to Lemma 3.19, we can define the following.
Definition 3.31 (moduli spaces on 4-manifolds with boundary).We define the moduli spaces in split slice: We also define a version of the moduli space using the full double Coulomb slice, which will be utilized in some of the proofs.From Lemma 3.28 it follows that Corollary 3.32.There is a well-defined restriction map A direct consequence of Proposition 3.29 is Corollary 3.33 (dependence on twistings).Given s and τ , there is an integral splitting s Z and a diffeomorphism

Properties of moduli spaces
In this section we prove that (Semi-infinite-dimensionality Theorem): • the moduli spaces of solutions to the Seiberg-Witten equations on X are Hilbert manifolds, • the restriction map to the boundary is "semi-infinite", i.e., Fredholm in the negative direction and compact in the positive direction, • if ∂X is disconnected, restriction to a single boundary component has dense differential.
This is done by analyzing the properties of the linearized Seiberg-Witten operator D SW.We start by investigating an extended version of this operator, D SW.The reason is that the standard Atiyah-Patodi-Singer boundary value problem as well as the elliptic theory developed in section 2 can be directly applied to the study of D SW.
Our understanding of the gauge action (subsection 3.2) will allow us to transfer these properties to D SW.

Extended linearized SW operator
Here we apply the Atiyah-Patodi-Singer boundary value problem to an extended version of the linearized Seiberg-Witten operator, D SW.The properties we prove are the direct analogues of the properties of D SW which are proved in the next section.
Definition 4.1 (extended linearized SW operator).We define the extended linearized Seiberg-Witten operator X by adding a component related to the linearization of gauge action: In order to study the Atiyah-Patodi-Singer boundary value problem we need to introduce the appropriate operator on the boundary and consider its Calderón projector.Denote Y = ∂X and define This is a first-order self-adjoint elliptic operator.Denote by , and by Π ± the projection onto H ± (Y, s) along H ∓ (Y, s).The proof of the following proposition follows a standard argument; we briefly recall it to set up the stage for the proofs in the rest of this section.Proposition 4.2 (semi-infinite-dimensionality of D SW).The operator Moreover, the positive part of the restriction map from the kernel of As explained in [Kha, Proposition 3.1], applying the Atiyah-Patodi-Singer boundary value problem [KM07, Theorem 17.1.3]to D proves that D ⊕ ΠR is Fredholm with index equal to (12).Furthermore, [KM07, Theorem 17.1.3]implies that for any bounded sequence The operator K is compact by Theorem A.1.Since D SW (A,Φ) = D + K, thus D SW (A,Φ) is Fredholm with the same index as D.Moreover, since K is compact, for any sequence (u i ) ⊂ ker D SW (A,Φ) we can choose a subsequence such that the By what we proved in the previous paragraph, the sequence The proof of surjectivity utilizes both of the analytical results of section 2 (cf.[Lip08, Theorem 2]).
Finally, we focus on the density of the restriction map from the kernel of D SW to one boundary component.The proof of this proposition utilizes some of the ideas we have just seen (cf.[Lip08, Lemma 5]).implying v L 2 1/2 = 0, which contradicts the assumption that v = 0.

