Desingularization of Coassociative 4-folds with Conical Singularities: Obstructions and Applications

We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for deformations of conically singular coassociative 4-folds, and thus relate them to the stability of the singularities. We use our results to describe the relationship between moduli spaces of coassociative 4-folds with conical singularities and those of their desingularizations. We also apply our theory in examples, including to the known conically singular coassociative 4-folds in compact holonomy G_2 manifolds.


Introduction
Coassociative 4-folds are calibrated 4-dimensional submanifolds in 7-manifolds with exceptional holonomy G 2 (and, more generally, in 7-manifolds with a G 2 structure). Studying gluing problems for calibrated submanifolds in manifolds with special holonomy has proven to be a rich and fruitful avenue of research, particularly in special Lagrangian geometry in the work of Joyce [9,10], Haskins and Kapouleas [5] and Pacini [26], as well as for associative [24] and coassociative geometry [17]. In particular, the desingularization problem for calibrated submanifolds with conical singularities naturally feeds into the understanding of the boundary of the moduli space of smooth calibrated submanifolds, and has crucial consequences for the construction of potential invariants for manifolds with special holonomy by suitable "counting" of calibrated submanifolds (see, for example, [7,8] for a discussion of these issues in the special Lagrangian case).
In any gluing problem one naturally has to tackle the issue of obstructions. This is normally achieved by making strong assumptions on the geometry of the submanifolds to be glued, for example in [9,17,26], or by restricting to situations where the obstructions can be identified and resolved in a natural way, either via topological conditions (as in [10]) or symmetries of the problem (as in [5]). In this article we extend the work in [17] and consider a gluing problem in coassociative geometry where we deal with both topological and analytic obstructions in desingularizing isolated conical singularities. We interpret the analytic obstructions geometrically using the deformation theories of asymptotically conical and conically singular coassociative 4-folds developed in [16,18]. We thus relate our obstructions to the notion of stability of coassociative conical singularities introduced in [20].
We use our desingularization results to help describe how the moduli spaces of asymptotically conical, conically singular and smooth compact coassociative 4-folds are related, and thus provide a greater understanding of the boundary of the moduli space of compact coassociative 4-folds. In the case of stable conical singularities, this enables us to construct a local diffeomorphism between the gluing data associated to the singular coassociative 4-fold and a neighbourhood of the coassociative smoothing "near the boundary" of the moduli space.
We also discuss examples where our desingularization theory applies, including the first known examples of coassociative 4-folds with conical singularities in compact manifolds with G 2 holonomy, which were constructed in [20].
The setting in this article is the following. We have a coassociative 4-fold N in an almost G 2 manifold M (a 7-manifold with a closed G 2 structure) and we suppose that N has a single conical singularity z modelled on a cone C. We also assume there exists a coassociative 4-fold A in R 7 which is asymptotically conical with rate λ < − 1 2 to the cone C. (See §2 for precise definitions.) In Definition 3.4 we give a matching condition between A and N . The matching condition is a mixture of topological and analytic constraints, which then allows us to deal with both topological and analytic obstructions.
To give a sense of the matching condition we make some observations. Let Σ be the link of C and let j A 2 : H 2 (A) → H 2 (Σ) and j N 2 : H 2 (N ) → H 2 (Σ) be the induced maps arising from inclusion of Σ in A andN = N \ {z} (the non-compact manifold given by removing the singularity from N ).
If ϕ 0 is the standard G 2 structure on R 7 , then since ϕ 0 is closed and ϕ 0 | A = 0 we have an element [ϕ 0 ] ∈ H 3 (R 7 , A) ∼ = H 2 (A), which we may also view as the cohomology class of the infinitesimal deformation of A corresponding to dilation. The natural topological constraint is therefore that j A 2 [ϕ 0 ] lies in Im j N 2 . We can relate the analytic obstructions to the obstruction theory of N and thus to the notion of C-stability of the cone C (see Definition 3.19) for a deformation family C of C -this condition is discussed in detail in [20].
Overall, we have the following interpretation of the matching condition.
∈ Im j N 2 and the cone C at the singularity is Cstable, then the matching condition is satisfied.
We show that the matching condition allows us to desingularize N using A via gluing, giving our main result. Theorem 1.2 If the matching condition is satisfied, there exists τ > 0 such that for all t ∈ (0, τ ) there is a smooth compact coassociative 4-fold N (t) in M , formed by gluing tA and N , such that N (t) converges to N in the sense of currents as t → 0.
By deforming N (t) we obtain a family of desingularizations of N whose dimension we can determine from the topology of A and N . Recall that for a noncompact 4-manifold we define b 2 + by considering the cup product on cohomology classes representing by compactly supported 2-forms. This shows that the moduli space of smooth compact coassociative 4-folds can be non-compact and that coassociative 4-folds with conical singularities can arise on the boundary of the moduli space. Moreover, we have a gluing map from the moduli space of matching pairs (N, tA) into the moduli space of smooth compact coassociative 4-folds, from which we may deduce the following. Proposition 1.4 If C is stable, the gluing map is a local diffeomorphism.
The organisation of the paper is as follows.
• In §2 we provide the basic definitions and notation which shall be used throughout the paper and discuss foundational results for coassociative 4-folds involving self-dual 2-forms and tubular neighbourhood theorems.
• In §3 we describe our gluing construction, identify the obstructions to the procedure and the necessary matching condition for the construction to succeed. We also set up the analytic framework for our problem and discuss the relationship between our matching conditions and the deformation theory of A and N , which allows us to prove Proposition 1.1.
• In §4, we realise our smoothing of the singular coassociative 4-fold as the fixed point of a map between Banach spaces, which we prove is a contraction by deriving appropriate analytic estimates on the smoothing using estimates on the "building blocks" A and N . We deduce Theorem 1.2 and Corollary 1.3 from this work.
• In §5 we compare the moduli space of "matching pairs" (N, tA) to the moduli space of smoothings and deduce Proposition 1.4. We conclude by applying our theory in examples.

Foundations
In this section we describe the basic foundational material we need to tackle our desingularization problem.
Definition 2. 1 We call a 3-form ϕ on an oriented 7-manifold M a G 2 structure if, for all x ∈ M , ϕ| x = ι * x (ϕ 0 ) for some orientation preserving isomorphism ι x : T x M → R 7 . A G 2 structure ϕ defines a metric g ϕ on M .
We denote a 7-manifold M endowed with a G 2 structure ϕ by (M, ϕ). An oriented 7-manifold will admit a G 2 structure if and only if it is spin. We now define special classes of G 2 structures which will be especially relevant.
Definition 2. 2 We say that (M, ϕ) is an almost G 2 manifold if dϕ = 0. We call (M, ϕ) a G 2 manifold if dϕ = d * ϕ = 0 with respect to g ϕ , which is equivalent to saying that the holonomy Hol(g ϕ ) ⊆ G 2 .
In 7-manifolds with a G 2 structure we have a distinguished class of submanifolds which shall form the basis for our study.
When (M, ϕ) is a G 2 manifold, coassociative 4-folds are volume-minimizing in their homology class. Although coassociative 4-folds lose this property in general almost G 2 manifolds, their geometry otherwise has essentially the same features as in the G 2 manifold case. Coassociative geometry is discussed in detail in [11].
Let B(0; r) denote the Euclidean ball of radius r about 0 in R 7 . For a cone C in R 7 (i.e. a dilation-invariant subset) such that C \ {0} is a smooth submanifold we let Σ = C ∩ S 6 with the induced metric g Σ , let ι : C ∼ = R + × Σ → R 7 be the inclusion map and let ∇ C be the Levi-Civita connection of the cone metric g C = dr 2 + r 2 g Σ on C.
We may now define the two types of submanifold which shall appear in our desingularization problem: namely conically singular and asymptotically conical. In each case we have a noncompact submanifold of (M, ϕ) which, outside a compact set, is diffeomorphic to a cone (or finite collection of cones) and converges to the cone with some prescribed rate. The two classes of submanifold will be dual in the sense that one converges towards the cone near its vertex (so has a singular point) whereas the other converges to the cone near infinity. Definition 2.4 Let N be a (singular) submanifold of (M, ϕ) and let z ∈ N . Choose local coordinates χ : B(0; ǫ M ) → V ∋ z, for some ǫ M ∈ (0, 1) and open V ⊆ M , such that χ(0) = z and (dχ| 0 ) * (ϕ| z , g ϕ | z ) = (ϕ 0 , g 0 ). (These are natural coordinates for a neighbourhood of z in (M, ϕ).) We say that N has a conical singularity at z if there exist a cone C ⊆ R 7 with link Σ ⊆ S 6 , constants ǫ ∈ (0, ǫ M ) and µ ∈ (1, 2), open U ⊆ V ∩ N containing z and a smooth map We call C the cone and µ the rate at the singularity. We say that N is a conically singular (CS) submanifold if N is compact and connected and smooth except for finitely many conical singularities. We call N a CS coassociative 4-fold if N is a CS submanifold whose nonsingular part is a coassociative 4-fold.

