by Simon Donaldson
I first met John during a short visit to New York in 1983. In the spring of 1985 we both attended a meeting in MSRI, Berkeley, which was an opportunity for more extensive mathematical discussions with John, which I have enjoyed over the many years since. This was an exciting time: as well as new 4-manifold invariants, Casson’s invariant of homology 3-sphere, Gromov’s work using holomorphic curves in symplectic geometry, the knot invariant of Jones and Floer’s new homology groups all appeared in the few years approximately 1984–87. Bob Friedman was also at the MSRI meeting and this was the beginning of the renowned Friedman–Morgan collaboration. In 1986 I visited Bob and John in Columbia for a week with much discussion of 4-manifolds, particularly complex surfaces, and instanton invariants. Some while after (perhaps 1988) John visited Oxford for a few days. Later, beginning in 2013, I was able to get to know John much better when he recruited me to the Simons Center in Stony Brook and I have many memories of his kindnesses in his role as Director of the Center over the following years.
One of the outstanding features of John’s mathematical work is that he has been able to move between completely different fields, making fundamental contributions to each. I witnessed that when, from around 1985, he quickly became expert in all facets of the new instanton invariants: he was one of the leaders in the development of that theory over the subsequent decade or so and, from 1994, of the Seiberg–Witten theory. I will not attempt to survey systematically here all of his work on 4-manifolds and gauge theory, and what follows is a series of remarks and comments about some parts of it.
1. Blow-ups and wall crossing
From the mid-1980s, questions in the area “4-manifolds and gauge theory” could be roughly divided into questions about smooth 4-manifolds and questions internal to the theory of the gauge theory invariants. Of course those are related because one hopes that progress on the latter will lead to progress on the former. In this section we are discussing two questions internal to the instanton theory.
Recall that, in outline and under suitable technical conditions, invariants of a compact smooth oriented 4-manifold \( X \) are obtained by evaluating cohomology classes on the moduli spaces \( \mathcal{M}_{k} \) of solutions of the Yang–Mills instanton equation over \( X \), with gauge group \( SU(2) \) or \( SO(3) \). (The instanton equation is a partial differential equation for a connection on a bundle \( E \) and depends on a choice of Riemannian metric.) These cohomology classes on the moduli space are obtained from the homology of \( X \) using the universal bundle over \( \mathcal{M}_{k} \times X \). One gets numbers
where \( \mu(\alpha)\in H^{2}(\mathcal{M}_{k}) \) corresponds to a class \( \alpha\in H_{2}(X) \) and \( u\in H^{4}(\mathcal{M}_{k}) \) corresponds to the point class in \( H_{0}(X) \). These numbers constitute a collection of polynomial functions on \( H_{2}(X) \).
The cohomology \( H^{2}(X) \) is equipped with a nondegenerate quadratic intersection form, so one writes the second Betti number as \( b^{2}_{+}+ b^{2}_{-} \) where \( b^{2}_{\pm} \) are the dimensions of the positive and negative parts of the form. A Riemannian metric \( g \) on \( X \) gives natural positive and negative subspaces \( H^{2}= H^{2}_{+,g}\oplus H^{2}_{-,g} \), via Hodge theory, as the spaces of \( \pm \) self-dual harmonic 2-forms. In the set of all connections on our \( SU(2) \) or \( SO(3) \) bundle there are typically reducible connections, where the structure group reduces to \( S^{1} \). These reductions are classified topologically by the first Chern class \( c \) in \( H^{2}(X,\mathbf{Z}) \) and there is a reducible instanton connection exactly when \( c \) lies in \( H^{2}_{-,g} \). The relevance of this is that the cohomology classes \( u, \mu(\alpha) \) are not defined over reducible connections and the definition (1) does not apply. When \( b^{2}_{+}=0 \), the condition \( c\in H^{2}_{-,g} \) holds for all metrics \( g \) and the instanton invariants are not generally defined. For \( b^{2}_{+} > 0 \), one shows that the set of metrics with a (nonzero) integral class in \( H^{2}_{-,g} \) has codimension \( b^{2}_{+} \) in the space of all metrics. For \( b^{2}_{+} > 1 \), this means that reducible connections do not affect the theory. The situation when \( b^{2}_{+}=1 \) is special. For generic metrics \( g \), the definition (1) applies but in a generic 1-parameter family of metrics \( g_{t} \) reducible instanton connections appear at a discrete set of times \( t \) and the pairings (1) change. The “wall-crossing problem” is to understand this change and give an explicit formula for it.
