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Four-manifold invariants from higher-rank bundles

Generalized Donaldson invariants of 4-manifolds are defined, using moduli spaces of anti-self-dual connections with structure group SU(N) or PSU(N). Some values of the invariants are calculated for the case that the 4-manifold arises by the knot-complement construction of Fintushel and Stern. The results are consistent with predictions of Marino and Moore.


Introduction (i) Background
Donaldson's polynomial invariants for a smooth 4-manifold X are defined in [4] using the moduli spaces of anti-self-dual connections on a vector bundle E → X. The structure group of the bundle E is generally taken to be either SU (2) or SO(3), both in [4] and in the subsequent mathematical literature, even though much of the underlying analysis is applicable to bundles with arbitrary compact structure group. In the physics literature, there is the work of Mariño and Moore, who argue in [15] that the SU (2) Donaldson invariants have a well-defined generalization for SU (N ) bundles. They also propose a formula for the SU (N ) invariants when X has simple type, which suggests that these higher-rank invariants contain no new topological information. The purpose of this paper is to give a mathematical definition of the SU (N ) Donaldson invariants, and to calculate some of them in an interesting family of examples arising from knot complements. The results we obtain are consistent with the predictions of [15].
We recall some background material on the Donaldson invariants. The notion of "simple type" was introduce in [14]. For 4-manifolds X with this property, the Donaldson invariants arising from the Lie groups SU (2) and SO (3) were shown to be determined by a rather small amount of data. To state this result, we recall that, in the case that E is an SU (2) bundle with c 2 (E)[X] = k, the corresponding polynomial invariant q k is defined whenever b + 2 (X) − b 1 (X) is odd and b + 2 (X) is at least 2; the invariant is a homogeneous polynomial function is the expected dimension of the instanton moduli space of instanton number k on X. If X has simple type and b 1 (X) = 0, then the structure theorem from [14] establishes that there is a distinguished collection of 2-dimensional cohomology classes K i ∈ H 2 (X; Z) (i = 1, . . . , r) and rational coefficients a i (independent of k) such that q k /(d/2)! is the term of degree d/2 in the Taylor expansion of the function D X : H 2 (X; R) → R given by Here Q is the intersection form on H 2 (X; Z). There is a similar formula for the SO (3) invariants, involving the same "basic classes" K i and coefficients a i , which means that there is no more information in the SO (3) invariants than is already contained in the SU (2) invariants.
Witten's conjecture [23] states that for simply-connected 4-manifolds of simple type, the coefficients a i and basic classes K i are determined by the Seiberg-Witten invariants of X: we should have a i = 2 2+(7χ(X)+11σ(X))/4 SW X (s i ) and K i = c 1 (S + s i ), where the s i are the spin-c structures on X for which the Seiberg-Witten invariant SW X (s i ) is non-zero and S + s i → X is the half-spin bundle corresponding to s i . This conjecture was extended to the case of SU (N ) Donaldson invariants in [15]: the conjectured formula for the SU (N ) invariants is a homogeneous expression of degree N − 1 in the quantities SW X (s i ).

(ii) Summary of results
In this paper we shall give a quite careful construction of the SU (N ) and PSU (N ) Donaldson invariants for 4-manifolds X with b + 2 ≥ 2. As is the case already with N = 2, the PSU (N ) invariants are best defined by choosing a bundle P → X with structure group U (N ) (and possibly non-zero first Chern class) and then studying a moduli space of connections A in the associated adjoint bundle. (Note that not every principal PSU (N ) bundle can be lifted to a U (N ) bundle, so there is a slight loss of generality at this point.) The new difficulties that arise in the higher-rank case are two-fold. First, we can expect singularities in the moduli space corresponding to reducible anti-self-dual connections. We shall bypass this difficulty initially by considering only the case that c 1 (P ) is not divisible by any prime dividing N in the lattice H 2 (X; Z)/torsion. In this case, if b + 2 (X) is non-zero, then for a generic Riemannian metric on X there will be no reducible solutions to the equations, as we explain in section 2(iii). If c 1 (P ) is not coprime to N in this way, we can replace X with the blow-upX = X#CP 2 , and then define the invariants of X using moduli spaces onX. (This is the approach taken in [14] for the SU (2) invariants.) The second difficulty is that for N > 2, one does not know that the irreducible parts of the SU (N ) or PSU (N ) moduli spaces will be smooth for generic choice of Riemannian metrics: we do not have the "generic metrics theorem" of Freed and Uhlenbeck [10], and the anti-self-duality equations must be artificially perturbed to achieve transversality.
Having defined the higher-rank invariants, we shall also calculate them in the case corresponding to a zero-dimensional moduli space on a particular family of 4-manifolds. Let K be a knot in S 3 , let M be the knot complement (a compact 3-manifold with torus boundary), let X be a K3 surface with an elliptic fibration, and let T ⊂ X be a torus arising as one of the fibers. The following construction is considered by Fintushel and Stern in [8]. Let X o denote the 4-manifold with boundary T 3 obtained by removing an open tubular neighborhood of T , and let X K be the closed 4-manifold formed as where φ : S 1 ×∂M → ∂X o is an identification of the two 3-torus boundaries, chosen so that φ maps (point) × (longitude) to a curve on ∂X o ⊂ X which is the boundary of a 2-disk transverse to T . It is shown in [8] that X K has the same homotopy type as X, but that the Seiberg-Witten invariants of X K encode the Alexander polynomial of K. If Witten's conjecture holds, then the function D that determines the SU (2) Donaldson invariants (see (1)) is given by where F is the cohomology class Poincaré dual to the torus fibers and ∆ K (t) = n k=−n a k t k is the symmetrized Alexander polynomial of the knot K.
We shall consider a particular U (N ) bundle P → X K with (c 1 (P ) · F ) = 1 and c 2 (P ) chosen so that the corresponding moduli space is zerodimensional. The zero-dimensional moduli space gives us an integer-valued invariant of X K (the signed count its points), which we are able to calculate when N is odd. The answer we obtain is the integer as long as this quantity is non-zero. It seems likely that this answer is correct also when N is even, up to an overall sign (the answer is consistent with Witten's conjecture when N = 2 for example); but we have not overcome an additional difficulty that arises in the even case: see Lemma 34 and the remarks at the end of section 6(iii).
Acknowledgement. This paper owes a lot to the author's continuing collaboration with Tom Mrowka. In particular, the discussion of holonomy perturbations in section 3 is based in part on earlier, unpublished joint work.

Gauge theory for SU (N )
(i) The configuration space Let X be a smooth, compact, connected, oriented 4-manifold. We consider the Lie group U (N ) and its adjoint group, PU (N ) or PSU (N ). Let P → X be a smooth principal U (N )-bundle, and let ad(P ) denote the associated principal bundle with structure group the adjoint group. Fix an integer l ≥ 3, and let A denote the space of all connections in ad(P ) of Sobolev class L 2 l . The assumption l ≥ 3 ensures that L 2 l ⊂ C 0 : we will use this assumption in Lemma 13, although at most other points l ≥ 2 would suffice. If we fix a particular connection A 0 , we can describe A as where su P is the vector bundle with fiber su(N ) associated to the adjoint representation. The gauge group G will be the group of all automorphisms of P of Sobolev class L 2 l+1 which have determinant 1. That is, G consists of all L 2 l+1 sections of the fiber bundle SU (P ) → X with fiber SU (N ), associated to the adjoint action of PSU (N ) on SU (N ). The group G acts on A, but the action is not effective. The subgroup that acts trivially is a cyclic group Z ⊂ G that we can identify with the center of SU (N ). Definition 1. For a connection A ∈ A, we write Γ A for the stabilizer of A in G. We say that A is irreducible if Γ A = Z.
We write A * ⊂ A for the irreducible connections. We write B and B * for the quotients of A and A * by the action of G. The space B * has the structure of a Banach manifold in the usual way, with coordinate charts being provided by slices to the orbits of the action of G. We write [A] ∈ B for the gauge-equivalence class represented by a connection A.
A U (N )-bundle P on X is determined up to isomorphism by its first and second Chern classes. We shall write w → X for the line bundle associated to P by the 1-dimensional representation g → det(g) of U (N ), so that c 1 (w) = c 1 (P ). We set κ = c 2 (P ) − ((N − 1)/(2N ))c 1 (P ) 2 , [X] The above expression is constructed so that κ depends only on ad(P ), and so that, in the case that X is S 4 , the possible values of κ run through all integers as P runs through all isomorphism classes of U (N )-bundles. We may label the configuration spaces B etc. using w and κ to specify the isomorphism class of the bundle. So we have B w, * κ ⊂ B w κ and so on. We refer to κ as the instanton number, even though it is not an integer.
There is a slightly different viewpoint that one can take to describe A. Fix a connection θ in the line bundle w. Then for each A in A, there is a unique U (N )-connectionÃ in the U (N )-bundle P such that the connections which A induces in ad(P ) and w are A and θ respectively. Thus we can identify A with the spaceÃ of U (N )-connections inducing the connection θ in w.
This viewpoint is helpful in understanding the stabilizer Γ A . Pick a basepoint x 0 ∈ X, and let A be a connection in ad(P ). An element g in Γ A is completely determined by its restriction to the fiber over x 0 , so we can regard Γ A as a subgroup of the copy of a copy of SU (N ), namely SU (P ) x 0 . LetÃ be the U (N )-connection in P → X, corresponding to A and θ as above, and letH be the group of automorphisms of P x 0 generated by the holonomy ofÃ around all loops based at x 0 . Then Γ A is the centralizer of H in SU (P ) x 0 . The structure of Γ A can then be described as follows. For any isomorphism of inner product spaces ψ : we get a representation ρ : for some such ψ. In particular, we make the important observation that Γ A has positive dimension unless r = 1 and n 1 = 1: that is to say, Γ A has positive dimension unless A is irreducible.
(ii) The moduli space Let M w κ ⊂ B w κ be the space of gauge-equivalences classes of anti-self-dual connections: → Ω + (X; su P ).
We denote by H 0 A , H 1 A and H 2 A the cohomology groups of this elliptic complex. If A is not smooth but only of class L 2 l , we can define the groups H i A in the same way after replacing the spaces of smooth forms above by Sobolev completions.
Proof. In general, H 0 A is the Lie algebra of Γ A . We have already observed that Γ A has positive dimension if A is reducible.
We say that A is regular if H 2 A vanishes. If A is regular and irreducible, then M w κ is a smooth manifold in a neighborhood of [A], with tangent space isomorphic to H 1 A . The dimension of the moduli space near this point is then given by minus the index of the complex above, which is We say that the moduli space is regular if A is regular for all [A] in M w κ .
Uhlenbeck's theorem allows us to compactify the moduli space M w κ in the usual way. We define an ideal anti-self-dual connection of instanton number κ to be a pair ([A], x), where A is an anti-self-dual connection belonging to M 2 κ−m , and x is a point in the symmetric product X m /S m . The space of ideal anti-self-dual connections is compactified in the usual way, and we define the Uhlenbeck compactification of M w κ to be the closure of the M w κ in the space of ideal connections:

(iii) Avoiding reducible solutions
We now turn to finding conditions which will ensure that the moduli space contains no reducible solutions. We shall say that a class c ∈ H 2 (X; R) is integral if it in the image of H 2 (X; Z) (or equivalently if it has integer pairing with every class in H 2 (X; Z)). We shall use H − to denote the space of real anti-self-dual harmonic 2-forms, which we may regard as a linear subspace of H 2 (X; R).
Proposition 3. If M w κ contains a reducible solution, then there is an integer n with 0 < n < N and an integral class c ∈ H 2 (X; R) such that Proof. Suppose that [A] is a reducible solution in M w κ , and letÃ be the corresponding U (N )-connection in the bundle P → X, inducing the connection θ in w. The connectionÃ respects a reduction of P to U (n 1 ) × U (n 2 ), with n 1 + n 2 = N : we have where P i is a bundle with structure group U (n i ), and n i > 0. LetÃ 1 and A 2 be the connections obtained fromÃ in the bundles P 1 and P 2 . Let ι : u(1) → u(N ) be the inclusion that is the derivative of the inclusion of U (1) as the center of U (N ). BecauseÃ induces the connection θ on w and A is anti-self-dual, we have where ι 1 : u(1) → u(n 1 ) is the inclusion again. Let w 1 be the determinant line bundle of P 1 , and θ 1 the connection that it obtains fromÃ 1 . From the equality above, we obtain Since F θ and F θ 1 represent −2πic 1 (w) and −2πic 1 (w 1 ) respectively, we have proved the proposition: for the class c in the statement, we take c 1 (w 1 ), and for n we take n 1 .
Definition 4. We will say that c 1 (w) is coprime to N if there is no n with 0 < n < N such that (n/N )c 1 (w) is an integral class. This is equivalent to saying that there is a class in H 2 (X; Z) whose pairing with c 1 (w) is coprime to N .
As corollaries to the proposition, we have: Corollary 5. Let w be a fixed line bundle on X, and suppose c 1 (w) is coprime to N . Let Met denote the set of all C r metrics on X, and let Ξ ⊂ Met denote the set of metrics for which there exists a κ such that M w κ contains reducible solutions. Then Ξ is contained in a countable union of smooth submanifolds of Met, each of which has codimension b + 2 in Met. Proof. If the coprime condition holds, then for each integral class c and each n less than N , the class c − (n/N )c 1 (w) is non-zero. The set of metrics g such that is therefore a smooth submanifold Met c,n ⊂ Met, of codimension b + 2 . (See [6], section 4.3, for example.) The set Ξ is contained in the union of these submanifolds Met c,n as c runs through the integral classes and n runs through the integers from 1 to N − 1.
Corollary 6. If c 1 (w) is coprime to N and b + 2 is non-zero, then for a residual set of Riemannian metrics g on X, the corresponding moduli spaces M w κ contain no reducible solutions.

Perturbing the equations
When N = 2, it is known that for a residual set of Riemannian metrics, the irreducible solutions in the corresponding moduli spaces M w κ are all regular. This is the generic metrics theorem of Freed and Uhlenbeck [10]. Unfortunately, no such result is known for larger N , and to achieve regularity we must perturb the anti-self-duality equations. The holonomy perturbations which we use follow a plan which appears also in [3], [9] and [22] among other places. A discussion of bubbles in the presence of such perturbations occurs in [5].

(i) A Banach space of perturbations
Let X be a Riemannian 4-manifold as before, let B be an embedded ball in X, and let q : S 1 × B → X be a smooth map with the following two properties: (a) the map q is a submersion; be a smooth complex-valued self-dual 2-form whose support is contained in B. For each x in B, let q x : S 1 → X be the map given by q x (z) = q(z, x). Given a smooth connection A in A, letÃ be the corresponding connection in P → X with determinant θ, and let Hol qx (Ã) denote the holonomy ofÃ around q x . As x varies over B, the holonomy defines a section of the bundle of unitary groups U (P ) over B.
We regard U (P ) as a subset of the vector bundle gl P , with fiber gl(N ). We can then multiply the holonomy by the 2-form ω, and extend the resulting section by zero to define a section on all of X: ω ⊗ Hol q (Ã) ∈ Ω + (X; gl P ).
We apply the orthogonal bundle projection π : gl P → su P to this section, and finally obtain a smooth section Recall that A is defined as the L 2 l completion of the space of smooth connections.
Proposition 7. For fixed q and ω, the map V q,ω extends to a smooth map of Banach manifolds, Furthermore, if we fix a reference connection A 0 , then there are constants K n , depending only on q and A 0 , such that the n-th derivative Proof. For smooth connections A and A 0 , consider first the pull-backs q * (A) and q * (A 0 ) on S 1 × B. Because q is a submersion, the pull-back map is continuous in the L 2 l topology: there is a constant c depending on q and A 0 , such that In other words, the pull-back map q * extends continuously to A. We wish to regard q * (A) as providing a family of connections on S 1 , parametrized by B. To this end, for each x ∈ B, let H x denote the Hilbert space of square-integrable su P -valued 1-forms on S 1 × {x}: These form a Hilbert bundle H → B, and q * (A 0 ) supplies H with a connection, A H 0 . For A in A, we have just observed that the 1-form q * (A) − q * (A 0 ) lies in L 2 l (S 1 × B); and we therefore obtain from it, by restriction, a section of the bundle H of class L 2 l : (This is just the statement that an element of L 2 l (X 1 × X 2 ) defines an L 2 l map from X 1 to L 2 (X 2 ).) Furthermore, we have an inequality . We can apply these constructions also to a unitary connectionÃ with determinant θ, to obtainã in a similar way.
Lemma 8. Let A 0 be a smooth connection in a bundle P → S 1 , and write a general connection as A = A 0 + a. Identify the structure group U (N ) with the group of automorphisms of the fiber of P over 1 ∈ S 1 . For each a, let Hol(a) ∈ U (N ) denote the holonomy of A 0 + a around the loop S 1 based at 1. Then Hol extends to a smooth map Furthermore, there are constants c n , independent of A 0 and A, such that the n-th derivative D n Hol| A satisfies Proof. The connection A 0 does not appear in the inequality and serves only to define the domain of Hol. As a statement about A and the a i , the inequality to be proved is gauge-invariant. It will suffice to show that, for each A, there is a connection c n (A) such that the inequality holds for all a i : the fact that c n can eventually be taken to be independent of A will follows automatically, because the space of connections modulo gauge is compact in the case of S 1 . So we are only left with the first assertion: that Hol extends to a smooth map on L 2 . This is proved in [22,7]. (In fact, L 1 suffices.) To return to the proof of Proposition 7, we apply the lemma to the family of circles parametrized by B. For each x in B, we regard the automorphism group of P x as subset of (gl P ) x , and have the holonomy map We regard this as giving us a smooth bundle map This map is nonlinear on the fibers; but the uniform bounds on derivatives of all orders mean that it still gives a map This Hol satisfies uniform bounds its derivatives also. (For example, if E and F are Hilbert spaces and h : E → F is a uniformly Lipschitz map, then h defines a uniformly Lipschitz map L 2 (B; E) → L 2 (B; F ).) The section V q,ω (A) is obtained by applying the map (5) to the section a from (4), then multiplying by the C l 2-form ω and applying a linear projection in the bundle. Now fix once and for all a countable collection of balls B α in X and maps q α : S 1 × B α → X, α ∈ N, each satisfying the two conditions laid out above. Let K n,α be constants corresponding to q α , as in the proposition above. Let C α be a sequence of positive real numbers, defined by a diagonalization, so that For each α, let ω α be a self-dual complex-valued form with support in B α , and suppose that the sum α C α ω α C l is convergent. Then for each n, the series α V qα,ωα (6) converges uniformly in the C n topology on bounded subsets of A. That is, then the series (6) converges in the uniform C n topology of maps B R (A) → L 2 l,A 0 (X : Λ + ⊗su P ). Thus the sum of the series defines a C ∞ map of Banach manifolds, V : A → L 2 l (X; Λ + ⊗ su P ).
Definition 9. Fix maps q α and constants C α as above. We define W to be the Banach space consisting of all sequences ω = (ω α ) α∈N such that the sum α C α ω α C l converges. For each ω ∈ W , we write V ω for the sum of the series α V qα,ωα , which is a smooth map The dependence of V ω on ω is linear, and the map (ω, A) → V ω (A) is a smooth map of Banach manifolds, W × A → L 2 l (X; Λ + ⊗ su P ). Given ω in W , we define the perturbed anti-self-duality equation to be the equation for A ∈ A. Note that the left-hand side lies in L 2 l−1 (X; Λ + ⊗ su P ). The equation is gauge-invariant, and we define the perturbed moduli space to be the quotient of the set of solutions by the gauge group: (7) holds }.

(ii) Regularity and compactness for the perturbed equations
If A is a solution of the perturbed equation (7), we can consider the linearization of the equation at A, as a map The derivative of the perturbation V ω defines a bounded operator from L 2 l to L 2 l ; so in the topologies of (8), the derivative of V ω is a compact operator. When we restrict the perturbed equations to the Coulomb slice through A, we therefore have a smooth Fredholm map, just as in the unperturbed case, and we have a Kuranishi theory for the perturbed moduli space. We write the Kuranishi complex at a solution A as where d + A,ω is the linearization of the perturbed equation, We denote the homology groups of this complex by H 0 A , H 1 A,ω and H 2 A,ω . If A is irreducible and H 2 A,ω is zero, then the moduli space M w κ,ω is smooth near [A], with tangent space H 1 A,ω . Understanding the Uhlenbeck compactness theorem for the perturbed equations is more delicate than the Fredholm theory: if the curvature of a sequence of connections A n concentrates at a point, then effect is seen in V ω (A n ) throughout the manifold, because the perturbations are non-local. To obtain a compactness result, we must work with weaker norms than L 2 l . To this end, we start by observing that Proposition 7 continues to hold in the L p k norms, for any p ≥ 1 and k ≥ 0. That is to say, if we let A p k denote the L p k completion of the space of smooth connections, the V q,ω defines a smooth map V q,ω : A p k → L p k (X; Λ + ⊗ su P ), and the derivatives of this map are uniformly bounded in terms of constants which depend only on q and the C k norm of ω. As long as the constants C α are chosen appropriately, we can define the Banach space of perturbations W as before, and for ω ∈ W we have again a smooth map V ω as long as k ≤ l. By a diagonalization argument, we may suppose if we wish that the constants C α are chosen so that V ω is smooth for all p and all k ≤ l.
We can also note that the constants C α are inevitably bounded below, and this implies that V ω (A) satisfies a uniform L ∞ bound, independent of A.
Consider now a sequence [A n ] in M w κ,ω . As just observed, the equations imply that the self-dual curvatures F + An are uniformly bounded in L ∞ , and F An is then uniformly bounded in L 2 (X) because of the Chern-Weil formula. After passing to a subsequence, we may therefore assume as usual that there is a finite set of points x and a cover of X \ x by metric balls Ω i , such that for all n and i. Here ǫ may be specified in advance to be smaller than the constant that appears in Uhlenbeck's gauge-fixing theorem. We may therefore put A n in Coulomb gauge relative to a trivial connection on the ball Ω i . For all p ≥ 2, we have a bound on F + An in L p , and the connection form A n in Coulomb gauge on Ω i is therefore bounded in L p 1 by a constant depending only on i.
By patching gauge transformations, we arrive at the following situation. There are gauge transformations g n on X of class L 2 l+1,loc , such that the sequence g n (A n ) is bounded in L p 1 (K) for any compact subset K in X \ x. We rename g n (A n ) as A n .
Lemma 10. Under the assumptions above, the sequence V ω (A n ) has a subsequence that is Cauchy in L p (X).
Proof. For r > 0, let K r ⊂ X \ x be the complement of the open metric balls around of radius r around the points of x. The uniform bound on the L p 1 norm of A n on K r means that we can pass to a subsequence which converges in L p (K r ). By diagonalization, we can arrange the subsequence so that convergence occurs in L p (K r ) for all r > 0. Rename this subsequence as A n . We claim that V ω (A n ) is Cauchy in L p (X). To see this, let ǫ > 0 be given. For any β ∈ N, let V ω(β) be the partial sum and let Z r = ∪ α≤β Z r,α . The volume of Z r goes to zero as r goes to zero; and because we have a uniform L ∞ bound on V ω (A), we can find r 0 sufficiently small that So we can find n 0 such that, for all n 1 , n 2 greater than n 0 , we have Adding up the inequalities, we have for all n 1 , n 2 greater than n 0 .
Bootstrapping and the removability of singularities now leads us to the following version of Uhlenbeck's theorem for our situation.
Proposition 11. Let A n be a sequence of connections in P → X representing points [A n ] in the moduli space M w ω,κ . Then there is a point x in X m /S m and a connection A ′ in a bundle P ′ → X representing a point of a moduli space M w ω,κ−m with the following property. After passing to a subsequence, there are isomorphisms The proof of the lemma above shows that (h n ) * (V ω (A n )), extended by zero to all of X, is Cauchy in L p (X) and has V ω (A ′ ) as its limit. It follows that A ′ solves the perturbed equations.
Note that, in an Uhlenbeck limit involving bubbles, we cannot expect anything better than L p convergence of the curvatures on compact subsets disjoint x, nor anything better than L p 1 convergence of the connection forms. This is in contrast to the unperturbed case, where the convergence can be taken to be C ∞ .