Remarks
(a) By [16,Proposition 3.6], if N is a CS coassociative 4-fold then the cones at the singularities are coassociative in R 7 .
(b) The stipulation that µ < 2 allows the definition of conical singularity to be essentially independent of the choice of local coordinates χ, as explained in [16, §3.2].
(c) Notice that if N is CS with rate µ 0 it is also CS with any rate µ ∈ (1, µ 0 ]. We are thus free to reduce the rate µ, so we always choose µ close to 1.
We say that A is AC with rate λ to C to emphasise the choice of C and λ.

Remarks
(a) By [18,Proposition 2.8], if A ⊆ R 7 is coassociative and AC to C then C is coassociative.
(b) Observe that A only genuinely converges to C at infinity if λ < 0, so allowing for λ ∈ [0, 1) permits weak decay.
(c) Note that if A is AC with rate λ 0 it is also AC with any higher rate λ ∈ [λ 0 , 1), so we are at liberty to increase the rate λ.
As we see, AC submanifolds are smoothings of cones and thus provide obvious models for desingularizing CS submanifolds via gluing. However, the challenge is to desingularize CS coassociative 4-folds so that the coassociative condition is preserved, so one would naturally require AC coassociative 4-folds in the gluing. Our problem is to study when this approach may be successfully applied and understand the obstructions to the coassociative gluing process.
To fix notation we describe the ingredients we wish to feed into our problem.
• Let N be a CS coassociative 4-fold in an almost G 2 manifold (M, ϕ) with a single conical singularity at z with rate µ and cone C.
• Let A ⊆ R 7 be a coassociative 4-fold which is AC with rate λ < 1 to C.
By Definition 2.5, there exist R > 0, compact K A ⊆ A, and a diffeomor- Whenever t is sufficiently small that t −1 ǫ > R, we set The subsets tÂ(t) ⊆ tA will be glued toN to resolve the singularity z.
• Let τ ∈ (0, 1) be small enough that τ R < ǫ and τÂ(τ ) ⊆ B(0; ǫ M ). Throughout we let t ∈ (0, τ ) be arbitrary and will make τ smaller a finite number of times, continuing to refer to this new smaller constant as τ . Our constraints on τ ensure that we can use the local coordinates χ to view the gluing of tÂ(t) toN , outside the compact set K N , as occurring in B(0; ǫ M ) ⊆ R 7 .
We shall occasionally refer toN \ K N and A \ K A as the end (or ends since they could be disconnected) ofN and A.

Self-dual 2-forms and tubular neighbourhoods
In geometric gluing problems, it is often crucial to know the relationship between deformations of the building blocks and those of the glued object. We shall therefore need to understand deformations of coassociative 4-folds, for which the key result is the following [23, c.f. Proposition 4.2].
Proposition 2.6 Let X be a coassociative 4-fold in (M, ϕ). There is an isometric isomorphism  X between the normal bundle ν(X) of X in M and Λ 2 + T * X given by v → (v ϕ)| X .
Note For any coassociative 4-fold X we will consistently use the notation  X to indicate the isomorphism in Proposition 2.6.
Using this identification, we can view nearby submanifolds to X as graphs of small self-dual 2-forms; that is, give open neighbourhoods of the zero section in Λ 2 + T * X and of X in M and a diffeomorphism between them which acts as the identity on X (identified with the zero section as usual). However, we must perform this construction carefully so as to ensure compatibility with  X and to take into account the asymptotic behaviour of A and N . We first make the compatibility property precise.
Definition 2.7 Let X be a coassociative 4-fold in (M, ϕ). Suppose we have a smooth map Υ X from an open neighbourhood of the zero section in Λ 2 + T * X to an open tubular neighbourhood of X in M , acting as the identity id X on X. We may then view dΥ X | X as a map from T X ⊕ Λ 2 where I : T X → T X is the identity and A : Λ 2 + T * X → T X is arbitrary.
We now construct our tubular neighbourhoods using self-dual 2-forms as in the author's earlier papers [16,17,18], however our presentation is different and more in the style of [10, §3-4] as it is more convenient.
There exist self-dual 2-forms α N on (0, ǫ) × Σ and α A on (R, ∞) × Σ such that Moreover, for all j ∈ N, Proof : Applying the Tubular Neighbourhood Theorem to Σ ⊆ S 6 , we can easily construct diffeomorphic dilation-invariant open neighbourhoods of C in ν(C) and R 7 . Using  C gives (a). We can certainly change ǫ and R as claimed given the asymptotic behaviour of Φ N and Φ A . Part (b) then follows from (a), the definition of N and A as CS and AC submanifolds and the fact that  C is an isometric isomorphism. Proposition 2.8 says we may effectively view the ends of A and N as graphs of the self-dual 2-forms α A and α N on the cone. We can then extend this result as in our earlier work to give neighbourhoods of A and N , which are adapted so that we may realize graphs of small self-dual 2-forms on the ends as graphs of small self-dual 2-forms on the cone. Proposition 2.9 Recall the notation of Proposition 2.8.  Having identified self-dual 2-forms α with nearby submanifolds X α to a coassociative 4-fold X we may ask: what is the condition on α which makes X α coassociative? By Definition 2.3 this is given by ϕ| Xα = 0, which leads to a fully nonlinear equation on α. By the calculation in [23, p. 731] we see that the linearisation of this equation is dα = 0 since dϕ = 0. (Here is where we use the condition that (M, ϕ) is an almost G 2 manifold, since otherwise the linearisation would have further terms.) We deduce the following well-known fact.
Proposition 2.10 Let X be a coassociative 4-fold in an almost G 2 manifold. Infinitesimal coassociative deformations of X are given by closed self-dual 2forms on X.
Closed self-dual forms are trivially also coclosed. Hence, if X is compact, Hodge theory implies that such forms uniquely represent cohomology classes in H 2 (X). In the non-compact setting we do not have such a result, but for AC and CS 4-folds we can say which cohomology classes are uniquely represented by L 2 closed self-dual 2-forms. This leads to our next definition.

Definition 2.11
For any Riemannian 4-manifold X, let Notice that H 2 (X) = H 2 + (X) ⊕ H 2 − (X) and that by elliptic regularity H 2 (X) consists of smooth forms.
If X is compact then dim H 2 (X) = b 2 (X) and dim H 2 ± (X) = b 2 ± (X). If X is an AC or (the nonsingular part of) a CS 4-fold and we let where J ± (X) are the maximal positive and negative subspaces of J (X) with respect to the cup product. (The subspaces J ± (X) are well-defined because the cohomology classes in J (X) are represented by compactly supported forms.) We thus define b 2 ± (X) = dim J ± (X).
By [23, §4], the deformation theory of compact coassociative 4-folds X is unobstructed, so infinitesimal deformations always extend to genuine deformations and thus we have the following.
Theorem 2.12 Let X be a compact coassociative 4-fold in an almost G 2 manifold. The moduli space of compact coassociative deformations of X is a smooth manifold near X of dimension b 2 + (X).
The author extended this result in [16] and [18] to the CS and AC settings, where various similarities and differences occur which shall be discussed later. These results will be crucial in understanding obstructions to the gluing problem.