The “blow-up problem” is to relate the invariants of a 4-manifold \( X_{1} \) and the connected sum \( X_{1}\sharp \overline{\mathbf{C}\mathbf{P}^{2}} \), where \( \overline{\mathbf{C}\mathbf{P}^{2}} \) is the complex projective plane with the opposite of the standard orientation. (The terminology arises from the fact that when \( X_{1} \) is a complex surface the blow-up at a point is this connected sum.) It is a part of a wider question of describing invariants of connected sums, or more generally of manifolds with a decomposition \( X_{1}\cup_{Y} X_{2} \) where \( Y \) is a 3-manifold and \( X_{1}, X_{2} \) have boundary \( Y \). (So the connected sum is the case \( Y=S^{3} \).)
The standard procedure for analysing the instanton invariants of a connected sum \( X=X_{1}\sharp X_{2} \) is to consider metrics with a very long neck between two components. Then the instantons on \( X \) can be described, roughly speaking, by gluing together instantons on \( X_{1} \) and \( X_{2} \). When
this leads to a vanishing theorem: the invariants of \( X \) are 0. In the blow-up problem \( X_{2}= \overline{\mathbf{C}\mathbf{P}^{2}} \) has \( b^{2}_{+}=0 \) and the vanishing argument does not apply because of the appearance of reducible instantons on \( \overline{\mathbf{C}\mathbf{P}^{2}} \).
In both of these problems it is not hard to work out the relevant formulae is some simple cases. The key difficulty in general is the interaction between reducible connections and the noncompactness of the instanton moduli spaces \( \mathcal{M}_{k} \). These are not usually compact because of the well-known “bubbling” phenomenon, whereby a sequence of instanton connections over \( X \) can fail to converge over a finite set of points in \( X \) at which the curvature becomes concentrated. This leads to the Uhlenbeck compactification in which one adjoins to \( \mathcal{M}_{k} \) additional strata
(where \( s^{l} \) is the \( l \)-fold symmetric product). The difficulty is, roughly speaking, to understand how these strata fit together, particularly when \( l \) is large and when there are reducible connections in \( \mathcal{M}_{k-l} \).
With this background in place, I go back to John’s visit to Oxford circa 1988. We worked at the blackboard over several days to try to understand blow-up formulae but without ultimate success. Later Peter Ozsvath (in his PhD thesis supervised by John) and Tom Leness obtained formulae for the first “hard” cases [e6], [e11]. Still later the problem was completely solved in a wonderful paper of Fintushel and Stern [e10]. Now, if my dim memory is correct, at the end of our Oxford discussions John and I saw that the blow-up formula we had tentatively arrived at could not be correct because it led to contradiction when one considered a double blow-up \( Z=X_{1}\sharp \overline{\mathbf{C}\mathbf{P}^{2}}\sharp \overline{\mathbf{C}\mathbf{P}^{2}} \). We should have pursued that line of thought more, because the use such a double blow-up was one key idea in Fintushel and Stern’s work. Another key component was their observation that \( Z \) contains a 2-sphere of self-intersection \( -2 \), so there is another decomposition \( Z=Z_{1}\cup_{\mathbf{R}\mathbf{P}^{3}}Z_{2} \) (with \( Z_{2} \) a neighbourhood of this 2-sphere) which could be used to analyse instantons on \( Z \) and obtain a recursion relation between different coefficients in blow-up formulae.
The significance of Fintushel and Stern’s result was not just in the resolution of the problem but in the form of their solution. The instanton invariants of \( X_{1}\sharp \overline{\mathbf{C}\mathbf{P}^{2}} \) are given by combinations of the invariants of \( X \) with coefficients which come from the power series of classical special functions associated with a family of elliptic curves. This came before the Seiberg–Witten theory but was later seen to fit in perfectly with that.