(iii) Transversality
Let A * again denote the irreducible connections of class L 2 l . Let W be the Banach space parametrizing the perturbations ω, and consider the perturbed equations (7) for an irreducible connection A and perturbation ω as the vanishing of a map given by . We want this map to be transverse to zero. To achieve this, we need the family of balls B α and maps q α to be sufficiently large: Condition 12. We require the balls B α and the submersions q α : S 1 ×B α → X to satisfy the following additional condition: for every x in X, the maps Lemma 13. Suppose that the balls B α and maps q α are chosen to satisfy the condition above. Then the map F is transverse to zero.
. We shall consider the restriction of L to the Coulomb slice: we write where K l is the Coulomb slice at A in the L 2 l topology. It will suffice to prove that L ′ 1 is surjective, because the regularity for the operator d + A allows us to use a bootstrapping argument to show that, if L ′ 1 (ν, a) belongs to L 2 l−1 , then a belongs to L 2 l . Our Fredholm theory already tells us that the cokernel of L ′ 1 has finite dimension. So we need only show the image of L ′ 1 is dense in L 2 . We shall show that, for a given irreducible A, the map ν → V ν (A) has dense image in L 2 .
In order to use the hypothesis that A is irreducible, we recall that if ρ : H → U (N ) is a unitary representation of a group H, then the hypothesis of irreducibility of H means that the complex span of the linear transformations So if x is any point of X, we an find loops γ i based at x such that the holonomies Hol γ i (A), regarded as elements of (gl P ) x , are a spanning set.
Because of Condition 12, we can therefore achieve the following. Given any x in X, we can find an r > 0 and N 2 elements α i (i = 1, . . . , N 2 ), such that B r 0 (x) ⊂ B α i for all i and such that the holonomies Hol qα i ,x (Ã) span (gl P ) x . Decreasing r if necessary, we may assume further that the sections e i = Hol qα i (Ã) of the bundle gl P are a basis for the fiber at every point of the ball B r (x). Here we have used the continuity of these sections, which follows from our assumption that L 2 l is contained in C 0 . If ω i are any complex-valued self-dual forms of class C l supported in B r (x), then the section π(ω i ⊗ e i ) of λ + ⊗ su P can be realized as V ν (A), by taking ν α i = ω i . (Here π is again the projection gl P → su P .) As the ω i vary, we obtain a dense subset of the continuous sections supported in the ball B r (x).
Because x is arbitrary and X is compact, this is enough to show that the sections V ν (A) are dense in C 0 as ν runs through W . They are therefore dense in L 2 also. Corollary 14. Suppose that Condition 12 holds. Then there is a residual subset of W such that for all ω in this set, and all w and κ, the irreducible part of the moduli space M w κ,ω is regular, and therefore a smooth manifold of dimension d given by (3).
Proof. As usual, the zero set of F is a C ∞ Banach manifold by the implicit function theorem. The projection of F −1 (0) to W is a smooth Fredholm map of index d between separable Banach manifolds, and the fiber over ω is the irreducible part of M w κ,ω ; so the result follows from the Sard-Smale theorem.

(iv) Reducible solutions of the perturbed equations
When b + 2 is non-zero, we would like to arrange that M w κ,ω contains no reducible solutions. For the unperturbed case, the relevant statement is Corollary 6 above; but the discussion that led to this corollary breaks down when the equations are perturbed. Using the compactness theorem however, we can obtain a result when the perturbation ω is small enough.
Proposition 15. Let κ 0 be given, and let g be a metric on X with the property that the unperturbed moduli spaces M w κ contain no reducible solutions for any κ ≤ κ 0 . Then there exists ǫ > 0 such that for all ω ∈ W with ω W ≤ ǫ, the perturbed moduli spaces M w κ,ω also contain no reducibles.
Proof. Suppose the contrary, and let [A n ] be a reducible connection in M w κ,ωn for some sequence ω n in W with ω n W → 0 as n → ∞. The proof of Uhlenbeck compactness for the perturbed equations extends to this case, so there is a weak limit ( The condition of being reducible is a closed condition, so A is reducible and we have a contradiction.
If we combine this proposition with the transversality results, we obtain the following.
Corollary 16. Suppose that b + 2 is non-zero and c 1 (P ) is coprime to N . Then given any κ 0 , we can find a Riemannian metric g and perturbation ω such that the moduli spaces M w κ,ω on (X, g) are regular for all κ ≤ κ 0 and contain no reducibles.

Orienting the moduli spaces (i) Conventions defining orientations
On the irreducible part of the configuration space B w κ , there is a line bundle Λ, obtained as the quotient by G of the line bundle on A given by the determinant of the family of operators acting on su P -valued forms. A trivialization of this line bundle provides an orientation of the moduli space M w κ,ω at all points where the moduli space is regular. The space of perturbations ω is contractible, so in studying trivializations of Λ, we may as well take ω = 0.
The first result is that Λ is indeed trivial: a proof for U (N ) bundles is given in [6, §5.4.3]. Our moduli spaces are therefore orientable manifolds. The trickier task is to isolate what choices need to be made to in order to specify a particular orientation, and to determine how the orientation depends on the choices made. For this task, we follow [3].
First, we orient the Lie algebra su(N ). Let s be the subalgebra of sl(N ) consisting of the traceless upper triangular matrices whose diagonal entries are imaginary. The orthogonal projection s → su(N ) is a linear isomorphism, so we can orient su(N ) by specifying an orientation of s. We write s as the direct sum of the diagonal algebra h and a complex vector space, the strictly upper triangular matrices. We take the complex orientation on the latter; so an orientation of h now determines an orientation of su(N ). To orient h, we write δ n for the matrix with a single 1 at the n-th spot on the diagonal, and we take the ordered basis i(δ n − δ n+1 ), n = 1, . . . , N − 1, to be an oriented basis of h.
Next we choose a homology orientation for the 4-manifold X: that is, we choose an orientation o X of the determinant line of the operator In the case that P → X is trivial and A is the trivial connection, the operator D A is the obtained from D by tensor product with the vector space su(N ).
In general, if V and W are oriented vector spaces, we orient a tensor product V ⊗ W by choosing oriented ordered bases (v 1 , . . . , v d ) and (w 1 , . . . , w e ) and taking as an oriented basis for the tensor product the basis If the dimension of W is even, this orientation of V ⊗ W is independent of the chosen orientation of V . Note also that if W is complex and has the complex orientation, then this convention equips V ⊗ W with its complex orientation also. Using this convention, our homology orientation o X and our preferred orientation of su(N ) together determine an orientation of the determinant line of D A at the trivial connection. This trivializes the line bundle Λ in the case of the trivial bundle P . Note that if N is odd, then the dimension of su(N ) is even; and in this case the orientation of Λ is independent of the choice of homology orientation o X . Moving away from the trivial bundle, we now consider a hermitian rank-N vector bundle E → X decomposed as and we take P to be the corresponding principal U (N ) bundle. In line with our use of the upper triangular matrices s, we can decompose the associated bundle su P as the sum of a trivial bundle with fiber h and a complex vector bundle j>i Hom(L j , L i ).
Using our preferred orientation of h and a chosen homology orientation o X , we obtain in this way an orientation for the determinant line Λ → B P . In the case that all the L n are trivial, this reproduces the same orientation as we considered in the previous paragraph. Again, this orientation is independent of o X when N is odd. As a special case, given a line bundle w → X, we define O(w, o X ) to be the orientation just defined, in the case that L 1 = w and L n is trivial for n > 1. The corresponding bundle P has κ = κ 0 given by κ 0 = −((N − 1)/(2N ))c 2 1 (w). In this way, for each w, we orient the determinant line over B w κ 0 . For different values of κ, the determinant lines over B w κ and B w κ ′ can be compared by the "addition of instantons", as in [3]. On S 4 , there is a standard SU (N ) instanton with κ = 1, obtained from the standard SU (2) instanton by the standard inclusion of SU (2) in SU (N ); using a concentrated instanton and excision, an orientation of the determinant line over B w κ determines an orientation over B w κ+1 . A precise convention can be fixed by requiring that this determination respects the complex orientations in the case of a Kähler manifold (see below). In this way, the orientation O(w, o X ) over B w κ 0 determines an orientation over B w κ for all κ. To summarize, given a homology orientation o X for the 4-manifold, we have fixed standard orientations for the smooth parts of all the moduli spaces M w κ . In the case that N is odd, the choice of o X is immaterial.

(ii) Comparing orientations
Suppose E is a hermitian rank-N bundle and E ′ = E ⊗ v is another, where v is a complex line bundle. The top exterior powers are related by w ′ = w+N v. (We use additive notation here for a tensor product of line bundles.) The configuration spaces B P and B P ′ for the corresponding principal bundles P and P ′ are canonically identified, as are the moduli spaces, because the adjoint bundles ad(P ) and ad(P ′ ) are the same. While this provides an identification of two moduli spaces M w κ and M w+N v κ , it is not necessarily the case that this identification respects the orientations of their smooth parts given by O(w, o X ) and O(w + N v, o X ) as in the previous subsection.
We follow the proof of [3, Proposition 3.25] closely to compare the two. The main step is to consider the case that X is Kähler, and compare the orientation O(w, o X ) to the complex orientation, where o X is the preferred homology orientation of the Kähler manifold defined in [3]. For a bundle E decomposed as (11), the operator decomposes as a sum of operators (−∂ * ⊕∂) acting on bundles , We treat these three summands in turn. In each case, the index of the operator has a complex orientation (coming from the complex structure of Ω 0,i ) which we wish to compare to our preferred orientation. In the first summand, one comparison that arises is between the complex orientation of V ⊗ R h and the tensor product orientation of V ⊗ C h defined in the previous subsection, when V is a complex vector space. These two differ by the sign irrespective of our convention for orienting h. This consideration contributes a sign to our comparison. As in [3], there is a further consideration in the first term, in that the preferred homology orientation of X uses the opposite orientation to that given by the complex structure of (−∂ * ⊕∂), and this contributes another (ind(−∂ * ⊕∂))(N − 1). Altogether, the first summand contributes a sign In the second summand, our convention for defining O(L 1 , . . . , L N , o X ) uses the complex structure of Hom(L j , L i ), and this is the same complex structure on the index as arises from the complex structure of Ω 0,i . So there is no correction here. In the third summand, our complex structure is opposite to the one inherited from Ω 0,i . We therefore get a correction We now specialize to the case that L 1 = w and the other L n are trivial, which is the case that we used to define O(w, o X ) above. The bundles Hom(L i , L j ) with j > i contribute N − 1 copies ofw and (N − 2)(N − 1)/2 trivial bundles. The formula (13) above (the contribution of the third summand) thus simplifies to Now we add the contribution (12) from the first summand, and we obtain where K is the canonical class of X. Thus we have proved: Proposition 17. If X is Kähler and o X is the preferred homology orientation of the Kähler manifold, then the distinguished orientation O(w, o X ) of the moduli space M w κ coincides with its complex orientation if N is odd. If N is even, then the two orientations compare by the sign (−1) (w·w+K·w)/2 . coincide. We consider the case that N is even. If we set Thus we have: are the same when N is odd or when N is zero mod 4. When N is 2 mod 4, they compare with the sign (−1) v·v .
Proof. We have just dealt with the case that X is Kähler. The general case is reduced to this one by excision, as in [3].