Regularity and semi-infinite-dimensionality of moduli spaces
We turn our focus to the operator D SW.The main difficulty in transferring the results of subsection 4.1 from D SW to D SW is the presence of the Coulomb condition on both X and ∂X.The split gauge condition introduces an additional twist to the story.
A key fact is that the differential of the gauge group action at e (cf.Lemma 3.5) preserves the kernel of D SW at a solution (A, Φ).
Proof.We compute: which finishes the proof.
Following the idea of [Kha, Proposition 3.1], we deduce the semi-infinite-dimensionality and compute the index of D SW from Proposition 4.2.These methods will be utilized to prove further results in this section, too.Proposition 4.6 (semi-infinite-dimensionality of D SW).The operator Moreover, the restriction Π + R : ker Proof.Firstly, we compare the respective polarizations.Recall that the decomposition of we see that L decomposes as This is enough to prove the statement about compactness.Indeed, the map where the map in the middle is compact by Proposition 4.2.
For Fredholmness, denote by ).This projection can be used to define the split Coulomb slice for the orthogonal splitting s ⊥ (Definition 3.21).Precisely, we have Khandhawit [Kha, Proposition 3.1] proves that im Π − 1 and ker Π C are complementary, and then [KM07, Proposition 17.2.6]implies that is an isomorphism onto im Π C , therefore the proof of [KM07, Proposition 17.2.6]implies that the index of (15) is the same as the index of (11).
The following lemma is a simple exercise in Fredholm theory.It implies that the operator Thus, we have proven the Proposition for a particular splitting, s = s ⊥ .For any splitting s, the inclusion X is a finite-dimensional subspace of the "full" double Coulomb slice L 2 1 iΩ 1 X ⊕ Γ S + X .Therefore the Fredholmness of (13) for s = s ⊥ implies the Fredholmness of and this, in turn, implies the Fredholmness of (13) for any splitting s.Moreover, the index of ( 16) is equal to b 0 (Y ) − 1 plus (14) for any splitting s, finishing the proof.
We turn to deducing the surjectivity of D SW from the surjectivity of D SW.
Proposition 4.8 (surjectivity of D SW).For any gauge splitting s, the differential Proof.We will prove a stronger statement, that this extended differential together with the exact part of the restriction to the boundary X is surjective.To prove surjectivity of (17) it thus remains to prove that We prove that Π d R| ker D SW (A,Φ) is surjective.Take any g ∈ L 2 3/2 (iΩ 0 (Y )) representing a given element dg ∈ L 2 1/2 (iΩ 1 C (Y )).Take the unique f ∈ L 2 2 (iΩ 0 (X)) such that ∆f = 0 and f | Y = g.Then Π d R(df, −f Φ) = df | Y = dg.The required surjectivity follows since (df, −f Φ) ∈ ker D SW (A,Φ) , which follows from d * df = ∆f = 0 and Lemma 4.5.
Similarly we prove Π Vs | ker D SW (A,Φ) ,Π d R is surjective.By (8) and definition of V s , the orthogonal projection d(H(X))) → V s is an isomorphism and therefore for any v ∈ V s there is f ∈ H(X) such that Π Vs (df ) = v s .Moreover D SW(df, −f Φ) = 0 and Π d R(df, −f Φ) = 0, as wished.
Finally, we prove the density of the restriction map to a connected component of Y .
Proposition 4.9 (density of moduli on one boundary component).The restriction R : ker Proof.Take any (b, Using the decomposition (9) we can write ãk = a k + df k for f k ∈ iΩ 0 (X) and a k ∈ Ω 1 s (X).Since Y 0 is connected, we can change f k by a constant to obtain The results of Proposition 4.6, Proposition 4.8 and Proposition 4.9 can be summarized as follows.
Semi-infinite-dimensionality Theorem.The moduli spaces M • s (X, ŝ) are Hilbert manifolds.The differential of the twisted restriction map R τ :

Gluing along a boundary component
This section is devoted to the proof of the main result of this article, the Gluing Theorem, which relates the moduli spaces of solutions on X 1 , X 2 and X = X 1 ∪ Y X 2 , where Y is a rational homology sphere, oriented as a component of the boundary of X 1 .Under the identification C cC (Y, s) ≃ C cC (−Y, s) there are S 1 -equivariant twisted restriction maps , where s = ŝ| Y .One can expect the fiber product to be diffeomorphic to M • s (X, ŝ), and this turns out to be true.One would also like to have this map intertwine the twisted restriction maps to ∂X, but this is a bit too much: the splittings and twistings (s, τ ), (s 1 , τ 1 ), (s 2 , τ 2 ), need to enjoy certain compatibility, and even then the restriction maps may not match on the nose but need to be homotoped to each other.This reflects the fact that we did not quotient by the action of S 1 on the configuration spaces.
The proof utilizes the following fact which is of independent interest.Let X ′ ⊂ X be a submanifold, the closure of which is contained in the interior of X.Then the restriction map from M • CC (X, ŝ) to L 2 k -configurations on X ′ is well-defined and smooth for any k ≥ 0. Well-definedness follows from a standard argument, but proving the smoothness of this map turns out to be a surprisingly delicate task which we tackle in subsection 5.1.
The same strategy should work to prove smoothness of restriction maps to interior submanifolds for other types of moduli spaces appearing in gauge theory, e.g., for the space of anti-self-dual connections on G ֒→ P → X.The key is the ellipticity of the equations together with the gauge fixing.