A short, brilliant paper of John and Tom Mrowka [5] used blow-ups to make a major simplification in the theory of instanton invariants. Above, we have discussed reductions of an \( SU(2) \) or \( SO(3) \) bundle to nontrivial \( S^{1} \) bundles but there is also the possibility of reduction to the trivial bundle. For the gauge group \( SO(3) \) this is not a problem (because the Stiefel–Whitney class \( w_{2} \) of the bundle is not affected by the development of point singularities) but for \( SU(2) \) it is. For example, it made up most of the work in the proof of the connected sum vanishing theorem in [e2]. Morgan and Mrowka’s idea was to define invariants of \( X \) by considering an auxiliary connected sum \( X\sharp \overline{\mathbf{C}\mathbf{P}^{2}} \). Starting with an \( SU(2) \) bundle over \( X \) they extend it to an \( SO(3) \) bundle over the connected sum which is nontrivial over the \( \overline{\mathbf{C}\mathbf{P}^{2}} \) summand. An easy case of the blow-up theory shows that \( SU(2) \) invariants of \( X \) defined by (1) can also be obtained from invariants of the connected sum. (More precisely, if \( e \) is the standard generator of \( H^{2}( \overline{\mathbf{C}\mathbf{P}^{2}}) \) we expand a polynomial on \( H_{2}(X\sharp \overline{\mathbf{C}\mathbf{P}^{2}}) \) in powers of \( e \) and take the term which is of degree 1: a polynomial on \( H_{2}(X) \).) Going backwards, Morgan and Mrowka take this formula as the definition of the invariant of \( X \) and then all the annoying complications involving the trivial bundle disappear.
We turn now to the wall-crossing problem. The paper [7] of Kotschick and Morgan put this on firm foundations, correcting an important gap in Kotschick’s previous paper [e3]. Let \( X \) be a smooth, compact oriented 4-manifold with \( b^{+}=1 \), \( b^{-}=m \) so \( H^{2}(X;\mathbf{R})= \mathbf{R}^{1,m} \). The set of positive 1-dimensional subspaces can be identified with the hyperbolic space \( \mathcal{H}^{m} \). A Riemannian metric \( g \) on \( X \) defines a point \( \omega_{g} \) in \( \mathcal{H}^{m} \), via the subspace \( H^{2}_{+,g} \). For each integral class \( c \) there is a codimension-1 “wall” \( c^{\perp} \subset \mathcal{H}^{m} \) of subspaces orthogonal to \( c \). For any fixed \( k \) the walls defined by classes \( c \) with \( -c^{2} < k \) form a locally finite collection in \( \mathcal{H}^{m} \) and divide \( \mathcal{H}^{m} \) into a system of chambers. If \( \omega_{g} \) lies in the interior of a chamber then reducible connections do not cause difficulties in the definition of the numbers (1). Now suppose that \( g^{\prime} \) is another metric on \( X \) with \( \omega_{g^{\prime}} \) in the same chamber as \( \omega_{g} \). If \( g \) and \( g^{\prime} \) can be joined by a path of metrics \( g(t) \) such that \( \omega_{g(t)} \) lies inside this same fixed chamber for all \( t \) then the usual theory shows that the numbers (1) defined using \( g \) and \( g^{\prime} \) are the same. If such paths always exist then it follows that this construction gives a map from the set of chambers to polynomial functions on \( H^{2}(X) \). However it is not known that these chamber-preserving paths always exist (although it is very plausible), and this was the gap in Kotschick’s earlier paper.
In [7] Kotschick and Morgan study the difference term appearing in a path of metrics when \( \omega_{g(t)} \) crosses a wall \( c^{\perp} \). This involves the topology of the compactified moduli spaces (2). They extend the picture to define an analogous space with strata \( \mathcal{B}_{k-l}\times s^{l}(X) \) where \( \mathcal{B}_{k-l} \) is the space of all connections (modulo gauge equivalence). In this way they were able to show that the difference term depends only on the wall \( c^{\perp} \). So if a path \( \omega_{g(t)} \) begins and ends in the same chamber the numbers (1) for the initial and final metrics agree, because the overall number of crossings of each wall, counted with sign, must be zero. Thus they proved that the construction does indeed give a map from the set of chambers to polynomial functions.