(iii) Dual bundles
The map δ : U (N ) → U (N ) given by g →ḡ allows us to associate to our principal bundle P a new principal bundleP . The corresponding vector bundles E andĒ obtained from the defining representation of U (N ) are dual. To each connection A in the adjoint bundle su P , we obtain a connection δ(A) in suP ; and this gives us a map In the case N = 2, this is the same as one obtains by tensoring E withw; but for other N it is different.
Let the smooth parts of M w κ and M −w κ be given the orienta- where o X is a chosen homology-orientation for X. Then the dualizing map δ respects these orientations in the case that N is odd. In the case that N is even, the map δ preserves or reverses orientation according to the sign (−1) w·w .
Proof. In the case that X is Kähler, the map δ respects the complex orientations of the moduli spaces, for it is a complex-analytic map between them. Using Proposition 17 and excision, we can again compare the orientations O(w, o X ) and O(−w, o X) ).

Integer invariants and product formulae
In this section, we use the gauge theory and perturbations that we have developed for PSU (N ) to define simple integer invariants of 4-manifolds. As a preliminary step for our later calculations, we also prove a product formula for such invariants, for a 4-manifold X decomposed along a 3-torus.

(i) Simple integer invariants
Let X be a smooth, compact oriented 4-manifold, with b + 2 (X) ≥ 2. Fix N and let P → X be a principal U (N ) bundle. Let w be the U (1) bundle det(P ), and let κ be the instanton number. If N is even, choose also a homology orientation o X for X. We impose the following simplifying condition.
Hypothesis 20. We shall suppose that c 1 (w) is coprime to N in the sense of Definition 4. We shall also suppose that the formal dimension d of the moduli space M w κ , as given by the formula (3), is zero.
By Corollary 6, we can choose a Riemannian metric g on X so that the moduli spaces M w κ ′ contain no reducibles solutions. More specifically, we want to choose g so that there is no integer class c and no n < N such that c − (n/N )c 1 (w) is represented by an anti-self-dual form. Having chosen such a g, we can find ǫ(g) as in Proposition 15 such that for all ω ∈ W with ω W ≤ ǫ(g), the perturbed moduli space M w κ ′ contains no reducibles for any κ ′ ≤ κ. Finally, we use Corollary 6 to find a ω with ω W < ǫ(g) such that the moduli spaces M w κ ′ are smooth for all κ ′ ≤ κ. We call any (g, ω) arrived at in this way a good pair.
Let (g, ω) be a good pair. The moduli space M w κ,ω is a smooth 0-manifold, and is compact because the moduli spaces M w κ−m,ω are empty, being of negative dimension. So M w κ,ω is a finite set of points. Using O(w, o X ), we can orient this moduli space: the points of the moduli space then acquire signs; and in the usual way we can count the points, with signs, to obtain an integer.
The integer which we obtain in this way is independent of the choice of good pair (g, ω), by the usual cobordism argument. In a little more detail, suppose (g 0 , ω 0 ) and (g 1 , ω 1 ) are good pairs. Because b + 2 is greater than 1, Corollary 5 allows us to choose a smooth path of metrics g t joining g 0 to g 1 , such that the moduli spaces M w κ (X, g t ) contain no reducibles. We can then find a continuous function epsilon(t) such that M w κ,ωt (X, g t ) contains no reducibles whenever ω t Wt ≤ ǫ(t). Here W t is the Banach space of perturbations on (X, g t ): we can take it that the maps q α and balls B α used in its definition are independent of t. Let W → [0, 1] be the fiber bundle with fiber the perturbation space W t . Over [0, 1], the maps F of (9) define a total map F : where L is the fiber bundle over [0, 1] with fiber L 2 l−1 (X; Λ + t ⊗ su P ). We can choose a section ω t of the W over [0, 1], transverse to F, and always smaller than ǫ(t). The family of moduli spaces M w κ,ωt (X, g t ) will then sweep out an oriented 1-dimensional cobordism between M w κ,ω 0 (X, g 0 ) and M w κ,ω 1 (X, g 1 ). We summarize our discussion.
Definition 21. Let X be a closed, oriented smooth 4-manifold with b + 2 (X) ≥ 2, equipped with a homology orientation o X . Let w be a line bundle with c 1 (w) coprime to N , and suppose there exists a U (N ) bundle P , with det(P ) = w, such that the corresponding moduli space M w κ has formal dimension 0. Then we define an integer q w (X) as the signed count of the points in the moduli space M w κ,ω (X, g), where (g, ω) is any choice of good pair. This integer depends only on X, w and o X up to diffeomorphism; and if N is odd it is independent of o X .
We extend the definition of q w by declaring it to be zero if there is no P for which the corresponding moduli space is zero-dimensional.
For given w, there may be no corresponding P for which the moduli space M w κ has formal dimension zero. Referring to the formula (3) for d and the definition of κ, we see that a necessary and sufficient condition is that the integer should be divisible by 4N . In the case that N is even, this implies in particular that b + 2 − b 1 should be odd, and that w · w has the same parity as (b + 2 − b 1 + 1)/2. From Proposition 19 we therefore derive: if N is odd. If N is even, then q w (X) and q −w (X) differ by the sign

(ii) Cylindrical ends
Let X be an oriented 4-manifold with boundary a connected 3-manifold Y , and let P → X be a principal U (N ) bundle. Let R w (Y ) denote the representation variety of Y : the quotient of the space of flat connections in su P → Y by the action of the group of determinant-1 gauge transformations of P | Y . We make the following simplifying assumption, in order to extend the definition of the integer invariant q w in a straightforward way to such manifolds with boundary.
Hypothesis 23. We shall suppose that R w (Y ) consists of a single point α = [A Y ]. We shall further suppose that α is irreducible, and non-degenerate, in the sense that the cohomology group H 1 (Y ; α) with coefficients in the flat bundle (su P , α) is trivial.
Once a particular representative connection A Y on su P | Y is chosen, we can extend A Y to all of su P to obtain a connection A X on X. But there is an essential topological choice in this process. The determinant-1 gauge group G(Y ) is not connected: its group of components is infinite cyclic; and the gauge transformations on Y that extend to X are precisely the identity component of G(Y ).
LetB Y denote the quotient of the space of connections A Y by the identity component G 1 (Y ) ⊂ G(Y ). LetR w (Y ) denote the subset ofB Y consisting of flat connections: this is an infinite set acted on transitively by the infinite cyclic group G(Y )/G 1 (Y ). Letα be a choice of a point inR w Y . Let A Y be a connection representingα, and let A X be any extension of A Y to all of X.
Up to this point, we have made no essential use of a metric on X. We now choose a Riemannian metric on X that is cylindrical in a collar of the boundary. We also suppose that A X is flat in the collar. Form a cylindricalend manifold X + by attaching a cylinder [0, ∞) × Y to X, and extend A X as a flat connection on the cylindrical part. We now introduce a space of connections on X + , and the quotient space B(X + ;α). The gauge group acts freely, and the quotient space is a Banach manifold. Inside this Banach manifold is the moduli space of anti-self-dual connections: We have the usual results concerning moduli spaces on cylindrical-end manifolds. Any finite-energy anti-self-dual connection A in the bundle P → X + is gauge-equivalent to a connection in M w (X + ;α), for some uniquẽ α ∈R w (Y ). Given [A] in this moduli space, there is an elliptic complex We write H i A for its cohomology groups. We always have H 0 A = 0, and again describe [A] as regular if H 2 A is zero, in which case the moduli space is smooth near [A]. The dimension of the moduli space at regular points is the index of the operator We write d(X + ;α) for this index. We write the transitive action of Z oñ R w (Y ) using the notationα →α + k. The sign convention is fixed so that We also have κ(X + ;α + k) = κ(X + ;α) + k, where κ is defined by the same Chern-Weil integral that would define the characteristic class (2) in the closed case: (The notation ad(F A ) denotes a 2-form with values in End(su P ), and tr is the trace on End(su P ).) We can construct a Banach space W parametrizing perturbations of the anti-self-duality equations on X + . In order to achieve transversality, we again choose a family of balls and maps q α satisfying the density condition (Condition 12). The constants C α in the definition of W can be chosen so that for all ω ∈ W , the perturbing term V ω (α) has rapid decay on the end of X + : for some constants K j , and t a function equal to the first coordinate on the cylindrical end, we can require for all A in A(X + ;α), all ω in W , and all j ≤ l. With such a condition, the term V ω (A) will contribute a compact perturbation to the linearized equations, and we have moduli spaces M w ω (X + ;α). For a residual set of perturbations ω in W , the perturbed moduli space M w ω (X + ;α) will be regular, and therefore a smooth manifold of dimension d(X + ;α). A pair (g, ω) consisting of a cylindrical-end metric and perturbation ω will be called a good pair if all the moduli spaces M w ω (X + ;α) are regular. The moduli space is orientable, and can be oriented by choosing a trivialization of the determinant line bundle on the connected Banach manifold B w (X + ;α). We make no effort here to define a canonical orientation for the moduli space.
The proof of the compactness theorem also adapts to the cylindrical end case. If [A n ] is a sequence in M w ω (X + ;α), then after passing to a subsequence there is an Uhlenbeck limit ([A]; x), where x ∈ Sym m (X + ) and [A] ∈ M w ω (X + ;α − m ′ ). The main difference from the case of a closed manifold is that we only have m ′ ≥ m, rather than equality, because some energy may be lost on the cylindrical end.
We can now define an integer invariant (with an ambiguous sign) whenever Hypothesis 23 holds. We define it by picking a good pair (g, ω), choosing an orientation for the determinant line, and then counting with signs the points in a zero-dimensional moduli space M w ω (X + ;α) if one exists. If there is noα for which the moduli space is zero-dimensional, we define q w (X; α) to be zero.
Note that there is no hypothesis on b + 2 (X) in this construction, because there are no reducible solutions in the moduli spaces.