Desingularization: geometry
In this section we tackle the more "geometric" aspects of our desingularization problem. The key part is to construct an appropriate connect sumÑ (t) ofN and tA such thatÑ (t) is a smooth compact 4-fold withÑ (t) → N as t → 0.
The crucial point will be to ensure thatÑ (t) is sufficiently "close" to being coassociative; i.e. that |ϕ|Ñ (t) | is "small enough" that one may hope to perturb N (t) to a nearby coassociative 4-fold N (t). Unlike in [17], where one was able to constructÑ (t) using a rather naive connect sum, here we have to use a more refined technique which requires a detailed understanding of the geometric obstructions to the coassociative gluing procedure. We discover that the obstructions which emerge are both topological and analytic in nature, and we can give natural interpretations for the obstructions.

Obstructions
Studying the argument in [17], one sees that for our problem we simply cannot use the same method of constructingÑ (t) since the analysis will fail. This is not a flaw with the analytic method, but rather it is a geometric phenomenon. Specifically, in [17] geometric assumptions were made precisely to ensure that the desingularization was unobstructed. In general, there are geometric obstructions to resolving the coassociative conical singularity, which we now identify.
We begin by examining the cone C. Consider a self-dual 2-form α on C which is homogeneous of rate υ say. We may write for a 2-form α Σ on the link Σ of C, noting that |α Σ | gC = O(r −2 ). (We use the notation * Σ to clarify that we are using the Hodge star on Σ.) The condition that α is closed is equivalent to d * Σ α Σ = (υ + 2)α Σ and dα Σ = 0.
Such closed forms α define infinitesimal coassociative deformations of C by Proposition 2.10. These forms will also naturally relate to deformations of A andN . To understand this relationship we first make a convenient definition. Since we may view A as a manifold with boundary Σ, we have the following exact sequence: The connection between deformations of C and A can now be succinctly expressed through one of the main results in [18].
Theorem 3.2 Suppose that the rate λ < 0 and let λ + ∈ (−2, 0) \ D be such that λ + ≥ λ. The moduli space of deformations of A as an AC coassociative 4-fold with rate λ + and cone C is a smooth manifold near A of dimension which is the dimension of The appearance of the term b 2 + (A) is clear by Definition 2.11. A key part of the proof and dimension count relies on showing that various closed self-dual 2forms on C lift to A, applying the theory in [15]. The forms on C corresponding to forms in H 2 + (A) are actually zero, but for the other terms in the dimension count one has non-trivial forms on C lifting to A.
Specifically, the harmonic representatives of the classes in Im j A 2 define the homogeneous closed self-dual 2-forms on C with order O(r −2 ) which lift to define closed self-dual 2-forms on A. Notice that such forms on A, given their decay rate on the ends, cannot lie in L 2 and so do not contribute to b 2 + (A). Moreover, the sum over d D (υ) counts the homogeneous closed self-dual 2forms on C which have rate between −2 and λ + , and the proof of Theorem 3.2 shows that these forms on C all lift to closed self-dual 2-forms on A.
The final key part of the proof of Theorem 3.2 is to show that, given a closed self-dual 2-form α 0 on A with appropriate decay on the ends, one can solve for a transverse self-dual 2-form α ′ on A so that ϕ 0 vanishes on the graph of α 0 + α ′ via the Implicit Function Theorem. Thus we can extend the infinitesimal AC coassociative deformation of A given by α 0 to a genuine deformation.
With these preliminaries we are now able to analyse A further.
Since A is coassociative, ϕ 0 vanishes on the graph of α A . By [23, Proposition 4.2] and the compatibility conditions we have imposed on Υ C , we see as in the proof of [16, Proposition 6.9] that where Here we have used the fact that r −1 |α A | and |∇ C α A | tend to zero as r → ∞.
We may view C as an AC coassociative deformation of A or vice versa, so from the discussion preceding the statement of the proposition we may decom- The main point is that any infinitesimal AC coassociative deformation of A of rate above −2 is given by the lift of an element in K C (λ).) Now dα 0 Remark We see that if λ < −2 then Proposition 3.3 is irrelevant. This proposition marks the significant departure from the work in [17].
We now make some observations to understand the obstructions to the gluing procedure. If we desingularize N using tA we will obtain a smooth 4-dimensional submanifoldÑ (t) of M which is diffeomorphic to the disjoint union of the compact sets tK A and K N and the portion of the cone (tR, ǫ) × Σ.
Suppose we constructÑ (t) so that This product must go to zero as t → 0 if we are to have any hope thatÑ (t) can be perturbed to a coassociative smoothing of N , so we require η < −3.
We can view the subset ofÑ (t) which is diffeomorphic to (tR, ǫ) × Σ as the graph of a self-dual 2-form α which is at best of order O(r λ ), since we are using A to constructÑ (t). Using a similar equation to (6), naively the behaviour of |ϕ|Ñ (t) | is dominated by |dα| = O(r λ−1 ), so it would appear that we need λ < −2 so that |ϕ|Ñ (t) | = O(t η ) for η < −3. (This is a way to interpret how this condition arises in [17].) Hence for rates λ ≥ −2 we should see obstructions to our gluing procedure, whereas for λ < −2 we should not.
However, if we can arrange α to be closed then, again using an equation like (6), we have that |ϕ|Ñ (t) | is now dominated by |r −1 α| 2 and |∇α| 2 , which are of order O(r 2λ−2 ). We have thus improved our estimate drastically as we now only require 2λ − 2 < −3 for our analysis to go through, which is equivalent to We deduce that for rates λ ∈ [−2, − 1 2 ), the obstructions to the desingularization arise purely from the ability to glue tA and N using a closed self-dual 2-form, which is a natural constraint in the context of coassociative geometry.
These considerations allow us to state the key condition that we require to overcome the obstructions. Notice that since α 0 A is homogeneous it is defined on the entire cone C.
We say that A and N satisfy the matching condition if there exists δ 0 > 0 and for i = 1, . . . , d there exists a closed self-dual form α i N onN such that Effectively, this says that each infinitesimal coassociative deformation α i A of the cone C extends to an infinitesimal coassociative deformation α i N ofN so that to "leading order" α i N tends to α i A as r → 0.
Remark As we shall see, the matching condition precisely allows us to define our desingularization so that over (tR, ǫ) × Σ it is the graph of a self-dual 2-form whose leading order term is closed.
We now give further geometric meaning for (part of) our matching condition.
The class [α 0 Σ ] is not mysterious but has a natural geometric interpretation. Recall the map  A given by Proposition 2.6 and (5).