Kotschick and Morgan also obtained a lot of information about the difference terms. They conjectured that these are given by universal formulae involving elementary homological invariants of \( X \), \( k \) and the cup-product \( c^{2} \). Finding the exact formulae was a problem of a similar nature to the blow-up problem — a direct attack requiring a detailed understanding of the structure of the compactified moduli spaces. In [e9] Göttsche showed that the blow-up formula of Fintushel and Stern could be used to obtain an explicit formula for the difference terms. (More precisely, Göttsche showed this assuming the Kotschick–Morgan conjecture on the existence of a universal formula.) The method was to consider wall-crossing on \( X \) and \( X\sharp{\overline{\mathbf{C}\mathbf{P}^{2}}} \) and use the blow-up formula to obtain recursive relations which ultimately determine all the difference terms, in a similar spirit to Fintushel and Stern’s work. As in that work the wall-crossing formulae which emerge involve classical special functions and modular forms and fit in with the Seiberg–Witten theory.
Remarks
\( \bullet \) Wall-crossing phenomena have appeared in many other contexts over the past few decades; see for example [e18].
2. Topology of complex algebraic surfaces
We now go back to the work of Friedman and Morgan, which they began in 1985 in Berkeley. To set the scene, recall that well-established theory in complex geometry shows that a simply connected compact complex surface is, up to deformation, either rational or elliptic or of general type. The first two classes overlap because if we blow up the projective plane at the nine-points of intersection of two generic cubic curves the resulting rational surface \( S \) has a fibration over the Riemann sphere induced from the pencil of cubics in the plane through these nine points. The Dolgachev surfaces \( S_{p,q} \) are defined by performing logarithmic transformations on two fibres, with multiplicities \( p \) and \( q \). (Topologically, a logarithmic transformation is the product with \( S^{1} \) of the operation which creates a multiple fibre in a Seifert-fibred 3-manifold. It is based on the fact that if a cyclic group acts in the obvious way on \( S^{1}\times D^{2} \), by rotation of both factors, then the boundary of the quotient is again \( S^{1}\times S^{1} \).) The surface \( S_{p,q} \) is again elliptic but there are two multiple fibres \( F_{p}, F_{q} \). If \( p,q \) are coprime (which we will assume from now on), Dolgachev showed that \( S_{p,q} \) is simply connected and it is then straightforward to see that it is homotopy equivalent to \( S \), hence, by the work of Freedman, homeomorphic. The general problem was to understand the diffeomorphism classification.
The main result of the first Friedman–Morgan papers [3], [4] is that the collection of manifolds \( S_{p,q} \) contain infinitely many diffeomorphism types. In other words, the same compact topological manifold \( S= \mathbf{C}\mathbf{P}^{2}\sharp 9 \overline{\mathbf{C}\mathbf{P}^{2}} \) supports infinitely many inequivalent smooth structures. This is in sharp contrast to the higher-dimensional case in which there are only finitely many smooth structures on the same topological manifold. I had shown, around the end of 1984, that \( S_{2,3} \) is not diffeomorphic to \( S \). This was an application of an instanton invariant in the case \( b^{+}=1 \), so involving wall-crossing, but the moduli spaces involved had low dimension (in fact 2) so the difficulties discussed in the previous section were mild. For complex algebraic surfaces the instanton moduli spaces can be identified with moduli spaces of stable holomorphic bundles, opening the way to doing calculations of invariants through algebraic geometry.