(iii) Perturbations with compact support
The following lemma is useful for the gluing formula in the subsection below.
Lemma 24. Let X + be a manifold with cylindrical end, as above, and suppose that w satisfies Hypothesis 23. Suppose M w κ (X + ;α) has formal dimension 0. Then we can find a cylindrical-end metric g and a perturbation ω making the moduli spaces M w κ,ω (X + ;α − m) regular for all m ≥ 0, with the additional condition that the sum (6) defining the perturbation has only finitely many non-zero terms: that is, only finitely many of the ω α are nonzero.
Proof. We already know that we can find ω in W such that the perturbed moduli spaces M w κ,ω (X + ;α−m) are regular. (For m strictly positive, regular means that this moduli space is empty.) We will show that the regularity of these moduli spaces for all m ≥ 0 is an open condition. This will suffice, because the perturbations given by finite sums are dense.
Suppose then ω n is a sequence converging to ω in W . Because of the Chern-Weil formula, we know that there exists m 0 such that M w κ,ωn (X + ;α − m) is empty (and therefore regular), for all m ≥ m 0 and all n.
If these are non-empty for infinitely many n, then we have a contradiction, arising from Uhlenbeck's compactness theorem and the emptiness of M w κ,ω (X + ;α − m) for m positive. In this way, we prove inductively that M w κ,ωn (X + ;α − m) is empty for all n ≥ n 0 and all m ≥ 1. Now consider the moduli spaces M w κ,ωn (X + ;α), which have formal dimension 0. Suppose these are non-regular for infinitely many n. After passing to a subsequence, we may suppose there is an irregular solution [A n ] in M w κ,ωn (X + ;α) for all n, and that these converge in Uhlenbeck's sense. The fact that the lower moduli spaces are empty means that this is in fact strong convergence of the connections A n after gauge transformation. Irregularity is a closed condition under strong limits, so the limit is an irregular point [A] in M w κ,ω (X + ;α), which contradicts our hypothesis.
Remark. One should expect to prove that irregularity is a closed condition under Uhlenbeck limits, and so extend the lemma to higher-dimensional moduli spaces. Because of the non-local nature of our perturbations, this would require some more work.

(iv) Gluing
Suppose X is a closed 4-manifold with b + 2 ≥ 2, so that the integer invariants q w (X) are defined. Suppose X contains a connected 3-manifold Y that separates X into two manifolds with common boundary, X 1 and X 2 , and suppose that the representation variety R w (Y ) satisfies Hypothesis 23. Let w i be the restriction of w to X i , so that we have integer invariants (with ambiguous sign) q w i (X i ; α) for i = 1, 2.
The sign-ambiguity can be partly resolved as follows. Let Λ be the orientation line bundle on B w (X), and let Λ i be the orientation bundles on B w i (X + ;α). Then there is a preferred isomorphism Suppose we choose a homology orientation o X for X, and so trivialize Λ using O(w, o X ). Then an orientation for Λ 1 determines an orientation for Λ 2 , using the above product rule. Thus a choice of sign for q w 1 (X 1 ; α) determines a choice of sign for q w 2 (X 2 ; α). In the next proposition, we assume that the signs are resolved in this way.
Proof. First, choose perturbation ω 1 and ω 2 for X + 1 and X + 2 satisfying the condition in the conclusion of Lemma 24. This condition means that there is a compact subset K i ⊂ X + i such that, for all A on X + i , the perturbing term V ω i (A) is supported in K i and depends only on the restriction of A to K i . Equip X with a metric g R containing a long cylindrical neck [−R, R] × Y , in the usual way. For R sufficiently large, X contains isometric copies of K 1 and K 2 , so we can regard ω 1 and ω 2 as defining a perturbation V ω of the equations on (X, g R ). Since this perturbation is supported away from the neck region, it does not interfere with the standard approaches to gluing anti-self-dual connections. The conclusion is that, for R sufficiently large, the moduli space M w ω (X, g R ) is regular and is the product of the two moduli spaces M w i ω i (X + i ;α), provided there exists a liftα such that these are zerodimensional.

(v) Other cases
We now consider a slightly more general setting, in which the representation variety R w (Y ) is no longer required to be a single point. We still ask that c 1 (w) is coprime to N on Y , but we suppose now that R w (Y ) consists of n irreducible elements α i (i = 1, . . . , n), all of which are non-degenerate. Rather than develop a complete Floer homology theory, we continue to make some simplifying assumptions, which we now lay out.
If γ : [0, 1] → B w (Y ) is a path joining α i to α j , we can associate to γ two quantities, both of which are additive along composite paths. The first is the spectral flow of the family of operators acting on (Ω 0 ⊕ Ω 1 )(Y ; su P ), where A(t) is a lift of the path γ to A(Y ). We call this integer d(γ). The second quantity is the Chern-Weil integral of the same kind as (15). We regard A(t) as defining a connection A on [0, 1] × Y in temporal gauge, and define Modulo Z, we can interpret κ(γ) as the drop in the suitably-normalized Chern-Simons invariant from α i to α j . The normalization is such that, for a suitably-oriented closed loop in B w (Y ) representing the generator of first homology, the value of κ(γ) is 1. For the same loop, d(γ) is 4N . If A(t) is a path of connections lifting γ, and A is the corresponding 4-dimensional connection as above, we may extend A to a connection A Z on the infinite cylinder Z = R × Y , just as we did for the cylindrical-end case. The connection A Z is flat on both ends of th cylinder. We can then define a space of connections l,a Z (Z; su P ) } and a quotient space B = B w (Z; γ). Inside the configuration space is the moduli space M w (Z; γ), and d(γ) is its formal dimension.
Hypothesis 26. Let γ denote a path from α i to α j as above. We impose two conditions: (a) d(γ) is not 1 for any such γ; (b) if d(γ) ≤ 0, then κ(γ) ≤ 0 also. Assume these two conditions hold, and consider a moduli space M w ω (X + ;α k ) on a cylindrical-end manifold X + . Suppose ω is chosen so that all moduli spaces are regular, and that this particular moduli space is zero-dimensional. The two conditions imply that if M w ω (X + ;α j ) is another moduli space, with κ(X + ;α j ) ≤ κ(X + ;α k ), then the formal dimension of this moduli space is −2 or less. This moduli space is therefore empty, and remains empty for a generic path of metrics an perturbations. It follows that there is a well-defined integer invariant q w (X; α k ) ∈ Z/ ± 1 counting points in this zero-dimensional moduli space. Once again, we extend the definition by declaring it to be zero if there is no liftα k for which the moduli space is zero-dimensional.
The product law from Proposition 25 then extends to this more general situation. In the situation considered in the proposition, we now have: 6. Calculations More specifically, we suppose these conditions are achieved by having where h is a primitive class of square 2(N − 1). These conditions ensure in particular that c 1 (P ) is coprime to N . The bundle P has κ = N − (1/N ), and the formal dimension of the corresponding moduli space M w κ,ω is zero. We choose the homology orientation o X arising from the Kähler structure, and Proposition 17 tells us that the moduli space has the same orientation as the one it obtains as a complex-analytic space. In this situation, there is an integer invariant q w (X) as in Definition 21, and our aim is to calculate it.
Proposition 27. The above invariant q w (X) for the K3 surface X is 1.
Proof. We use a familiar circle of ideas to evaluate the invariant: see [4] for the calculation in the case N = 2, on which our argument is based. We assume N ≥ 3 for the rest of this proof.
We take X to be an algebraic surface embedded in CP N by the complete linear system of a line bundle with first Chern class h. This can be done in such a way that h generates the Picard group of X: all holomorphic line bundles on X are powers of the hyperplane bundle H.
For the inherited metric g, the corresponding moduli space M w κ can be identified with the moduli space of poly-stable holomorphic bundles with the topology of P . The condition on c 1 (P ) ensures that there are no reducible bundles, so poly-stable means stable in this case. As usual on a K3, irreducible stable bundles are regular, so the moduli space M w κ is smooth and 0-dimensional; and the moduli spaces M w κ−m are empty for m positive. We can therefore use the stable bundles to calculate the integer invariant. By the remark about orientations above, the integer q w is simply the number of stable bundles on X with the correct topology.
Mukai's argument [16] shows that the moduli space is connected, so to prove the proposition we just have to exhibit one stable bundle. The Chern classes of P have been specified so that the restriction to X of the holomorphic tangent bundle T CP N as the correct topology. We will show that the holomorphic vector bundle E = T CP N | X is stable, under the hypothesis that H generates the Picard group. To do this, we recall that E has a description as a quotient where V is a vector space of dimension N + 1. The bundle E has rank N and determinant (N +1)H; so if it is unstable then there is subsheaf F with rank n and determinant at least (n + 1)H, where n lies in the range 1 ≤ n ≤ N . We can suppose that E/F is torsion-free. LetF be the inverse image of F under the quotient map q. ThenF has rank n + 1 and determinant at least (n + 1)H; and the quotient of V ⊗ H byF is torsion-free. This forces where U is a non-trivial proper subspace of V . However, the image of the section s in the above sequence is not contained in U ⊗ H, for any proper subspace U . (Geometrically, this is the statement that X is not contained in any hyperplane in CP N .) SinceF contains s(O) by construction, we have a contradiction.

(ii) Flat connections on the 3-torus
We consider the 3-torus T 3 = R 3 /Z 3 , and take x, y and z to be the simple closed curves which are the images of the three coordinate axes in R 3 . We regard these as oriented curves, defining elements of the fundamental group π 1 (T 3 , z 0 ), where z 0 is the base point. We take w → T 3 to be a line bundle with c 1 (w) Poincaré dual to z, and we take P to be a principal U (N ) bundle having w as determinant.
Proof. Let z ′ ⊂ T 3 be a parallel copy of z, disjoint from the curves x and y and meeting the 2-torus T 2 × {0} spanned by x and y transversely in a single point. Let η be a 2-form supported in a tubular neighborhood of z ′ , representing the dual class to z. The support of η should be disjoint from x, y and z. Equip the line bundle w with a connection θ whose curvature is −(2πi)η and which is trivial on the complement of the tubular neighborhood. Use θ to trivialize w away from z ′ . This trivialization of the determinant reduces the structure group of P to SU (N ) on the complement of the tubular neighborhood of z ′ .
Using θ, we identify R w (T 3 ) with the moduli space of flat SU (N ) con-nectionsÃ on T 3 \ z ′ such that the corresponding homomorphism The last condition implies that, up to similarity ρ(x) and ρ(y) can be taken to be the pair where ζ = e 2πi/N and the sign in the top row of ρ(y) is negative if N is even (so that ρ(y) has determinant 1). As an element of π 1 (T 3 \ z ′ , z 0 ), the class of z commutes with x and y, and this tells us that The elements x, y and z generate the fundamental group, so there are exactly N elements in the representation variety, as claimed. Let ρ k be the representation with ρ(z) = ζ k 1 N , and let α k be the corresponding point of R w (T 3 ).
We have already remarked that the only elements commuting with ρ k (x) and ρ k (y) are the central elements; so each ρ k is irreducible. Each ρ k determines a representationρ k of π 1 (T 3 , z 0 ) in the adjoint group PSU (N ); and all these are equal. To see that they are non-degenerate, we look at the cohomology H 1 (T 3 ; (su P ,ρ k ) with coefficients in the adjoint bundle. Any element of this group is represented by a covariant-constant su P -valued form, because the torus is flat; and there are no non-zero covariant-constant forms because the bundle is irreducible.
Lemma 29. For each k, there is a path γ joining α k to α k+1 in B w (T 3 ) such that he spectral flow d(γ) is 4. For the same path, the energy κ(γ) is 1/N .
Proof. The Chern-Simons invariant of α k in R/Z is calculated in [1], where it is shown to be −k/N mod Z. It follows that there is a path γ 0 joining α 0 to α 1 with κ(γ 0 ) = 1/N . The connections represented by α k and α k+1 are gauge-equivalent under the larger gauge group consisting of all PSU (N ) gauge transformations of ad(P ). It follows that we can lift γ 0 to a path of connections A : [0, 1] → A w (T 3 ) with A(1) = g(A(0)) for some automorphism g of ad(P ). Applying g to this whole path, we obtain a path of connections A : [1,2] → A w (T 3 ) joining A(2) to a representative A(3) for some other point of R w (T 3 ), which must be α 3 on the grounds of its Chern-Simons invariant.
In this way, we construct paths γ k from α k to α k+1 , each of which has the same energy and the same spectral flow. The composite path is a loop δ based at α 0 with κ(δ) = 1. It follows that d(δ) is 4N . So each γ k has d(γ k ) = 4.