Proposition 3.5
Let v be the dilation vector field on R 7 and let u be the projection of v onto the normal bundle of A. We have that d A (u) = 0, Remark Proposition 3.5 says that the cohomology class of the infinitesimal dilation deformation of A is a multiple of the class of Hence, the corresponding infinitesimal deformation is the closed part of 3α A which, to leading order (that is, for order at least O(r −2 )), is given by 3α 0 As for A in (5), we have an exact sequence forN : We can now interpret part of the matching condition in topological terms.
There exists a closed self-dual 2-form α onN and δ 0 > 0 such that, for all j ∈ N, we can pull it back to the end ofN and extend it smoothly to define a closed 2-form β (c.f.
Since µ − 3 > −2 we would be done except that γ is not necessarily closed. Now, dγ lies in the space of exact forms which decay at rate O(r µ−4 ), so lies in the image of d acting on 2-forms which decay with rate O(r µ−3 ). As we shall see in §3.5, given k ≥ 4 and δ 0 > 0 such that (−2, −2 (see [17, §3.4] for example for the definition of the weighted Sobolev spaces, which control the decay rate of forms near the singularity).
We deduce from Proposition 3.6 that part of the matching condition is purely topological; that is, we can replace the condition that a closed self-dual 2-form exists onN asymptotic to the rate −2 part of α 0 A with the assumption that [α 0 Σ ] lies in Im j N 2 . This motivates the following definition for convenience.
Definition 3.7 Recall (5) and (7). We say that A and N satisfy the topological matching condition if We have identified part of the matching condition as a topological constraint, but the remainder is analytic and still needs to interpreted geometrically. As we remarked, homogeneous closed self-dual 2-forms on C with rates in (−2, 0) always extend to A. However, this is not the case forN , and such forms which do not extend correspond to obstructions to the deformation theory ofN (c.f. [16]). Therefore if the deformation theory ofN is unobstructed, the analytic part of matching condition will hold. We shall discuss these ideas in detail later.
The work in this subsection leads us to impose the following conditions on A and N from now on.

Conditions Assume that
• the rate λ of convergence of A to C satisfies λ < − 1 2 and • A and N satisfy the matching condition in Definition 3.4.
In particular, the topological matching condition in Definition 3.7 is satisfied.
From our discussion it is clear that, unless we make further assumptions or develop an even more sophisticated construction, the conditions we have imposed will be essential for our analysis to go through.
We shall see later that the condition λ < − 1 2 allows for far more examples of conical singularities than the situation in [17] where λ < −2. One could conceivably impose further conditions on α for λ ≥ − 1 2 to ensure that |ϕ|Ñ (t) | has the required decay property for our analysis to work, but these appear to be less geometrically natural so we choose not to pursue this.

Construction
We now define our (approximately coassociative) desingularizationsÑ (t) of N using tA. Recall the notation of the matching condition in Definition 3.4 and suppose without loss of generality that δ 0 is small enough that Choose τ sufficiently small so that Observe that Therefore, if we let then, by Proposition 2.9,Ñ (t) is a smooth compact 4-fold so thatÑ (t) → N as t → 0 in the sense of currents in Geometric Measure Theory.
Remark The choice of ν in (9) will remain mysterious until late in the argument but, roughly, we need to choose ν close to 1 so that the interpolation region r ∈ [ 1 2 t ν , t ν ], where ϕ potentially has the worst behaviour, is small as t → 0.
Our aim is to solve the following problem.
Problem DeformÑ (t) to a nearby coassociative 4-fold N (t), so that N (t) → N as t → 0 in the sense of currents.
Informally, if |ϕ|Ñ (t) | is sufficiently small we hope to make it vanish after a small perturbation. It is clear from the observations in §3.1 that the conditions we imposed precisely ensure that obstructions to this procedure can be overcome. Our problem involves deforming the non-coassociativeÑ (t). As we have seen, infinitesimal deformations of coassociative 4-folds are defined by closed self-dual 2-forms, resulting in a deformation theory determined by solutions to an elliptic problem. To exploit this fact we want to be able to identify normal deformations toÑ (t) with self-dual 2-forms, even thoughÑ (t) is not coassociative. We achieve this as in [17] by defining the self-dual 2-forms with respect to a different metric onÑ (t) from the induced one. This follows from an easy modification of [17, Proposition 2.9 & Lemma 4.3], since all one requires is that ϕ|Ñ (t) C 0 is smaller than some universal constant, and we can in fact make this norm arbitrarily small by choosing τ sufficiently small sinceÑ (t) → N .
If τ is sufficiently small, there exists a unique metricg(t) onÑ (t) such that Note From now on we shall calculate all quantities onÑ (t) with respect to the metricg(t) given in Proposition 3.9, unless stated otherwise, and we shall use the notation of Definition 3.8.

Weighted spaces
We wish to define spaces of forms onÑ (t) whose behaviour on the piece we have glued into N is controlled, since this is where the geometry is degenerating. We achieve this using Banach spaces with weighted norms, as discussed in detail in [25]. To define these spaces we need an appropriate radius function.
Definition 3.10 Recall Definition 3.8 and let R ′ , ǫ ′ be constants so that We define a radius function ρ t :Ñ (t) → [tR, ǫ] as a smooth map such that and ρ t on Υ C (Γ αC (t) ) is a strictly increasing function of r such that In other words, ρ t is a small perturbation of the piecewise smooth function which is constant on χ(tK A ) and Υ N (Γ α 0 N (t)|K N ) and equal to r on Υ C (Γ αC (t) ). The existence of such a function ρ t onÑ (t) is clear.
These weighted spaces are Banach spaces since the norms are equivalent to the usual norms for each fixed t.
These spaces have the nice feature, unlike their "unweighted" counterparts, that they are equivariant under dilations in t. Thus, with respect to these weighted norms, we can understand the behaviour of quantities as t → 0, which is crucial for our analysis. For a detailed discussion of these issues see [25].
On CS and AC submanifolds we can define Sobolev, C k and Hölder spaces of forms, denoted L p k,υ , C k υ and C k,a υ , which depend on a weight υ ∈ R. Informally, these Banach spaces consist of forms whose restriction to any compact set lies in the usual Sobolev, C k or Hölder space, but which also have controlled rate of decay on the ends determined by υ. This is achieved using weighted norms as in Definition 3.11, replacing ρ t by an appropriate radius function -we refer the interested reader to [17, §3.3] or [25] for details. We point out that L 2 0,−2 = L 2 .

Topology
Clearly we need to know b 2 + Ñ (t) to understand deformations ofÑ (t). This is a topological invariant which we can determine using the topology ofN and A.
Theorem 3.12 Using the notation of Definition 2.11, (5) and (7), we have that Proof : We can clearly choose a pair of open subsets ofÑ (t), diffeomorphic to A andN , which coverÑ (t) and whose intersection is diffeomorphic to C. By Mayer-Vietoris we then have the following exact sequence: We first calculate dim Im∂ 1 . Using (5) we see that dim Im ∂ A 1 = dim Im j A 2 and the same result holds for N by (7). The fact that these spaces have the same dimension is a consequence of Poincaré duality, which further allows us to construct an isomorphism between them. Now∂ 1 is defined so that∂ 1 [α Σ ] can be simultaneously viewed as Hence, Im∂ 1 is dual to the intersection of Im j A 2 and Im j N 2 . We conclude that We now determine dim Kerj 2 . By definition,j 2 = j A 2 − j N 2 , so may also calculate, recalling Definition 2.11, We deduce from (13), (14) and (15) that There is no obstruction to elements of Im j N 2 ⊆ H 2 (Σ) lifting to closed 2forms onN which are either self-dual or anti-self-dual by Proposition 3.6, and a similar result holds on A (as noted after Theorem 3.2). We conclude that and deduce (11).
Remarks From the proof of Theorem 3.12 we deduce a special case of the Novikov additivity theorem, namely that This is unsurprising since our proof essentially follows the argument in [1] for proving the Novikov additivity theorem.