Smoothness of restrictions
Assume X ′ ⊂ X is a submanifold with closure contained in X.The goal is to show (cf.Theorem 5.2) that the restriction map R : We restrict ourselves to the case k = 2, but the same strategy may be used iteratively, bootstraping the result to any k, if needed.
Due to Theorem A.2 we may assume, without loss of generality, that X ′ is of codimension 0. Since we require the closure of X ′ to be contained in the interior X, we may as well assume that X ′ is a closed submanifold.along H is a diffeomorphism.Similarly, the Implicit Function Theorem implies that there is a neighborhood V of p such that the affine projection where the middle identity map is smooth by Lemma 5.4 and the two other maps are smooth since they are parametrizations coming from the Implicit Function Theorem, as we just showed.

Proof of the gluing theorem
We are ready to prove the gluing theorem.Let X = X 1 ∪ Y X 2 with Y connected and b 1 (∂X i ) = 0. Let ŝ be a spin c structure and A 0 be a reference spin c connection on X.Let restrictions of ŝ be the spin c structures used to define configuration spaces and the restrictions of A 0 to be the reference connections.Denote Y 1 = ∂X 1 \ Y , Y 2 = ∂X 2 \ Y , ŝi = ŝ| X i .Fix gauge splittings s, s 1 , s 2 and twistings τ, τ 1 , τ 2 on X, X 1 , X 2 .
Denote by s Z , s 1,Z , s 2,Z the associated integral splittings given by Proposition 3.29.We say they are compatible if s Z corresponds to (s 1,Z , s 2,Z ) under the following identification.
Proposition 5.5 (integral splittings on a composite cobordism).There is a canonical identification between the set of integral splittings on X and the set of pairs of integral splittings on X 1 and X 2 .
Proof.Choose an integral splitting s.Take any a ∈ H 1 D (X).Denote by ãi its restriction to X i .For each i there is a unique and therefore R H coincides with doing the gauge fixing of Lemma 3.16 on both components, i.e., a → (Π CC (a| X 1 ), Π CC (a| X 2 )).