(iii) Knot complements
Let K be a knot in S 3 , and let M be the knot complement, which we take to be a 3-manifold with 2-torus boundary carrying a distinguished pair of oriented closed curves m and l, the meridian and longitude, on its boundary.
We write Z K for the 4-manifold S 1 × M . The boundary of Z K is a 3torus. We identify m and l with curves 1 × m and 1 × l on ∂Z K , and write s for the curve S 1 × z 0 on ∂Z K , where z 0 is the point of intersection of m and l, which we take as base-point in Z K . The three classes generate the first homology of the boundary. We orient K, so as to orient the longitude l. We suppose m is oriented so that the oriented tangent vectors of s, m and l (in that order) are an oriented basis for the tangent space to ∂Z K at z 0 .
The first homology of Z K itself is generated by [s] and [m], as the class l is the boundary of an oriented Seifert surface for K. We take Σ to be such a Seifert surface, regarded as a 2-dimensional submanifold with boundary in (Z K , ∂Z K ), and we take w → Z K to be the line bundle with c 1 (w) dual to [Σ, ∂Σ] ∈ H 2 (Z K , ∂Z K ).
We identify ∂Z K with the standard 3-torus T 3 in such a way that s, m and l are identified with x, y and z (the curves considered in the previous subsection). From the discussion above, we have connections α k in R w (∂Z K ) for k = 0, . . . , N − 1, with spectral flow 4 along paths from each one to the next. Because of the lemmas above, the conditions of Hypothesis 26 hold for the 3-torus ∂Z K , so we can consider the integer-valued invariants q w (Z K ; α k ), as defined in section 5(v), for the manifold-with-boundary Z K . Here is the result: Proposition 30. When N is odd, the integer invariant q w (Z K ; α 0 ) for Z K = S 1 × M can be expressed in terms of the Alexander polynomial ∆(t) of the knot K by the formula provided this quantity is non-zero. For k non-zero the integer invariant Proof. We prove the proposition in a series of lemmas. The main idea is that we can calculate this invariant by counting certain flat connections on Z K , so reducing the problem to one about representations of the fundamental group, rather than having to understand non-flat solutions to the anti-selfduality equations.
Lemma 31. There is a liftα 0 ∈R w (T 3 ) of α 0 ∈ R w (T 3 ) such that the moduli space M w (Z + K ;α 0 ) on the cylindrical-end manifold Z + K consists only of flat connections and has formal dimension 0.
Proof. The moduli space M w (Z + K ;α 0 ) will consist entirely of flat connections if κ(Z + K ;α 0 ) is zero, which in turn holds if there is at least one flat connection in the moduli space. The PSU (N ) representationρ 0 of π 1 (∂Z K ; z 0 ) which corresponds to α 0 sends l to 1, so it extends as an abelian representation of the fundamental group of Z K . This representation defines a PSU (N ) connection on the non-trivial bundle ad(P ), so it defines a liftα 0 of α 0 of the sort required.
By excision, the formal dimension of the moduli space M w (Z + K ;α 0 ) with κ = 0 is independent of the knot K, so we may consider just the unknot, in which case the formal dimension is easily seen to be zero (for example by using the product formula for the integer invariants, as in the section below).
Lemma 32. The number of flat connections in M w (Z + K ;α 0 ) can be expressed in terms of the Alexander polynomial ∆(t) by the formula provided that this quantity his non-zero.
Proof. Because of the previous lemma, we can regard M w (Z + K ;α 0 ) as a moduli space of flat connections on the compact space Z K . More exactly, it is the space of flat PSU (N ) connections in ad(P ) → Z K , divided by the determinant-1 gauge group. (Any such flat connection must define the representation α 0 at the boundary, rather than α k for non-zero k, because the latter have non-zero Chern-Simons invariant.) We proceed as we did for the 3-torus T 3 previously. Let l ′ be a parallel copy of l, just as z ′ was a parallel copy of z in our earlier language. We take l ′ to be the boundary of Σ ′ , which is a parallel copy of Σ in the 4-manifold Z K = S 1 × M . The surface Σ ′ should lie in p × M , where p is distinct from the basepoint 1 ∈ S 1 . We equip w → Z K with a connection θ that is trivial outside the tubular neighborhood of Σ ′ and whose curvature is (−2πi)η, where η represents the Poincaré dual class to Σ ′ and is supported in the tubular neighborhood. Using θ, we can lift a flat connection A in ad(P ) lifts to a U (N ) connectionÃ in P that is flat outside the neighborhood of Σ ′ . The trivialization of w provided by θ reduces the structure group of P to SU (N ) away from the tubular neighborhood, andÃ becomes a flat SU (N ) connection there.
Suppose, then that A andÃ are such connections. Let be the representation defined by the holonomy ofÃ, and let be defined by the holonomy of A. We have already seen, from our discussion of the boundary T 3 , that after change of basis we must have and ρ(l) = 1 N .
In π 1 (Z K ; z 0 ) the class of the circle s is central. So the image ofρ lies in the centralizer ofρ(s). LetJ denote this centralizer, and let J be its inverse image in SU (N ). We can describe the group J as a semi-direct product where H is the standard maximal torus of SU (N ) and V is the cyclic group of order N generated by the matrix ρ(m) above. We identify the 3-manifold M with the submanifold 1×M in the product Z K = S 1 × M . Then, by restriction, ρ determines a homomorphism and σ(l) = 1 N . Conversely, σ determines ρ and henceρ. We are therefore left with the task of enumerating such homomorphisms σ.
Because M is a knot complement, the quotient π/π (1) is infinite cyclic, generated by the coset represented by the meridian m. The abelian group π (1) /π (2) becomes a Z[t, t −1 ]-module when we let the action of t on π (1) /π (2) be defined by the action of conjugation, x → mxm −1 , on π (1) . In a similar way, the abelian group H becomes a Z[t, t −1 ]-module if we define the action of t to be given by conjugation by the element v ∈ SU (N ) that appears in (21). In this way, σ determines (and is determined by) a homomorphism of Z[t, t −1 ]-modules, We are left with the task of enumerating such Z[t, t −1 ]-module homomorphisms. The structure of π (1) /π (2) as a Z[t, t −1 ] module is described by the Alexander polynomial ∆: there is an isomorphism (See [19] for example.) So the homomorphism σ ′ is entirely determined by the element h = σ ′ (1) ∈ H. We switch to additive notation for the torus H. We write τ v for the automorphism of H given by conjugation by v. In the ring of endomorphisms of H, we can consider the element ∆(τ v ). The description (22) of π (1) /π (2) means that the element h = σ ′ (1) must lie in the kernel of ∆(τ v ). Thus we have shown that the moduli space M w (Z + K ;α 0 ) is in one-to-one correspondence with the elements h in the abelian group LetH be the universal cover of H (or the Lie algebra of this torus), and letτ v be the lift of τ v to this vector space. Then the order of the kernel of ∆(τ v ) is equal to the absolute value of the determinant det ∆(τ v ) of the linear transformation ∆(τ v ), provided that this determinant is nonzero. If the determinant is zero, then the kernel is infinite, for it is a union of tori, whose dimension is equal to the dimension of the null-space of ∆(τ v ). As a linear transformation of the (N − 1)-dimensional real vector spaceH, the operatorτ v is diagonalizable after complexification, and its eigenvalues onH ⊗ C are the non-trivial N -th roots of unity, ζ k for k = 1, . . . , N − 1. The eigenvalues of ∆(τ v ) are therefore the complex numbers ∆(ζ k ), so This completes the proof of the lemma.
Remark. The formula (18) has another, closely-related interpretation. It is the order of the first homology group H 1 (Y N ; Z) of the 3-manifold Y N obtained as the N -fold cyclic cover of S 3 branched over the knot. By Poincaré duality, this group is the same as H 2 (Y N ; Z), which (when finite) classifies flat complex line bundles on Y N . Let p : M ′ → M be the N -fold cyclic cover of the knot complement. If we take a flat complex line bundle on Y N and restrict it to M ′ , we can then push it forward by the map p to obtain a flat rank N bundle on M . In the case that N is odd, this construction produces a flat SU (N ) bundle on M for each element of H 2 (Y N ; Z). In another language, we are constructing N -dimensional representations of π 1 (M ) as induced representations, starting from 1-dimensional representations of the subgroup π 1 (M ′ ). (In the case N even, a slight adjustment is needed, corresponding to the negative sign in the top-right entry of ρ(m) above.) Lemma 33. If the quantity in Lemma 32 is non-zero, then the moduli space M w (Z + K ;α 0 ) is regular.
Proof. Let A be a flat connection in ad(P ) representing a point of this moduli space. The formal dimension of the moduli space is zero, and A is irreducible. So to establish regularity, it is sufficient to show that H 1 A is zero, or equivalently that ker d + A is equal to imd A in L 2 l,A (Z + K ; su P ). Harmonic theory on the cylindrical end manifold means that we can rephrase this as saying that the kernel of d + A ⊕ d * A is zero. Any element of the kernel of this Fredholm operator is exponentially decaying on the cylindrical end, and we can integrate by parts (using the fact that A is flat), to conclude that an element a in the kernel must have d − A a = 0. Thus a is in the kernel of the The kernel of d A ⊕ d * A represents the first cohomology group with coefficients in the flat bundle su P with connection A: There is no difference here between the absolute or relative group, because the connection A is irreducible on the boundary of Z K . We must therefore only show that this cohomology group is zero.
Under the adjoint action of the matrix ρ(s) from (20), the complexified Lie algebra sl(N, C) decomposes as a sum where E k is the ζ k -eigenspace of Ad(ρ(s)), which is the span of the elementary matrices e ij with a 1 in row i and column j, with i − j = k (mod N ). There is a corresponding decomposition of the bundle su P which is respected by the flat connection, because s is central in the fundamental group.
So we look now at the individual summands H 1 (Z K ; (E k , A)). The manifold Z K is a product, and s is the element of π 1 corresponding to the circle fiber. On the circle H 1 (S 1 , λ) is zero if λ is a local system with fiber C and non-trivial holonomy. By the Kunneth theorem, it follows that H 1 (Z K ; (E k , A)) is zero for k non-zero.
We are left with the summand H 1 (Z K ; (E 0 , A)). The holonomy of A on E 0 is trivial in the circle factor of Z K ; so we may as well consider H 1 (M ; (E 0 , A)). On the knot complement M , the bundle E 0 has rank N − 1 and the holonomy of A factors through the abelianization of the fundamental group, H 1 (M ; Z), which is generated by m. Under the action of Ad(ρ(m)), the bundle E 0 decomposes into N − 1 flat line bundles L k (k = 1, . . . N ); and the holonomy of L k along the meridian is ζ k .
To complete the proof, we have to see that H 1 (M ; (L k , A)) is zero unless ∆(ζ k ) = 0. This is a standard result, and can be proved as follows. If the cohomology group is non-zero, let α be a non-zero element, and let α ′ be the pull-back to the N -fold cyclic cover p : M ′ → M . This pull-back is an element of H 1 (M ′ ; C) and has trivial pairing with the loop m ′ = p −1 (m). It therefore pairs non-trivially with a loop γ ⊂ M ′ that lifts to the Z-covering M . Thus the pull-back of α to the Z-covering is a non-zero classα ∈ H 1 (M ; C). By constructionα lies in the ζ k -eigenspace of the action of the covering transformation. Up to units in the ring C[t, t −1 ], the characteristic polynomial of the covering transformation acting on H 1 (M ; C) is ∆(t). So ∆(ζ k ) is zero.
Lemma 34. Suppose N is odd. Then under the same hypothesis as Lemma 33, all the points of the moduli space M w (Z + K ;α 0 ) have the same sign.
Proof. We wish to compare the orientations of two points [A 0 ] and [A 1 ] in the moduli space M w (Z + K ;α 0 ). Let [A t ] be a 1-parameter family of connections in A(Z + K ;α 0 ) joining A 0 to A 1 . Let D t be the corresponding Fredholm operators on the cylindrical-end manifold Z + K , as in (14). The operators D 0 and D 1 are invertible, and so there are canonical trivializations of the determinant lines det(D 0 ) and det(D 1 ). The lemma asserts that these canonical trivializations can be extended to a trivialization of the determinant line det(D • ) on [0, 1] .
Let p : M ′ → M be again the N -fold cyclic cover of the knot complement, and let p : Z ′ K → Z K be the corresponding cyclic cover of Z K = S 1 × M . Let Z ′ + K be the corresponding cylindrical-end manifold. Set These are connections on Z ′ + K , asymptotic to p * (α 0 ) on the cylindrical end. There are corresponding Fredholm operators det(D ′ t ), for t ∈ [0, 1]; we use a weighted Sobolev space with a small exponential weight to define the domain and codomain of D ′ t , because the pull-back ofα 0 under the covering map p is not isolated in the representation variety. The operators D ′ 0 and D ′ 1 are invertible, as one can see by a calculation similar to that in the proof of Lemma 33 above. The covering group Z/(N Z) acts on the domain and codomain of D ′ t , which therefore decompose into isotypical components according to the representations of this cyclic group. The restriction of D ′ t to the component on which Z/(N Z) acts trivially is just D t (with the inconsequential change that D t is acting now on a weighted Sobolev space). The non-trivial representations of Z/(N Z) are all complex, because N is odd, and their contribution to the determinant is therefore trivial. We can therefore replace D t by D ′ One should expect to prove Proposition 30 also in the case that the expression that appears there is zero. In this case, the moduli space is a union of circles or higher-dimensional tori, and one should try to use a holonomy perturbation to make the perturbed moduli space empty.

