Celebratio Mathematica — Morgan

John Morgan’s work in gauge theory

by Tomasz Mrowka and Daniel Ruberman

Introduction

John Morgan’s work in gauge theory is intimately tied to the early years of gauge theory as a focus of low-dimensional topologists. Gauge theory had been studied by physicists and geometers for more than a decade when Simon Donaldson stunned the mathematical world in 1982 by proving that some simply connected topological manifolds with a definite intersection form could not admit smooth structures. Such topological manifolds had only recently been constructed by Mike Freedman. Donaldson’s argument was based on an analysis of the moduli space of self-dual connections on an $ \operatorname{SU}(2) $ bundle over an oriented 4-manifold with positive-definite intersection form. Donaldson’s investigations of manifolds with indefinite form soon produced many examples of compact 4-manifolds with more than one smooth structure. Over a short period, 4-manifold topology was transformed, as topologists learned to use this powerful new machinery. John was at the forefront of this effort and helped set the direction of the field for the next decade. We’ll intersperse a description of John’s contributions with some personal recollections of that period and our interactions with John.

Early days

In 1984–85, MSRI hosted a full-year program in low-dimensional topology, organized by Bob Edwards, Rob Kirby, Bill Thurston, and John. In the fall, there was a well-attended learning seminar on gauge theory. The goal was to understand Donaldson’s new paper [e14] constructing and calculating an invariant $ \Gamma $ that distinguished the Dolgachev surface $ S(2,3) $ from the rational elliptic surface

\[ S = \mathbb{C}P^2 \, \# 9 \overline{\mathbb{C}P}^2 \]

as smooth manifolds. (To conform better with complex geometry, conventions had by then switched to consider anti-self-dual connections.) Both being simply connected manifolds with odd intersection form of type $ (1,9) $, they are homeomorphic by the work of Freedman [e6]. John was not around much in the fall, but late in the semester he visited Berkeley and sat in on a few meetings. Arriving for the spring term, he brought a collection of fantastic new results, joint with Bob Friedman, about the smooth topology of elliptic surfaces. It was astonishing that they had gotten so far in such a short time, when the rest of the participants were still struggling to understand the flood of new ideas from analysis and geometry. John’s lectures about his developing work made clear the central issues, and were highly influential on both of us.

The first two Friedman–Morgan papers extend Donaldson’s work considerably, and deal with the classification and automorphisms of the general Dolgachev surfaces $ S(p,q) $. By definition, these are given by log transforms of orders $ p $ and $ q $ on the rational elliptic surface. When $ p $ and $ q $ are relatively prime, they are simply connected and, as above, homeomorphic to $ \mathbb{C}P^2\, \# 9 \overline{\mathbb{C}P}^2 $. The canonical divisors of $ S(p,q) $ and $ S(p^{\prime},q^{\prime}) $ show that they are distinct as complex manifolds if $ \{p,q\} \neq \{p^{\prime},q^{\prime}\} $, but the smooth classification was a complete mystery at the time. (See, for instance, [e12] for the pre-gauge theory state of the art.) Friedman and Morgan didn’t quite classify the Dolgachev surfaces, but they showed that the divisibility of the canonical class, $ pq-p-q $, is a differentiable invariant, and deduced that the map from pairs $ \{p,q\} $ to manifolds $ S(p,q) $ is finite-to-one. Moreover, their rough classification was preserved by blowing up. This was a remarkable achievement, giving the first examples of compact 4-manifolds with infinitely many smooth structures. (See also [e13].) One can readily discern many of the main themes of John’s subsequent work in gauge theory in the two papers [4], [5]: the classification of elliptic surfaces, the degree to which a priori algebro-geometric invariants are actually $ \mathcal{C}^{\infty} $ invariants, the key role of blowing up and the study of embedded spheres in 4-manifolds.

The gauge-theoretic input for these results comes from Donaldson’s paper on $ S(2,3) $, but Friedman and Morgan go much further. Donaldson’s invariant $ \Gamma(X) $ is defined by evaluating the first Chern class of a line bundle over a 2-dimensional moduli space of anti-self-dual $ \operatorname{SU}(2) $ connections on a bundle $ E $ with $ c_2(E)=1 $. The invariant has several complicating features that made it a particularly tricky starting point. To define $ \Gamma(X) $, the moduli space has to be compactified to have a fundamental class. Because $ b_2^+(X) = 1 $, the invariant has a dependence on the choice of the period point $ \omega_g \in H^2(X;\mathbb{R}) $ corresponding to a $ g $-self-dual 2-form of norm 1. This dependence is dealt with via a wall-crossing formula that explains how $ \Gamma $ changes when $ \omega_g $ passes through a “wall” $ W_\alpha $, the codimension-one set of self-dual forms that are orthogonal to an integral cohomology class $ \alpha $ with $ \alpha \cup \alpha = -1 $. The condition that $ \omega_g $ is on a wall $ W_\alpha $ implies that there is a solution to the ASD equations on $ E $ whose structure group reduces to $ \operatorname{U}(1) $; such solutions are called reducible. So rather than being a single integer, $ \Gamma(X) $ is a function on the chambers, defined as the components of the complement of the walls in the positive cone of $ H^2(X;\mathbb{R}) $.