The cohomology class
is the identity, where ι * is the inverse of the restriction map and by Proposition 3.23 there is a unique integral splitting s inducing s H . Proposition 4.8 guarantees that whenever ∂X = ∅, the moduli M • s (X, ŝ) is a smooth Hilbert manifold, and the same follows for M • s i (X i , ŝ).We want to include the case when X is a closed manifold, when it is well-known that to achieve surjectivity one, in general, needs to perturb the metric on X or perturb the Seiberg-Witten equations.Therefore, we assume that for any (A, Φ) ∈ SW −1 (0) the operator D SW (A,Φ) is onto.
Remark 5.6.The careful reader may notice that we do not assume any transversality of the maps R τ i ,Y and thus the fiber product may not a priori be a manifold.That it is a manifold follows from the proof of the theorem.What is not proven here, but may be useful in other contexts, is that the transversality is indeed equivalent to D SW (A,Φ) being surjective for all (A, Φ) ∈ M • s (X, ŝ).
Gluing Theorem.Assume s Z and (s 1,Z , s 2,Z ) are compatible.Then there is an S Proof.By Corollary 3.33 we can assume, without loss of generality, that s = s Z and s i = s i,Z , as well as τ ≡ 1 and τ i ≡ 1.We thus drop τ 's from the notation entirely.
The plan is as follows.We will construct a map and a homotopy , and that H is G h,• s (X)-invariant; thus, both F and H descend to M • s (X, ŝ).Furthermore, we will prove F and H are continuous and smooth, and that F has image in the fiber product.Finally, we will show that F is a smooth immersion onto M ).The S 1 -equivariance of F and H will be apparent from the construction.
Step 1.We start by constructing F , proving its smoothness and that its image lies in the fiber product.Take (A, Φ) ∈ M • s (X, ŝ).We would like to simply restrict it to the components X i and then put into split Coulomb slice.By Lemma 3.16 modulo S 1 there a unique way of doing that using a contractible gauge transformation.Here we make a different choice than in Lemma 3.16, requiring where g = G d Π d b (A,Φ) depends smoothly on (A, Φ) as an element of L 2 5/2 .Moreover, the map g → f ∂ i is linear and continuous as a map L 2 s+1/2 → L 2 s+1 for s ≥ 0, so f ∂ i depend smoothly on (A, Φ) as elements of L 2 3 .Furthermore, since a| X i − df ∂ i ∈ L 2 1 (iΩ 1 CC (X i )), f s i are the unique elements of H 1 D (X i ) such that a| X i − df ∂ i − df s i ∈ L 2 1 (iΩ 1 s (X i )) and f i | Y = 0.By Remark 3.17, f s i depend continuously on a| X i − df ∂ i ∈ L 2 1 .Which proves that f i ∈ L 2 3 depend continuously and linearly on g ∈ L 2 5/2 , thus depend smoothly on g, so they depend smoothly on (A, Φ).
This establishes the well-definedness and smoothness of F .That its image lies in the fiber product follows directly from the construction.
Step 2. We proceed to constructing H.By (18), the functions f i are locally constant on Y i .We also have by the construction of F .Thus we can define H(A, Φ, t) = e tf 1 | Y 1 , e tf 2 | Y 2 R(A, Φ) which at t = 0 coincides with R(A, Φ) and at t = 1 coincides with (R Y 1 , R Y 2 ) • F (A, Φ).
Step 3. We now investigate the equivariance of F under the actions of gauge groups.Let u ∈ G h,• s (X).From Lemma 3.16 if follows that there is exactly one contractible ũi = e fi ∈ G e (X i ) which puts −u −1 du into L 2 1 iΩ 1 s X i with Y fi = 0. Equivalently, this is the unique ũi = e fi ∈ G e (X i ) with Y fi = 0 such that ũi u| X i ∈ G h,• s i (X i ).Define u i = ũi u| X i .Since ũi is contractible, thus [ι * X i (u −1 du)] = [u −1 i du i ] in H 1 (X i ; 2πiZ).Therefore the map u → (u 1 , u 2 ) provides the canonical isomorphism which agrees with the isomorphism coming from the Mayer-Vietoris sequence H 1 (X; Z) ≃ H 1 (X 1 ; Z)×H 1 (X 2 ; Z).From the construction of F (A, Φ) it follows that (u| −1 X 1 , u| −1 X 2 )F (u(A, Φ)) differs from F (A, Φ) exactly by the factor of (e f1 , e f2 ).Thus (u −1 1 , u −1 2 )F (u(A, Φ)) = F (A, Φ).This proves that F commutes with the gauge group action as identified in (19).
Step 4. We prove the invariance of H under G h,• s (X).By the construction of H, and since u| ∂X = 1 for u ∈ G h,• s (X), we have H(u(A, Φ), t) = (e t f1 | Y 1 , e t f2 | Y 2 )H(A, Φ, t) where fi are as in the previous paragraph.Since R H (H 1 D (X)) ∈ (im s H 1 ) × (im s H 2 ) we get that R H (u −1 du) is already in the split Coulomb slice on X 1 and X 2 .Moreover, a i = ι * X i (u −1 du) is coclosed on X i and Y i .Since there are functions fi satisfying the uniqueness of fi implies fi = fi and therefore fi | Y i = 0. Thus H(u(A, Φ), t) = H(A, Φ, t), as wished.
Step 5. We show that F is bijective onto the fiber product, following the argument in [Lip08].Let These would give a configuration on X if the normal components of connections A 1 and A 2 agreed on Y .Let h 1 dt and h 2 dt be the dt-components of (A 1 − A 0 )| Y and (A 2 − A 0 )| Y .We want to find harmonic functions f i ∈ L 2 2 (X i ; iR) such that e g 1 − e g ′ 1 L ∞ ([−δ,0]×Y ) ≤ 2.Moreover, we have via a direct computation (or by interior regularity estimates following from Theorem A.3).This finishes the proof that the inverse map is continuous.