In [3], [4] Friedman took these ideas much further. In the algebraic geometry formulation, they needed to study stable rank 2 holomorphic bundles \( E \) over \( S_{p,q} \) with Chern classes \( c_{1}=0 \), \( \,c_{2}=1 \). For a suitable line bundle \( L \) they find that \( E\otimes L \) has a unique nontrivial holomorphic section \( s \), up to scale. They show that \( s \) has a single zero in \( S_{p,q} \) and this is constrained to lie on one of the exceptional fibres \( F_{p}, F_{q} \). In this way they found that each connected component of the moduli space can be identified as a set with either \( F_{p} \) or \( F_{q} \). There is a serious complication however in that, regarded as spaces of solutions of the instanton equation, these moduli spaces are not always “cut out transversely”. They are endowed with multiplicities which need to be taken into account in computing the invariants. However the multiplicities are always positive integers and this sufficed for Friedman and Morgan’s purposes. Another serious complication is that the invariants depend on chambers, so to deduce that \( S_{p,q} \) is not diffeomorphic to \( S_{p^{\prime},q^{\prime}} \) they had to show that the invariants are different for any matching of the chambers. Overcoming these difficulties, they proved that there is an integer function \( n(p,q) \) with \( n(p,q)\geq pq-p-q \) such that if \( S_{p,q} \) is diffeomorphic to \( S_{p^{\prime},q^{\prime}} \) then \( n(p,q)= n(p^{\prime},q^{\prime}) \). This immediately implies the statement about infinitely many diffeomorphism types. (The multiplicities were calculated later by Bauer [e4] and the complete diffeomorphism classification of Dolgachev surfaces was obtained later by Friedman [e8] and Bauer, in the nonsimply connected case, or via Seiberg–Witten theory.)
Another early paper of Friedman and Morgan was their survey [2] with influential “conjectures and speculations” concerning the differential topology of algebraic surfaces. These include the following statements (here somewhat simplified):
Conjecture 1. The map from deformation classes of surfaces to diffeomorphism classes is finite to 1.
Speculation A. The above map is actually one-to-one.
Conjecture 2. For a minimal surface with Kodaira dimension \( \kappa \geq 0 \) the canonical class is preserved up to sign by oriented diffeomorphisms.
Speculation B. If there is an embedded 2-sphere with self-intersection \( -1 \) in a surface with \( \kappa\geq 0 \) then there is a holomorphic sphere in the same homology class.
Speculation C. Any compact simply connected 4-manifold is diffeomorphic to a connected sum of complex surfaces, with possibly reversed orientations.
The work of Friedman and Morgan discussed above established Conjecture 1 for Dolgachev surfaces. Around the same time as the survey, Friedman, Morgan and Moishezon [1] established Conjecture 2 in many cases (for example, complete intersections of general type). Speculation A was disproved by Manetti [e14] using branched covers of rational surfaces. (Manetti’s examples suggested that there could be some correct variant of the speculation, with an extended notion of deformation equivalence.) Speculation C was disproved by an example of Gompf and Mrowka [e5] who constructed a manifold homotopy-equivalent to the K3 surface but not itself a complex surface. (Their construction used the differentiable version of logarithmic transformation applied to nonholomorphic 2-tori in a K3 surface.) Speculation B was proved using Seiberg–Witten theory.
There was much work on these conjectures in the ensuing decade, by Friedman–Morgan and other mathematicians. From complex surface theory, Conjecture 1 reduced to the case of elliptic surfaces, and Friedman and Morgan attacked this using the instanton invariants and a general algebro-geometric study of holomorphic bundles over elliptic surfaces. Their results are of considerable interest, independent of the application to instanton invariants and differential topology. Holomorphic vector bundles over an elliptic curve were classified in the 1950s by Atiyah. The generic bundle is a direct sum of line bundles. Friedman and Morgan applied this fibrewise on an elliptic surface: roughly, for a vector bundle of rank \( r \) the line bundles in the direct sum define an \( r \)-fold cover of the Riemann sphere, embedded in the associated Jacobian fibration. But there are many subtleties and difficulties arising from the singular fibres and fibres over which the vector bundle is nongeneric. Overcoming these, Friedman and Morgan were able to describe Zariski open subsets in moduli spaces of stable bundles, which were sufficient to calculate certain of the invariants (1). This work, and much else, appeared in the monograph [8]. Combined with work of Friedman and Qin [e7] the results of Friedman and Morgan in [8] completed the proof of the “Van de Ven conjecture”: the Kodaira dimension of a complex surface is a differentiable invariant. (This conjecture is closely related to the Friedman–Morgan Conjectures and Speculations above.)