Stability
Theorem 3.2 allows us to conclude that infinitesimal deformations of C which are homogeneous of rate (−2, 0) always extend to genuine deformations of A, so the deformation theory of A is unobstructed. This unobstructedness follows from the fact that, for λ + ≥ λ such that λ + ∈ (−2, 0) \ D, In contrast onN we find that the images of d on 2-forms and self-dual 2-forms in L 2 4,µ can differ.
We call O(N, µ) the obstruction space since one of the main results in [16] states that if O(N, µ) = {0} then N has a smooth moduli space of deformations as a CS coassociative 4-fold; that is, its deformation theory is unobstructed.
We shall now show that the obstruction space corresponds to closed self-dual 2-forms on C which do not lift toN . From our matching condition in Definition 3.4, we see that these are exactly the sort of obstructions we need to overcome in order to solve our gluing problem. This allows us to draw a direct link between obstructions to the smoothing of N and obstructions to deformations of N .
We begin with the following result from [16].
Proposition 3.14 Recall Definition 3.1, let µ 0 be the least element of (−2, 0)∩ D ∪ {0} and let µ − ∈ (−2, µ 0 ). If µ + ∈ (−2, 0) \ D with µ + > µ − then the dimension of the kernel of d in L 2 4,µ+ (Λ 2 The reason for the appearance of Ker(d * + + d) υ is that it is isomorphic to the cokernel of the map We need to relate Ker(d * + +d) υ to the space of closed and coclosed 3-forms, since O(N, µ + ) is isomorphic to the subspace of Ker(d * + + d) µ+ which is orthogonal to these forms by the work in [16]. We begin with the following observation. Proof : Since d * γ ∈ L 2 3,−2 ֒→ L 2 we can calculate using the fact that d * γ is anti-self-dual and the decay properties of γ to ensure the integration by parts is valid.
As previously mentioned, the work in [16] shows that Using the notation of Proposition 3.14, applying Proposition 3.16 gives: for arbitrarily small ε > 0, to finish we need to calculate how the obstruction space changes as the rate crosses −2. The next lemma states that it does not change.
Combining the results in this section we deduce the following.
We can therefore check that the matching condition is satisfied purely using a topological criterion and the spectrum of the curl operator on Σ. The analytic part of the matching condition therefore relates to the work in [20] on stability of coassociative conical singularities.
We now recall the notion of stability index for a coassociative cone from [20].

Definition 3.19
Let C denote a deformation family of coassociative cones in R 7 containing C which is closed under the action of translations and G 2 transformations. The C-stability index of C is If the family C consists solely of the G 2 ⋉R 7 transformations of C, we simply write ind C (C) = ind(C) and call ind(C) the stability index of C.
Since translations of C trivially provide coassociative deformations of C of order O(1), they define homogeneous closed self-dual 2-forms on C of rate 0, and thus d D We say that C is C-stable (or stable) if ind C (C) = 0 (or ind(C) = 0).
The C-stability index is a non-negative integer invariant and it follows from [20, Proposition 4.11] that the deformation theory of N as a CS coassociative 4fold, where we allow the singularity to move in M and the cone at the singularity to deform in C, is unobstructed if ind C (C) = 0. In particular, dim O(N, µ + ) = 0 for all µ + ∈ (−2, 0) if ind C (C) = 0. We thus conclude with the following.

Desingularization: analysis
In this section we apply analytic techniques to prove our main result (Theorem 1.2). We begin by deriving a key Sobolev embedding inequality, which involves the construction of an "approximate kernel" for the exterior derivative on selfdual 2-forms on our glued manifold. We then view our desingularization problem as a fixed point problem for a certain map, so we show that this map is a contraction using further analytic estimates. To derive the embedding inequality and the estimates we shall make crucial use of the geometric preliminaries of §3.
For the whole of this section we let δ satisfy and be such that [−2 − δ, −2 + δ] ∩ D = {−2} if −2 ∈ D and let δ = 0 if −2 / ∈ D. This is possible by our assumption that λ < − 1 2 and the properties of D.

The approximate kernel
Here we obtain our Sobolev embedding inequality, closely following the work in [17] with improvements in light of Pacini's work in [25]. The result is a bound (depending on t in an explicit way) for the norm of self-dual 2-forms α onÑ (t), transverse to the closed forms, by the norm of dα. Since we have scale-invariant Sobolev embedding inequalities on A andN , the idea is to use closed self-dual 2-forms on A andN to build a subspace of the self-dual 2-forms onÑ (t) which "approximates" the kernel of the exterior derivative on self-dual 2-forms. If the closed self-dual 2-forms on A andN decay sufficiently fast on the ends, we can simply cut them off and approximate them using a compactly supported 2-form which can easily be viewed as an approximate kernel form oñ N (t). However, at the critical decay rate, namely at the L 2 growth rate −2, this cut off procedure will not work and so one needs to define approximate kernel forms by interpolating between L 2 kernel forms on A and L 2 kernel forms onN , when this is possible. Using the topological calculations in §3, we find that these forms define an approximate kernel of equal dimension to the actual kernel.
However, the interpolation between L 2 kernel forms on A andN is not always possible, and one can detect this topologically by the work in §3.1. When this occurs, we have closed self-dual 2-forms on A which do not define approximate kernel forms onÑ (t) and so give potential obstructions. These forms will cause the Sobolev embedding constant to blow up as t → 0 but because we can identify the forms explicitly we can determine the rate at which the blow up occurs.
and let K A 0 be such that Moreover, this estimate holds for the same constant C(A) on tA for all t > 0.
Proof : The map is elliptic and Fredholm by choice of δ (c.f. [18,Proposition 5.4] and the remarks preceding the statement), and therefore has a finite-dimensional kernel of smooth forms by elliptic regularity. Since K A ± is contained in this kernel it is also necessarily finite-dimensional and consists of smooth forms.
We can cut off the forms in K A − appropriately at infinity to define a space K A ap of compactly supported self-dual 2-forms on A, L 2 -orthogonal to (This is a manifestation of the fact that, by definition, C ∞ cs is dense in L 2 4,−2−δ .) In other words, we can ensure that the L 2 -orthogonal complement of K A ap in L 2 4,−2+δ is transverse to K A − and contains K A 0 . The theory of elliptic operators on weighted Sobolev spaces as in [22] applied to (22)  We can also prove the following analogue of Proposition 4.1 in a similar (easier) manner which we omit. There is a subspace K N ap ⊆ C ∞ cs (Λ 2 + T * N ), with dim K N ap = dim K N , and a constant C(N ) > 0 such that if α ∈ L 2 4,−2+δ (Λ 2 + T * N ) satisfies α, β L 2 = 0 for all β ∈ K N ap then α L 2 4,−2+δ ≤ C(N ) dα L 2 3,−3+δ . We now wish to define our approximate kernel. We begin with K A ap and define a diffeomorphism If τ is sufficiently small we may identify the metrics, hence the self-dual 2-forms, on tÂ(t) and Ψ A,t tÂ(t) . This allows us to view the open subset tÂ(t) of tA as a subset of the desingularizationÑ (t).
Let β A 1 , . . . , β A mA be a basis for K A ap . Since ν < 1, we can choose τ so that for i = 1, . . . , m A , so we may identify the β A i with forms on tÂ(t) in the obvious manner, which we denote by the same symbols. Using Ψ A,t we can define for each For τ sufficiently small we can identify the metrics, hence the self-dual 2-forms, onN (t) and Ψ N,t (N (t)). As above, this allows us to view the open subsetN (t) ofN as a subset ofÑ (t). Let β N 1 , . . . , β N mN be a basis for K N ap . Since ν > 0 we can ensure, by making τ smaller if necessary, that for all i. Using Ψ N,t we can then define for each β N i a corresponding self-dual 2-form ξ N i onÑ (t) which vanishes outside Ψ N,t (supp β N i ).
Definition 4.4 LetK N ap (t) = Span{ξ N 1 , . . . , ξ N mN }. It will be important to identify the closed self-dual 2-forms on A which extend toN and those which do not, as we saw from our discussion of the obstructions to the gluing problem in §3.1. Motivated by Proposition 3.6 we can split K A 0 in the following useful way.