A Appendix
This section presents some standard analytical results which are used repeatedly in the article.We recall the Gårding inequality, Sobolev multiplication and trace theorems and the Implicit Function Theorem.
Theorem A.1 (Sobolev multiplication theorem).Let M be a manifold with compact boundary and cylindrical ends.
What is more, whenever it is continuous, it restricts to a compact map on {f } × L q l (M ) → L r m (M ) provided that l > m and l − n/q > m − n/r.In particular, the projection

Implicit Function
along C is a local diffeomorphism.
We will also utilize the Gårding inequality.where the middle inequality follows from [Shu92].
Remark A.4.The author's understanding is that without the use of twistings there is (in general) no choice of s, s 1 , s 2 making F commute with the restriction maps on the nose.This problem does not show up in the construction of monopole Floer homology since there one quotients the moduli and configuration spaces by S 1 after blowing up.

Definition 3. 2 (
Coulomb slice).The Coulomb slice on Y with respect to the reference connection B 0 is the space of configurations

Lemma 4. 7 .
Let (F, G) : H → A ⊕ B be Fredholm.Then F = F | ker G : H = ker G → A is Fredholm and has index equal to ind((F, G)) + dim coker G.
Theorem.Suppose A, B are Hilbert spaces and F : A → B is a smooth map such that the derivative D p F at p is surjective and that its kernel splits with C as a complementary subspace, A = ker(D p F)⊕C.Let Π : A → ker(D p F) denote the projection onto the kernel along C. Then there are open neighborhoods U ⊂ A of p, V ⊂ B of F(p) and W ⊂ ker(D p F) of 0 and a smooth diffeomorphismG : V × W → U such that V × W G − → U (F ,Π) −−−→ B ⊕ ker(D p F)is the identity map (i.e., G = F −1 ).

Theorem A. 3 (
Gårding inequality).Let D a first-order elliptic operator with smooth coefficients on a compact manifold M (possibly with boundary) and M ′ ⊂ M be open with compact closure.Then there is a constant C such that for any γ ∈ L p k+1 (M ) we haveγ L p k+1 (M ′ ) ≤ C( Dγ L p k (M ) + γ L p k (M ) ).Proof.This follows from [Shu92, Appendix 1, Lemma 1.4] by extending D to the cylindrical-end manifold M * = M ∪ (∂M ) × [0, ∞).Indeed, taking a smooth bump function ρ such that ρ| M ′ = 1 and ρ| M * \M = 0 we getγ L p k+1 (M ′ ) ≤ ργ L p k+1 (M * ) ≤ C( D(ργ) L p k (M * ) + ργ L p k (M * ) ) ≤ CC ′ γ ( D(γ) L p k (M ) + γ L p k (M ) ) by Mayer-Vietoris).We can thus choose (s H 1 , s H 2 ) to be the composition R H •s H •ι * .Since s H was integral, thus s H i are integral and by Proposition 3.23 there exist unique integral splittings s 1 , s 2 inducing s H 1 , s H 2 .On the other hand, given integral splittings s 1 , s 2 on X 1 , X 2 , we can choose s H to be