3. Seiberg–Witten invariants
The Seiberg–Witten invariants of 4-manifolds were defined by Witten in 1994. This transformed the field, providing much simpler proofs of nearly all of the results obtained using the instanton theory, completing the solution of many outstanding problems which had been attacked using that theory and leading to new unexpected results, such as connections to positive scalar curvature and symplectic topology.
For simplicity, suppose that \( X \) is compact 4-manifold with a spin structure. The Seiberg–Witten equations are partial differential equations for a connection on a complex line bundle \( L \) over \( X \) and spinor field with values in \( L \). The moduli space \( \mathcal{M} \) of solutions is compact. Invariants are defined by the same general procedure as in the instanton case: if the moduli space has dimension \( 2d \) then there is an invariant \( \langle \mu^{d}, \mathcal{M}\rangle \) defined using a class \( \mu\in H^{2}(\mathcal{M}) \) coming from the universal bundle. The dimension \( 2d \) is given by a formula involving \( c_{1}(L)^{2} \). As before, the theory is different in the case \( b^{+}=1 \) due to wall-crossing. When \( b^{+} > 1 \) the case of primary interest is for a line bundle \( L \) such that \( d=0 \), so we are just counting (with sign) the points in the moduli space. A manifold is called of simple type if all the invariants for \( d > 0 \) vanish: one of the main open conjectures in the field is that for all manifolds with \( b^{+} > 1 \) are of simple type.
John Morgan was one of the leaders in the rapid developments of this theory. He wrote a lecture note volume [e14] which is a standard text in the subject, and he contributed to a drive to elucidate Quantum Field Theory, and in particular the physics background to the Seiberg–Witten theory, for mathematicians [e13]. He wrote many research papers, of which I will just discuss a small selection below.
(i) One of the main strands in this mathematical area (gauge theory and low-dimensional topology) is the description of invariants (either instanton or Seiberg–Witten) for manifolds \( X= X_{1} \cup_{Y} X_{2} \): i.e., \( X_{1} \) and \( X_{2} \) have the same 3-manifold boundary \( Y \) and the closed manifold \( X \) is obtained by gluing them along their boundaries. The standard strategy is to consider Riemannian metrics on \( X \) containing a long cylinder \( (-T,T) \times Y \). The paper [11] of Morgan, Mrowka, Szabó is an important contribution to this strand: it treats the Seiberg–Witten invariants in the case when \( Y \) is a 3-torus. From the point of view of analysis one main component is to show that finite-energy solutions of the Seiberg–Witten equation on a manifold with an infinite cylindrical end have a well defined limit at infinity. The analysis arguments are related to those in earlier work of Morgan, Mrowka and Ruberman [6] in the instanton theory. A prominent application of the Seiberg–Witten gluing formula of Morgan, Mrowka and Szabó came in the work of Fintushel and Stern on “knot surgery” [e12], producing a huge collection of manifolds homotopy equivalent to the K3 surface.
(ii) The paper [12] of Morgan and Szabó bears on the fundamental issue of the failure of the h-cobordism theorem in dimension four. Let \( W \) be a h-cobordism between simply connected 4-manifolds \( X_{0}, X_{1} \). One can arrange the following picture, going back to work of Wall in the 1960s. There is a middle level \( X_{1/2}\subset W \) containing ascending a collection of ascending 2-spheres \( A_{1}, \dots \), \( A_{n} \) and descending spheres \( B_{1},\dots \), \( B_{n} \) with algebraic intersection numbers \( A_{i}\cdot B_{j}=\delta_{ij} \). If the geometric intersections are the same as the algebraic ones then \( W \) is the trivial cobordism and \( X_{0} \) is diffeomorphic to \( X_{1} \). Morgan and Szabó define the complexity of the cobordism to be the minimum over all such pictures of the excess of the geometric over algebraic intersections. So the complexity is zero if and only if the \( h \)-cobordism is trivial. Their main result is that Seiberg–Witten invariants of \( X_{0}, X_{1} \) defined by moduli spaces of dimension sufficiently large compared to the complexity are equal. This is not very useful if \( X_{0}, X_{1} \) are of simple type (and so for all known examples with \( b^{+} > 1 \)) but when \( b^{+}=1 \) there are examples of pairs of 4-manifolds which are distinguished by invariants defined by high-dimensional moduli spaces. In such cases Morgan and Szabó deduce that the complexity of an h-cobordism between them must be large, refining the statement that it is nonzero.