Definition 4.5 Recall (5) and (7). Let
The notation I and O reflects the fact that elements in K I extend to infinitesimal deformations ofN and henceÑ (t), whereas K O gives potential obstructions. Let β I 1 , . . . , β I mI and β O 1 , . . . , β O mO form bases of K I and K O . Since every element of Im j A 2 lifts to a form in K A 0 by the work in [18], we have that We shall now define a part of the approximate kernel using K I and we explain the idea. Since each β ∈ K I is asymptotic to a closed self-dual 2-form β C on C whose cohomology class lies in the image of j N 2 : H 2 (N ) → H 2 (Σ) ∼ = H 2 (C), we can find a closed self-dual 2-form γ onN which is also asymptotic to β C . We then interpolate between β and γ to define a self-dual 2-form onÑ (t) which is "almost" closed, i.e. it is closed except on some small compact set.
We can now define a self-dual 2-form ξ I i onÑ (t), for i = 1, . . . , m I , so that on χ(tK A ) it equals β I i (using the identification Ψ A,t ), on Υ N (Γ tα 0 N |K N ) it equals γ I i (using the identification Ψ N,t ), and on Υ C (Γ αC (t) ) it interpolates between these definitions in the following way: where f inc is given in Definition 3.8. Notice that ξ I i is closed except on the region where tA is connected toN to formÑ (t).
We now turn to β ∈ K O which do not define approximate kernel forms. We cut off β to define a self-dual 2-form ξ onÑ (t) which vanishes onN (t) and is closed except on a small compact set so that, as t → 0, ξ converges back to β on A, after re-scaling. Although ξ will again be "almost" closed, this time there is no corresponding closed self-dual 2-form onÑ (t) which it approximates. Moreover, since ξ converges to a non-trivial closed form as t → 0 it is clear that ξ will contribute to the blow up of the Sobolev embedding constant as t → 0.
Define f O : A → [0, 1] to be a smooth function such that again using the identifications as before. We can achieve this essentially by cutting off the form β O i using f O . Notice that ξ O i is closed except on the interpolation region between tA and N inÑ (t).
Observe that, by construction, the sums inK ap (t) are direct and therefore that using (25) and Theorem 3.12.
We may now state our Sobolev embedding inequality.
Proof : The idea of the proof is first to show that the only way that the Sobolev embedding constant can blow up as t → 0 is if we have a sequence converging to a closed self-dual 2-form on A or N . The construction ofK ap (t) means that the only non-trivial limit that can occur is an element of K O . We can therefore just study the behaviour of forms inK O (t) as t → 0 to deduce our estimate.
Suppose, for a contradication, that there exists a decreasing sequence of positive numbers t n → 0 and a sequence α n ∈ L 2 4,−2+δ,tn Λ 2 + T * Ñ (t n ) such that α n , ξ L 2 = 0 for all ξ ∈K ap (t n ) and Therefore, the sequences are bounded. So, by the compact embedding theorem for weighted Sobolev spaces [14,Theorem 4.9], after passing to a subsequence, both sequences converge in L 2 3,−2+δ ′ , to ξ A and ξ N say, where δ ′ > δ for ξ A and δ ′ < δ for ξ N . Using the bounds for dα n L 2 3,−3+δ,tn we see that Hence dξ A = 0 and similarly dξ N = 0. Elliptic regularity implies that ξ A and ξ N are smooth and, as −2 + δ / ∈ D, the work in [15] shows that the space of closed self-dual 2-forms is the same at rates −2+δ and −2+δ ′ if δ ′ is sufficiently close to δ. We conclude that Ψ * A,tn α n and Ψ * N,tn α n converge in L 2 4,−2+δ to ξ A and ξ N respectively.
The fact that α n is L 2 -orthogonal toK ap (t n ) means that ξ A is L 2 -orthogonal to K A ap ⊕ K I and ξ N is L 2 -orthogonal to K N ap . From Proposition 4.2 we deduce that ξ N = 0. If ξ A is L 2 -orthogonal to K O then ξ A = 0 by Proposition 4.1, so we must have that Overall α n is a sequence of forms such that Ψ * N,tn α n → 0 and Ψ * A,tn α n → ξ A ∈ K O . Thus, for all n sufficiently large, α n is well approximated by elements inK O (t n ), so we now analyse these forms.
By definition, elements of K O are kernel forms on A which do not extend to corresponding kernel forms on N and thus do not define kernel forms onÑ (t). Hence,K O (t) must be transverse to the closed self-dual 2-forms onÑ (t) for τ small. Therefore there exist some (t-dependent) constants C t (Ñ ) > 0 such that, for all α ∈K O (t), α L 2 4,−2+δ,t ≤ C t (Ñ ) dα L 2 3,−3+δ,t . Recall Definition 4.6 and the discussion preceding it. Any α ∈K O (t) is identified with f O β for some β ∈ K O . Using the definition of f O , the facts that dβ = 0 and satisfies |∇ j C Φ * A β| = O(r −2−j ) for all j ∈ N as r → ∞, together with the assumption that δ < 1 in (20) we calculate Notice that this norm tends to zero as t → 0 as we would expect. We deduce that for some constant C > 0. Hence, for n >> 1, there exists some other constant C > 0 so that This contradicts (27).
Using our estimate we have the following crucial result.
Theorem 4.9 Recall Definition 4.7. The exterior derivative is a bounded invertible linear map between Banach spaces with bounded linear inverse P d satisfying P d ≤ C(Ñ )t −δ(1−ν) .
Proof : Recall that Im d| Λ 2 + = Im d| Λ 2 on compact Riemannian 4-manifolds (c.f. [17,Proposition 2.10]). An immediate consequence of Theorem 4.8 is that the L 2 -orthogonal projection of H 2 + Ñ (t) , given in Definition 2.11, toK ap (t) is injective. It is also surjective since the dimensions of the two spaces are equal by Theorem 3.12.
The domain and range of d given are clearly Banach spaces and the existence of the bounded inverse P d is now clear. The bound on the operator norm of P d is simply a restatement of the estimate in Theorem 4.8.

The contraction map
Recall the tubular neighbourhood constructions for C, A andN in Propositions 2.8-2.9 which identified nearby deformations with graphs of self-dual 2forms. Using the isomorphism ν(Ñ (t)) ∼ = Λ 2 + T * Ñ (t) given by Proposition 3.9, a straightforward adaptation of the work in [17, §5.1-5.2] shows that we can construct a tubular neighbourhoodT (t) ofÑ (t) in M which is identified with theǫ-ball about the zero section in C 1 1,t Λ 2 + T * Ñ (t) for someǫ > 0, in a manner which is compatible with the constructions in Propositions 2.8-2.9. Rather than repeating the details here we refer the interested reader to [17, §5.1-5.2]. We can use this construction to describe coassociative deformations ofÑ (t).
Using the construction discussed above, we can define a nearby deformationÑ α (t) ⊆T (t) ofÑ (t) with a natural diffeomorphism f α (t) : . By the coassociativity of N and A, F t (α) is exact.
By construction, the zeros of F t correspond exactly to nearby coassociative deformations ofÑ (t).
As in [17, Proposition 6.2], we can say more about the deformation map F t .

Proposition 4.11
For α ∈ C 1 Λ 2 + T * Ñ (t) with α C 1 1,t <ǫ, we may write for a smooth map Q t depending on α and ∇α. Moreover, if α ∈ L 2 4 Λ 2 + T * Ñ (t) with α C 1 1,t <ǫ, then From Proposition 4.11 we see that solving F t (α) = 0 is equivalent to solving Since the right-hand side lies in d L 2 4 Λ 2 + T * Ñ (t) , we can use Theorem 4.9 and try to solve for α ∈K ap (t) ⊥ ⊆ L 2 4 Λ 2 + T * Ñ (t) . The idea is to apply the Contraction Mapping Theorem to give a solution to (28), which will in turn define a zero of F t and hence a coassociative deformation ofÑ (t).
As observed, fixed points of C t define elements of Ker F t . Moreover, given a fixed point α of C t , we may apply the Implicit Function Theorem and parameterise the elements of Ker F t near α byK ap (t) ∼ = H 2 + Ñ (t) .
Given the estimate on the norm of P d in Theorem 4.9, to show that C t is a contraction on some neighbourhood of zero in L 2 4,−2+δ,t , it is enough to obtain estimates on the L 2 3,−3+δ,t norm of ϕ|Ñ (t) and Q t (α) − Q t (β) for α, β ∈ L 2 4 . We begin with the estimate on the norm of ϕ|Ñ (t) .