(iv) The Fintushel-Stern construction
Let X be a closed oriented 4-manifold with b + 2 (X) ≥ 2. Let w be a line bundle with c 1 (w) coprime to N , and suppose that the integer invariant q w (X) is non-zero.
Let T be an embedded torus in X, with trivial normal bundle, and suppose that the pairing of c 1 (w) with [T ] is 1. The torus T has a closed tubular neighborhood N ⊂ X of the form T × D 2 . If U is the unknot in S 3 , then the 4-manifold Z U obtained from the unknot is also T × D 2 . We choose an identification φ of Z U with N , in such a way that the longitudinal curve l on the boundary of Z U is matched by φ with the curve (point) × ∂D 2 that links the embedded torus. Using the standard curves l, m and s, the boundary of Z U is canonically identified with the boundary of Z K , for any other knot K in S 3 . Thus the choice of φ gives us preferred diffeomorphisms ψ : ∂Z K → ∂N for all K. We now form a closed 4-manifold X K by removing N = Z U and replacing it with Z K : This is Fintushel and Stern's construction, from [8]. The Mayer-Vietoris sequence tells us that the cohomology rings of X and X K are canonically isomorphic.
The line bundle w on X \ N extends to X K . The extension w is unique if we demand the additional property that c 1 (w) 2 [X K ] and c 1 (w) 2 [X] are equal. The moduli spaces M w κ (X K ) and M w κ (X K ) have the same formal dimension, and there is a potentially non-zero integer invariant q w (X K ). We can calculate this invariant for odd N : Proposition 35. When N is odd, the integer invariant q w (X K ) can be expressed in terms of q w (X) and the Alexander polynomial ∆ of K as where ζ = e 2πi/N .
Proof. Up to an overall ambiguity in the sign, this proposition is now a formal consequence of the calculation of the invariants q w (Z K ; α k ) and q w (Z U ; α k ) which are provided by Proposition 30, and the product law (17). These tell us that where n = q w (X \ N • ; α 0 ) is a quantity independent of K. Because X U is X, and the Alexander polynomial of the unknot is 1, it follows that n = ±q w (X).
It remains to check the overall sign. Certainly, the sign is correct if K is the unknot. Note also that since the complex numbers ∆(ζ k ) come in conjugate pairs when N is odd, their product is positive; so our task is to check that the sign of the invariant is independent of K. To this end, we observe that among the flat connections on the cylindrical-end manifold Z + K that are enumerated in Lemma 32, there is a unique connection, say [A * ], distinguished by the fact that the corresponding representationρ of the fundamental group (see (19)) factors through the abelianization H 1 (Z K ; Z). Let [B] be an isolated solution in the moduli space M w ω ((X \ N • ) + ;α 0 ), and let [C] be the connection on X K obtained as the result of gluing [B] to [A * ] with a long neck. It is not important that C is actually a solution of the equations: we can instead just use cut-off functions to patch together the connections A * and B; but it is important that the operator D C is invertible (10), so that we can ask about the sign of the point [C].
To complete the proof we will show that the sign of [C] is independent of the knot K. If K is any knot and K 0 is the unknot, then we can transform Z K = S 1 × M into Z K 0 by a sequence of surgeries along tori where in each surgery the curve δ is a null-homologous circle in M (compare [8]). The connection A * (and therefore also C) is flat on such a torus and lifts to an abelian representation ρ of π 1 (T ), sending the S 1 generator to the first matrix in (20) and the generator δ to 1. Thus C is compatible with a decomposition of the associated rank-N vector bundle as a sum of flat line bundles. It now follows from excision and the material of section 4(i) that the surgery does not alter the sign of [C].
As a particular case, we can take X to be a K3 surface. With w as in section 6(i), we calculated the invariant q w (X) to be 1, for the preferred homology-orientation of X. The value of c 1 (w) in our calculation was (N + 1)h; but we could equally have taken c 1 (w) to be just the primitive class h (with square 2(N − 1)). In this case, the invariant q w (X) would have again been 1, by Proposition 18. A primitive cohomology class on a K3 surface has pairing 1 with some embedded torus T . So choose T with h[T ] = 1. We are then in a position to apply the proposition above, to obtain for odd N . As Fintushel and Stern observe in [8], the 4-manifold X K is a homotopy K3-surface if we take T to be a standardly embedded torus (so that X \ T is simply connected).

(i) Irreducibility on open sets
Let Ω ⊂ X be a connected, non-empty open set, chosen so that π 1 (Ω; x 0 ) → π 1 (X; x 0 ) is surjective, for some x 0 ∈ Ω. The unique continuation argument from [6] shows that if an anti-self-dual connection [A] ∈ M w κ (X) is irreducible, then its restriction to Ω is also irreducible. For the perturbed equations, it may be that this result fails. The next lemma provides a suitable substitute for our purposes. It is only a very slight modification of Proposition 15, and we therefore omit the proof.

Lemma 36.
Let Ω ⊂ X be as above. Let κ 0 be given, and let g be a metric on X with the property that the unperturbed moduli spaces M w κ contain no reducible solutions for any κ ≤ κ 0 . Then there exists ǫ > 0 such that for all ω ∈ W with ω W ≤ ǫ, and all [A] in M w κ,ω (X), the restriction of A to Ω is irreducible.