Let \( N \) be a neighbourhood in \( X_{\tiny{1/2}} \) of the union of the ascending and descending spheres, with smooth boundary \( Y \). Then there are surgery descriptions
In other words, \( X_{1} \) is obtained from \( X_{0} \) by cutting out a subset \( N_{0} \) and gluing in \( N_{1} \), with the same boundary \( Y \). The proof of the statement about Seiberg–Witten invariants goes through artful gluing arguments with these descriptions. The bound “sufficiently large” for the dimension in the statement depends on the solutions of the Seiberg–Witten equations on \( Y\times \mathbf{R} \), but for a fixed complexity only finitely many different manifolds \( Y \)can appear.
The question of whether there are bounds on the complexity of h-cobordisms for manifolds with \( b^{+} > 1 \) seems to be open.
(iii) The spin representation in 4-dimensions is quaternionic. This implies that the Seiberg–Witten solutions for line bundles \( L, L^{*} \) can be identified and the invariants are the same. For the trivial bundle we get an involution \( J:\mathcal{M}\rightarrow \mathcal{M} \) on the moduli space of solutions. If \( X \) is simply connected the only fixed point is the solution given by the trivial connection and zero spinor field. The consequences of this for the Seiberg–Witten invariants were worked out by Morgan and Szabó in the note [10]. The condition that the dimension (more precisely the “virtual dimension”) of the moduli space for the trivial line bundle is zero gives
for some integer \( m\geq 0 \). To find the contribution of the trivial solution to the Seiberg–Witten count one has to study the local “Kuranishi model”. Without much loss of generality we can suppose that the space of positive-spinor solutions of the Dirac equation has quaternionic dimension \( m+1 \) (and is zero for the negative spinor solutions). Then the Kuranishi map is a map
which is equivariant for the group actions of \( S^{1} \) acting trivially on \( \mathbf{R}^{3+4m} \) and \( \{1,J\} \) with \( J \) acting as quaternion multiplication on \( \mathbf{H}^{m+1} \) and \( -1 \) on \( \mathbf{R}^{3+4m} \). The main case discussed in [10] is when \( m=0 \), so the manifold \( X \) is a homotopy K3 surface. Morgan and Szabó show that the contribution from the trivial solution is then odd and deduce that the Seiberg–Witten invariant of \( X \) is odd. For example, in the standard case of a K3 surface with a Calabi–Yau metric, the Kuranishi map is
(identifying \( \mathbf{H}=\mathbf{C}^{2} \), \( \mathbf{R}^{3}= \mathbf{R} \times \mathbf{C} \)). To find the contribution we need to count the points in \( \kappa^{-1}(\eta)/S^{1} \) for generic \( \eta \). Taking \( \eta=(1,0) \) shows that this count is 1.
At the end of the note, Morgan and Szabó mention that the situation is different when \( m > 1 \): the contribution of the trivial connection is even, so the same for the Seiberg–Witten invariant. They omit the proof, but this was written down by Bauer in [e16], in a more general setting. This result has an interesting consequence in symplectic topology (as pointed out by Bauer). An intriguing question there is the classification of simply connected compact symplectic 4-manifolds with zero first Chern class. The only known examples are the standard symplectic structures on K3 surfaces. If \( X \) is such a manifold it falls into the class discussed above. The Morgan–Szabó result implies that the Seiberg–Witten invariant would be even if \( m > 1 \) but this contradicts one of Taubes’ results that the invariant is 1. So we deduce that \( X \) is homotopy equivalent to a K3 surface (but possibly with a different differentiable structure, or a nonstandard symplectic structure on the standard smooth structure).