Proposition 4.13
There exists a constant C(ϕ) > 0, independent of t, such that Remark It is in the proof of this proposition that we finally use the constraint on ν in (9).
The key idea in the proof is that our matching condition and the assumption that λ < − 1 2 ensure that the terms which should naively give the largest contribution to |ϕ|Ñ (t) | in fact are zero or effectively cancel.
Following [17,Proposition 6.3] we can estimate the norm of Q t .
Proposition 4.14 There exists a constant C(Q), independent of t, such that if α, β ∈ L 2 4 Λ 2 Before proving this we have the following lemma, which explains the appearance of the factor of t −3+δ in Proposition 4.14 as the t-dependence of the Sobolev embedding constant between weighted Sobolev spaces of different weights.
Then α ∈ C 1 Λ 2 + T * Ñ (t) and there exists a constant c > 0, independent of α and t, such that Proof : The Sobolev Embedding Theorem gives a continuous embedding L 2 4 ֒→ C 1 . Examination of the definition of the weighted norms shows there exists a t-independent constant c 0 such that, for all α ∈ L 2 4 , i.e. the embedding constant L 2 4,−2+δ,t ֒→ C 1 −2+δ,t is independent of t. We now calculate for some constant c 1 independent of t and α. Combining (40) and (41) proves the lemma.
Proof of Proposition 4.14.
In the proof of [17, Proposition 6.2] and following [17,Proposition 6.3], it is explained that that is, Q t is dominated by quadratic terms in ρ −1 t α and ∇α when α C 1 1,t < ǫ. Therefore, an inequality of the type (39) must hold for some (possibly tdependent) constant C(Q) (see, for example, [9, Proposition 5.8] for a detailed description of the type of argument involved). It suffices therefore to show that C(Q) can be chosen to be independent of t.
We may calculate using (42) with β = 0 to show that Using Lemma 4.15 and (43) shows that for some t-independent constant c 2 . We deduce that (44) can be improved to give (39) with constant C(Q) independent of t.
We may now show that C t is indeed a contraction.
Using Theorem 4.9, Proposition 4.13 and Proposition 4.14, we calculate Taking τ such that which is possible by the choice of κ, ensures that C t (α) ∈ B t κ .
Using Theorem 4.9 and Proposition 4.14 again, we deduce that . We finally take τ such that  Proof : Since L 2 4 ֒→ C 1,a by the Sobolev Embedding Theorem, we can apply the method of proof of [17,Proposition 7.16] to show thatα(t) is smooth. The result is now immediate by definition ofÑ (t).
We can now deduce Corollary 1.3 since we can parameterize the zeros of F t nearα(t) using closed self-dual 2-forms onÑ (t).

Proposition 4.18
There is a smooth family of compact coassociative smoothings of N of dimension b 2 + (A) + b 2 + (N ) + dim(Im j A 2 ∩ Im j N 2 ).

Applications
In this section we give some applications of our main results. We first use these results to describe the relationship between the moduli space of "matching pairs" of AC and CS coassociative 4-folds which can be used in our desingularization and the moduli space of the smooth compact coassociative 4-fold we construct. This work leads us to deduce Proposition 1.4 which gives evidence, in the stable case, for local surjectivity of our gluing; i.e. that all nearby smooth coassociative 4-folds to the given CS coassociative 4-fold arise from our desingularization method. We then discuss examples where our theory applies and consequences.

Moduli spaces
We now describe how the moduli spaces of the CS and AC building blocks "fit together" with the moduli space of smoothings, in a similar manner to [8, §8]. Suppose we have an almost G 2 manifold M and a matching pair of a CS coassociative 4-fold N ⊆ M and an AC coassociative 4-fold A ⊆ R 7 with asymptotic cone C ∼ = R + × Σ to which Theorem 1.2 applies. Hence we have τ > 0 and smooth compact coassociative 4-folds N (t) for t ∈ (0, τ ) such that N (t) → N as t → 0. We can make τ canonical by taking the supremum, which will be finite. Moreover, since t ∈ (0, τ ) determines the scale of A glued into N to form N (t), and we are free to re-scale A initially and maintain the AC convergence to C, we may choose A such that τ = 2. All of the N (t) are diffeomorphic to the same compact coassociative 4-fold, so we set X = N (1) for definiteness.
We make a definition for convenience. Recall that for any coassociative 4-fold . Let M(X) denote the moduli space of compact coassociative deformations of X, the coassociative 4-fold arising from gluing N and A as described above. Theorems 2.12 and 3.12 state that M(X) is a smooth manifold of dimension b 2 The main point of this definition is that pairs (N ′ , A ′ ) ∈ M(N ) × M(A) satisfy the topological matching condition as well as the constraint on the AC rate of convergence to C. Therefore, the set of gluing data near (N, A) for which we can apply Theorem 1.2 is a subset of M(N ) × M (A). Moreover, we have a natural map from the gluing data into M(X) given by Theorem 1.2. Since we desingularize N using A to get X it is natural to ask whether we can construct all compact coassociative 4-folds near X via gluing; that is, whether the gluing map is a local diffeomorphism. In general this should not be possible, and the first thing to compare is the dimensions of M(N ), M(A) and M(X).
We begin by recalling the description of M(N ) from [16].
where the stability index ind(C) is given in Definition 3.19.