(ii) Cohomology classes and submanifolds
Fix as usual a U (N ) bundle P → X, and let B * = B * (X) denote the irreducible part of the usual configuration space. There is a universal family of connections, carried by a bundle ad(P) → X × B * .
This universal bundle can be constructed as the quotient of ad(P ) × A * by the action of G. Note, however, that we cannot form a universal U (N ) bundle P in this way, and our notation is not meant to imply that such a U (N ) bundle exists. We define a 4-dimensional cohomology class on X × B * by taking the first Pontryagin class of the adjoint bundle, with the nowfamiliar normalization: ∈ H 4 (X × B * ; Q).
Using the slant product, we now define by the formula µ(α) = c/α. We will concern ourselves here only with evendimensional classes on X, so we define A(X) to be the polynomial algebra A(X) = Sym(H even (X; Q)), and we extend µ to a ring homomorphism We regard A(X) as a graded algebra, defining the grading of H i (X; Q) to be 4 − i so that µ respects the grading. where the notation means that we couple∂ * to the PSU (N ) connections in the family over B * . The following lemma and Corollary 38 below play the same role as Lemma (5.2.9) of [6]. There is a slight extra complication in our present setup: we need to deal with the weaker L p 1 norms, because of the way our perturbations behave under Uhlenbeck limits. (See Proposition 11.) Lemma 37. Fix an even integer p > 2. Then there is a complex Banach space E and a continuous linear map s : E → Γ(B * ; L) with the following properties.
(a) For each e ∈ E, the section s(e) is smooth, and furthermore the map Proof. Let [A] belong to B * , and let S ⊂ A be the Coulomb slice through A. Using the definition of L as a determinant line bundle, construct a nonvanishing local section t for L on some neighborhood U of A in S. The section t can be constructed to be smooth in the L q 1 topology for all q; so we can take it that U is open in the L p 1 topology. Because p is an even integer, the L p 1 norm (or an equivalent norm) has the property that The above corollary can be applied directly to the situation that M is a regular moduli space M w κ,ω of perturbed anti-self-dual connections, containing no reducibles. In this case, the zero set of s on M is a cooriented submanifold of codimension 2 whose dual class is (2N )µ([Σ]). As with Donaldson's original construction however, we must modify the construction to obtain a codimension-2 submanifold which behaves a little better with respect to the Uhlenbeck compactification.
Let Σ be an embedded surface as above, and let ν(Σ) be a closed neighborhood that is also a 4-dimensional submanifold with boundary in X. By adding 1-handles to ν(Σ), we may suppose that this neighborhood is connected and that π 1 (ν(Σ)) → π 1 (X) is surjective. Let B * (ν(Σ) be the configuration space of irreducible connections on ν(Σ). We suppose w and g are such that the moduli spaces M w κ ′ contain no reducible solutions for all κ ′ ≤ κ. Lemma 36 tells us that we may choose a small regular perturbation ω so that there are well-defined restriction maps The determinant line bundle L is defined on B * (ν(Σ)). As in [6], we choose a section s of the L → B * (ν(Σ)) such that the section s • f is transverse to zero. This can be done by repeating the proof of the lemma above with B * (ν(Σ)) replacing B * (X). We can even choose s so that transversality holds for all κ ′ ≤ κ simultaneously.
As is customary, we write V Σ ⊂ B * (ν(Σ)) for the zero-set of a section s of this sort; and we write M w ω,κ ∩ V Σ for the zero set in the moduli space, suppressing mention of the restriction map f . Proof. If x is disjoint from ν(Σ), then the A n converge to A ′ in the L p 1 topology on ν(Σ) after gauge transformation. The set V Σ is closed in B * (ν(Σ)) in the L p 1 topology, because of the last condition in Lemma 37, so [A ′ ] belongs to V Σ .
We now wish to represent (2N )µ([x]) (x is a point in X) by a codimension-4 intersection M w ω,κ ∩ V x in the same manner. As in [14], we must now take V x to be a stratified space with smooth manifolds for strata, rather than simply a smooth submanifold.
To obtain a suitable definition of V x , we observe that (2N )µ([x]) is represented on B * by c 2 (sl P ), where P x → B * is the base-point fibration: the restriction of P to {x} × B * . To represent c 2 geometrically, we take N 2 − 2 sections s 1 , . . . s N 2 −2 of the complex vector bundle sl Px and examine the locus on which they are linearly dependent. We must require a transversality condition: the following lemma and corollary are essentially the same as Lemma 37 and Corollary 38 above.
Lemma 40. Fix an even integer p > 2, and write V → B * for the bundle Hom(C N 2 −2 , sl Px ). Then there is a complex Banach space E and a continuous linear map s : E → Γ(B * ; V) with the following properties. As with the case of 2-dimensional classes, we now take a closed neighborhood ν(x) that is also a manifold with boundary. We require that π 1 (ν(x), x) → π 1 (X, x) is surjective. We can again suppose that for κ ′ ≤ κ, the restriction of every solution in M w ω,κ ′ to ν(x) is irreducible, so that we have maps f : M w ω,κ ′ → B * (ν(x)). We take a section s satisfying the transversality condition of the above corollary, taking M to be the union of the M w ω,κ ′ and replacing B * by B * (ν(x)) in the statement. Write for the locus where s does not have full rank. Then the intersection M w ω,κ ∩ V x is a stratified subset, with each stratum a smooth, cooriented manifold, and top stratum of real codimension 4 in M w ω,κ . The dual class to V x is (2N )µ([x]). Lemma 39 continues to hold with V x in place of V Σ .
(iii) Polynomials in the coprime case We now have the necessary definitions to construct the PSU (N ) generalization of Donaldson's polynomial invariants, in the case that w = c 1 (P ) is coprime to N .
We suppose as usual that b + 2 (X) is at least 2. Because of the coprime condition, we can choose a Riemannian metric g on X such that the moduli spaces M w κ contain no reducible solutions. Given some κ 0 , we may find ǫ so that for all perturbations ω with ω W ≤ ǫ, the moduli spaces M w ω,κ contain no reducibles for κ ≤ κ 0 . Amongst such small perturbations, we may choose one so that the moduli spaces are regular. If N is even, choose a homology orientation for X so as to orient the moduli spaces.
Let d be the dimension of M w ω,κ , and suppose that d is even. (This parity condition holds if N is odd, or if b + 2 (X)−b 1 (X) is odd.) Consider an element of A(X), given by a monomial Suppose that 4r + 2t = d, so that z is of degree d. We may assume that the surfaces Σ i intersect only in pairs, and we can represent the class [x] by r distinct points x 1 , . . . , x r all disjoint for the surfaces. Choose neighborhoods ν(x 1 ), . . . , ν(x r ), ν(Σ 1 ), . . . , ν(Σ t ) satisfying the surjectivity condition on π 1 as above. We may assume that these satisfy the same intersection conditions: the only non-empty intersections are possible intersections of pairs ν(Σ i ) ∩ ν(Σ j ). Construct submanifolds (or stratified subspaces) V x i and V Σ j as above. By a straightforward extension of the transversality argument, we may arrange that all multiple intersections In the case of the V x i , transversality means transversality for each stratum. The intersection has dimension zero: it is an oriented set of points, contained entirely in the top stratum of the V x i . The next lemma is proved by the same dimensioncounting that is used for the case N = 2: Lemma 42. Under the stated transversality assumptions, the zero-dimensional intersection (25) is compact.
Finally, we can state: Proposition 43. Fix N , and let X be a closed, oriented smooth 4-manifold with b + 2 (X) ≥ 2, equipped with a homology orientation o X . Let w be a line bundle with c 1 (w) coprime to N , and let P be a U (N ) bundle on X with det(P ) = w, such that the corresponding moduli space M w κ has formal dimension an even integer d. Let V x i and V Σ j be as above. Then the signed count of the points in the intersection (25) has the following properties: (a) It is independent of the choice of Riemannian metric g, perturbation ω, the neighborhoods ν(x i ) and ν(Σ j ), and the sections defining the submanifolds V x i and V Σ j , subject to the conditions laid out in the construction above.
(b) It depends on the Σ j only through their homology classes [Σ j ].
(c) It is linear in the homology class of each Σ j .
Proof. The proof can be modeled on the arguments given in [4] and [6].
Staying with our assumptions that b + 2 (X) ≥ 2, that c 1 (w) is coprime to N and that a homology orientation o X is given, we now define the polynomial invariant q w (X) : A(X) → Q by declaring its value on the monomial z given in (24) to be given by where κ is chosen (if possible) to make the moduli space d-dimensional.
If there is no such κ, we define q w (X)(z) to be zero for this monomial of degree d. The definition of q w (X) is set up so that the integer invariant that we defined earlier can be recovered from this more general invariant as the value q w (X)(1).
(iv) The non-coprime case Thus far, we have defined q w (X) only in the case that c 1 (w) is coprime to N , because this condition allows us to avoid the difficulties arising from reducibles. In the case that c 1 (w) is not coprime to N , however, we can still give a useful definition of q w (X) by a formal device previously used in [14].
Given X and w (with b + (X) ≥ 2 as usual), we consider the blow-up A maximal positive-definite subspace of H 2 (X; R) is also a maximal positivedefinite subspace of H 2 (X; R) (which is canonically a direct sum); so a homology-orientation o X determines a homology-orientation oX for the blow-up. We setw = w + e, where e = PD(E) is the Poincaré dual of a chosen generator of E in H 2 (CP 2 ; Z). Notice thatw is coprime to N onX, so qw(X) is already well-defined. We now define q w (X) in terms of qw(X) by the formula for z ∈ A(X) ⊂ A(X). (The result is independent of the choice of generator E.) The rationale for this definition, as in [14], is that the formula (26) is an identity in the case that c 1 (w) is already coprime to N (so that the left-hand side is defined by our previous construction). This is a blow-up formula, which in the case N = 2 was first proved in [13]. We now outline the proof of this blow-up formula, for general N .
On the manifoldCP 2 , let Q be the U (N ) bundle with c 1 (Q) = e and c 2 (Q) = 0. The characteristic number κ(Q) for this bundle is (N − 1)/2N , so the formal dimension d of the moduli space is 2(N − 1) − (N 2 − 1). If we pick a base-point y 0 ∈CP 2 and consider the framed moduli space (the quotient of the space of solutions by the based gauge group), then the formal dimension is 2(N − 1). The fact thatCP 2 has positive scalar curvature and anti-self-dual Weyl curvature means that the framed moduli space is regular; and since its dimension is less than N 2 − 1, the moduli space consists only of reducibles. The framed moduli space therefore consists of a single CP N −1 , which is the gauge orbit of a connection [A] compatible with the splitting of Q as a sum Q = l ⊕ Q ′ .
(The first is the line bundle with c 1 (l) = e, and the second summand is a trivial U (N − 1) bundle.) The stabilizer Γ A in the gauge group G is S(U (1) × U (N − 1)). Let us write N for this framed moduli space. There is a universal bundle ad(Q) →CP 2 × N and a corresponding slant-product map We can describe Q as the quotient ad(Q) = ad(Q) × SU (N ) S(U (1) × U (N − 1)) , which makes it the adjoint bundle of the U (N ) bundle .
The decomposition of Q gives rise to a decomposition where τ → N = CP N −1 is the tautological line bundle and τ ⊥ is its complement. On the submanifold CP 1 × CP 1 ⊂CP 2 × N , the bundle Q has c 2 1 = 0 and c 2 is a generator. So µ(E) is an integral generator for H 2 (N ) = Z. In particular, µ(E) N −1 , [N ] = ±1.
Now we consider the blow-up formula in the case that w is coprime to N on X and that the moduli space M w ω,κ (X) is zero-dimensional, leading to an integer invariant q w (X). Let x 0 be a base-point in X. Using Lemma 24 and the conformal invariance of the anti-self-duality equations, we can find a good pair (g, ω) with the property that the support of the perturbation is disjoint from a neighborhood of x 0 . We can then form a connected sum X = X#CP 2 , attaching theCP 2 summand to X in a neighborhood of x 0 .
We equip the connected sum with a metric g R containing a neck of length R, in the usual way. OnX, we form the bundleP as the sum of the bundles P → X and Q →CP 2 . As in 5(iv), we may regard ω as defining a perturbation onX, with support disjoint from the neck and from theCP 2 summand. The formal dimension of the moduli space onX is 2(N − 1), and the moduli space is compact once R is sufficiently large. Standard gluing If j i ≤ N − 1 for all i, and if the V j i ,x i are chosen so that all relevant intersections are transverse, then this zero-dimensional intersection will be compact. So we can define Donaldson-type invariants using the characteristic class ν j of the base-point fibration, as long as j ≤ N − 1.
For j ≥ N , the dimension-counting fails. For example, suppose that M w ω,κ is 4N -dimensional, so that M w ω,κ−1 is 0-dimensional, and suppose that w is coprime to N and that the moduli spaces are regular and without reducibles. We can try taking j = N , and considering the 0-dimensional intersection M w ω,κ ∩ V N,x , so as to arrive at an evaluation of p N of the base-point fibration on the (potentially non-compact) moduli space. The problem is that the above zero-dimensional intersection need not be compact. We cannot rule out the possibility that there is a sequence [A n ] of solutions belonging to this intersection, converging to an ideal connection ([A ′ ], x ′ ), with [A ′ ] ∈ M w ω,κ−1 and x ′ in the neighborhood ν(x).

(ii) Other Lie groups
The approach we have taken to define the SU (N ) polynomial invariants (even in the case of a bundle with c 1 = 0) makes essential use of a U (N ) bundle on the blow-up, in order to avoid reducible solutions. This approach does not seem to be available in general: we cannot define E 8 Donaldson invariants this way, for example. It is possible that by taking the instanton number to be large, one could sidestep the difficulties that arise from reducible solutions, and so arrive at an alternative construction. This was Donaldson's original approach, in [4]. It seems likely that this approach, if successful, would be technically more difficult than what we have done here for SU (N ).

(iii) Mahler measure
The large-N behaviour of the SU (N ) Donaldson invariants is briefly considered in [15]. In the particular examples we have studied in this paper, the leading term of the invariant for large N is easily visible, and turns out to have a rather interesting interpretation.
We turn again to the manifold Z K with torus boundary, obtained by taking the product of the circle and a knot-complement, and we look again at the integer-valued invariant q w (Z K ; α 0 ). The formula for this invariant in Proposition 30 relates it to the geometric mean of the Alexander polynomial ∆ K on the unit circle: as long as there are no zeros of ∆ K on the unit circle, we have q w (Z K ; α 0 ) N ∼ α N , as N increases through odd integers, where This asymptotic result is very easily obtained by considering the expression for (1/N ) log |q w (Z K ; α 0 ) N | arising from Proposition 30: it is a Riemann-sum approximation to the integral of the smooth function |∆ K (z)| on the circle. The remaining point is then that the Riemann sum approximates the integral of a periodic function with an error which is at most O(1/N 3 ) when the integrand is C 2 .
If ∆ K has zeros on the unit circle, the situation is much more delicate. We have only the weaker statement as N runs through those odd integers for which q w (Z K ; α 0 ) N is non-zero. The proof of (29) is given in [11] and in [18], in answer to a question raised by Gordon in [12]. A different proof and an extension of the result are contained in [20]. The quantity α is known as the Mahler measure of the polynomial. It can also be expressed as where a is the leading coefficient of the polynomial and λ runs through the roots in the complex plane. The Mahler measure of the Alexander polynomial, as well as the Mahler measures of the Jones and A-polynomials, have appeared elsewhere in the literature, in connection with interesting quantities such as the hyperbolic volume of the knot complement. See [21,17] and [2], for example.