Remarks
(a) In [16] a lower bound for the expected dimension is given, but Proposition 3.18 allows us to improve the lower bound to an equality here.
(b) As discussed in [20], it is possible to generalize the deformation theory of N so that the cone at the singularity deforms in a family C and the analogous result to Theorem 5.2 holds with the stability index replaced by the C-stability index.
We can easily calculate the expected difference in dim M(X) and dim M(N ): We deduce that the higher the stability index of C, the "less likely" a compact coassociative 4-fold is going to develop a conical singularity modelled on C.
We now use the deformation theory for A from [18] to describe M(A).
Proof : The moduli spaceM(A) of AC coassociative deformations of A with cone C and rate λ 0 is a smooth manifold by Theorem 3.2. Moreover, the tangent space T AM (A) is isomorphic to the closed self-dual 2-forms on A in L 2 4,λ0 . We may define a smooth map π : is surjective by the work in [18], as explained after Theorem 3.2. Thus π is a submersion, so it follows that is a smooth manifold of the claimed dimension.
We deduce from Definition 3.19 that the quantity in brackets is non-negative and vanishes if and only if C is stable. We conclude that, unless C is stable, our gluing method can only at most generate a subset of M(X) near X.
We therefore from now on restrict our attention to the situation where C is stable, which corresponds, in some sense, to the most probable type of conical singularity to occur. It follows from Theorem 5.2 that M(N ) is smooth. Proposition 1.1 and Theorem 1.2 also imply that for any (N ′ , A ′ ) ∈ M(N )× M(A) there exists τ (N ′ , A ′ ) > 0 and smooth compact coassociative 4-folds N ′ (t) for 0 < t < τ (N ′ , A ′ ) formed by gluing N ′ and A ′ which converge to N ′ as t → 0. (We can make τ (N ′ , A ′ ) canonical by taking the supremum again.) Observe further that A ′ ∈ M(A) implies that tA ′ ∈ M(A) for all t > 0 and we may choose τ (N ′ , A ′ ) such that tτ (N ′ , tA ′ ) = τ (N ′ , A ′ ). We can thus make the following definition.
Definition 5.4 Using the notation above, let which is the moduli space of "matching pairs". We can define a smooth map Having defined our "gluing map" G, we can show Proposition 1.4. and that the proof of Theorem 2.12 implies that we can use these closed self-dual 2-forms to define natural coordinates on the moduli space M(X).
A consequence of the work in [16], Theorem 5.3 and the stability of C is that, for δ > 0 such that (−1, −1 + δ) ∩ D = ∅, we have Moreover, we can use these spaces of closed self-dual 2-forms to define natural coordinates on M(N ) and M(A) respectively. In our construction of the approximation of the closed self-dual 2-forms on X in Definition 4.7, we used the same spaces of forms on N and A as above except with the weighted Sobolev space L 2 4,−2+δ . The crucial fact was that the topological condition that j A 2 [α] ∈ Im j N 2 enabled us to match closed self-dual 2-forms α on A with decay of order O(r −2 ) to closed self-dual 2-forms onN with the same decay, and thus effectively interpolate between them to construct our desired self-dual 2-form which is "almost" closed. Since C is stable, the analytic matching condition in Definition 3.4 is always satisfied, meaning in particular that any closed self-dual form on A of order O(r υ ) for υ ∈ (−2, −1] can be matched with a corresponding closed self-dual 2-form onN . Moreover, we have from Theorems 5. We conclude therefore, in the same way as for our approximate kernel in Definition 4.7, that we may define a natural isomorphism between T N M(N ) ⊕ T A M(A) and T X M(X). We may thus identify the product of the natural coordinates on M(N ) and M(A) with the natural coordinates on M(X). With this identification dG| (N,A) becomes the identity map and hence G is a local diffeomorphism.
Thus all compact coassociative 4-folds near X arise via gluing in the stable case. Moreover, Proposition 5.5 suggests that all elements of M(X) "sufficiently close" to M(N ), thought of as lying in the "boundary" of M(X), arise via the desingularization given by Theorem 1.2.
Remark It is also possible to extend our discussion of the stable case to where C is C-stable, as long as one knows that for every deformation C ′ of C in C there is a corresponding deformation A ′ of A which is AC to C ′ .

Examples
We now wish to discuss applications of our theory in examples. We recall that to apply our results we need • a coassociative 4-fold N , in an almost G 2 manifold M , with a conical singularity z modelled on a cone C ∼ = R + × Σ and • a coassociative 4-fold A ⊆ R 7 asymptotically conical with rate λ < − 1 2 to C such that A and N satisfy the matching condition given in Definition 3.4.
A particular criterion for when this matching condition is satisfied is given in Proposition 1.1, namely that the topological matching condition holds (see Definition 3.7) and the cone C is C-stable in the sense of Definition 3.19. Recall that stability is related to the exceptional rates D given in Definition 3.1. Moreover, in the case when λ ≤ −2 the matching condition is equivalent to the topological matching condition.
We now give examples of situations where we can apply our results and begin with a degenerate case.
Example 5.6 Suppose we make the perverse choice that N is smooth and z is any point. Then C = R 4 and Σ ∼ = S 3 so b 1 (Σ) = 0 and the topological matching condition will hold trivially. We can take A = R 4 which is obviously AC with any negative rate, so the matching condition is satisfied. Since b 2 + (A) = 0 and b 2 + (N ) = b 2 + (N ), applying Corollary 1.3 gives that there is a b 2 + (N )-dimensional deformation family of coassociative "smoothings" of N . This corresponds to the fact that our gluing construction will just give back N in this case. matching condition (as λ < −2), but this is trivially satisfied since b 1 (Σ) = 0. Applying Theorem 1.2 gives nothing but the main result in [17] and we have a deformation family of coassociative smoothings of N of dimension b 2 + (A)+b 2 + (N ) by Corollary 1.3.
In fact, if we drop the assumption that b 1 (Σ) = 0, our topological matching condition is still met since λ < −2 means that α 0 A , given in Proposition 3.3, vanishes. Hence Theorem 1.2 applies and clearly extends the work in [17].
Example 5.9 If we assume that b 1 (Σ) = 0 (so −2 ∈ D) and λ ≤ −2, our matching condition is potentially non-trivial and equivalent to the topological matching condition. If we then assume further that the topological criterion holds, we may apply Theorem 1.2 and deduce that we can smooth N using A via gluing. This situation is directly analogous to the material in [10] on desingularization of special Lagrangian conical singularities in what is described as "the obstructed case".
Remark Example 5.9 shows that the work in this article provides an extension of the desingularization theory in the coassociative world which has no current analogue in special Lagrangian geometry, but which should surely follow by adapting the ideas presented here.
Perhaps the best known example of a non-trivial coassociative cone is the Lawson-Osserman SU(2)-invariant cone (see [13] or [20,Example 4.2], for example), originally exhibited because it gives an example of an area-minimizing Lipschitz submanifold which is not smooth. This cone gives a natural model for a coassociative conical singularity. Therefore the topological matching condition is trivially satisfied, so the matching condition is equivalent to the dilation deformation of A extending to an infinitesimal deformation of N , which is obviously essential for the smoothings of N to exist via gluing. The stability of C [20, Corollary 5.8] implies that this always occurs by Proposition 1.1.
Applying Corollary 1.3 gives a family of coassociative smoothings X of N of dimension b 2 + (N ). Notice that the stability of C means that N has a smooth moduli space of deformations as a CS coassociative 4-fold of dimen- Example 5.12 If C is complex, [20,Theorem 6.5] shows that (−2, 0)∩D = {−1} and d D (−1) can be determined by the degree of the holomorphic curve in CP 2 which is the complex link of C. Thus for any AC smoothing of C either λ ≤ −2 or λ = −1. We can therefore apply our theory whenever the matching condition holds, which is non-trivial to check in general. A case of particular interest is discussed in the next example.
Remark The same situation as Example 5.12 holds if the link Σ of C is a tube of radius π 2 in the second normal bundle of a null-torsion pseudoholomorphic curve in S 6 (see [19,Example 6.12] for a description of such Σ).

Example 5.13
In [20, Theorem 1.3] the author constructed CS coassociative 4-folds N in holonomy G 2 manifolds which are diffeomorphic to K3 surfaces, so b 2 + (N ) = 3. The cone C at the singularity is complex with Σ ∼ = RP 3 .
There is a 2-parameter family of AC smoothings A of the cone at the singularity which have rate −1 and b 2 + (A) = 0. Moreover, d D (−1) = 2, corresponding to the choices for A. Thus, since the topological matching condition is trivially satisfied, the matching condition holds if and only if the 2-parameter family of deformations of A extend to infinitesimal deformations of N .
By [20,Corollary 6.11], C is C-stable for some natural choice of family of cones C, so Proposition 1.1 implies that we may apply our theory and Corollary 1.3 gives us a 3-dimensional family of smoothings of N . In fact, since the coassociative 4-folds arise initially in a fibration (see [12] and [20, §7]), we see that this family of smoothings is maximal. Notice also that the C-stability of C implies that N has a smooth moduli space of CS deformations of dimension b 2 + (N ) − d D (−1) = 1. Thus, in the notation of Definition 5.1, dim M(N ) = dim M(X) − 2, which agrees with the fact that singular fibres in the fibration arise in S 1 -families and are thus codimension two in the space of smooth fibres. Moreover, for every deformation of C in C there is a corresponding deformation of A which is AC to the deformed cone. Therefore, as remarked after Proposition 5.5, the gluing map defines a local diffeomorphism G : M(N ) × B(0; τ ) → M(X) where B(0; τ ) ⊆ R 2 . Thus every smooth fibre near X arises via gluing and comes in a 2-parameter family which degenerates to a singular fibre in M(N ).