Donaldson calculated the value of $ \Gamma $ for the rational surface $ S $ and $ S(2,3) $ in a particular chamber, using his recently proved correspondence [e11] between the ASD moduli space and a moduli space of stable vector bundles on these algebraic surfaces. The wall-crossing formula shows that $ \Gamma(S) $ and $ \Gamma(S(2,3)) $ can’t agree on any chamber, proving that the two manifolds are not diffeomorphic. Friedman and Morgan took this line of argument and ran with it at an astonishing pace. Much of their work deals with the fine structure of the set of chambers and geometry of the walls. As they point out in the introduction to [4] (still very much worth a read) this is in principle an arithmetic question, but the analysis is done using methods of algebraic geometry. More algebraic geometry goes into the description of the algebro-geometric moduli spaces of stable vector bundles, carried out in [5].

The outcome of all of this work largely classifies the Dolgachev surfaces and their blowups, up to diffeomorphism. Blowing up at a point increases $ b_2^{-} $ by one. When $ b_2^{-} > 9 $, the chamber structure becomes combinatorially very complicated, with strong consequences. In particular, since the chamber structure is acted on by diffeomorphisms, this imposes strong restrictions on automorphisms of the intersection form that are realized by diffeomorphisms. In particular, Friedman and Morgan show that for $ \mathbb{C}P^2 \,\# n \overline{\mathbb{C}P}^2 $ with $ n > 9 $, the group of automorphisms realized by diffeomorphisms has infinite index in the automorphism group of the form. Similar (but less precise) results hold for any 4-manifold of this homotopy type.

The underlying theme of the Friedman–Morgan work was laid out in an influential AMS Bulletin article [3] and other articles [2], [6]: to what degree are algebro-geometric properties of a smooth complex surface determined by its smooth topology? This question had been considered since the earliest days of 4-manifold topology, but most of the progress lay in the topological description of algebraic surfaces. See [e4], for example, for a summary of what was known just before Donaldson’s and Freedman’s work. The main properties considered in the Friedman–Morgan articles were the deformation type, the minimal model and smooth invariance of minimality, the canonical class (up to sign), the Kodaira dimension, and the validity for general 4-manifolds of restrictions on the intersection form known to hold for algebraic surfaces (and connected sums thereof). Although not discussed in [3], in lectures around that time John (motivated by the work we did with him) advertised another important problem: Is there a constraint on the genus of smoothly embedded representative, $ \Sigma $ of a two dimensional homology class in a four manifold $ X $ in analogy to the adjunction formula for complex curves in complex surfaces. The prototypical example of this was the conjecture attributed to Thom asserting that a complex curve minimizes the genus of an embedded surface in a given 2-dimensional homology class in $ \mathbb{C}P^2 $. John asked if, under suitable hypothesis on $ X $, one had

\[ \Sigma\cdot \Sigma \le 2g(\Sigma)-2. \]

Shortly after the appearance of [e14], Donaldson introduced [e16] his polynomial invariants for manifolds with $ b_2^+ > 1 $, using higher-dimensional moduli spaces of connections on bundles with larger values of $ c_2 $. This allowed him to avoid reducible solutions and to deal more effectively with issues of compactness. Kotschick [e17] (see also [e15]) extended these ideas to the case of $ b_2^+ =1 $, where again there is wall-crossing to account for reducibles. John’s article with Kotschick [12] explored this case in greater depth, and in particular posed the influential Kotschick–Morgan conjecture that the wall-crossing formula depends only on the homotopy type of the underlying manifold. In the hands of John (and many others) the polynomial invariants proved to be an effective tool for investigating the topology of algebraic surfaces. However, this was not at all straightforward, as the polynomial invariants proved to be difficult to compute. The main available tools were computations with stable vector bundles, and the beginnings of cut/paste methods; even a single coefficient of a Donaldson polynomial could require a hundred or more pages of strenuous calculations. John made numerous contributions to both of these directions.

Completing the differentiable classification of elliptic surfaces became an important test for this new method. In a series of papers, starting with his work with Friedman (for the case $ b_2^+=1 $) and culminating in [9], [15], [11], John found enough coefficients of the Donaldson polynomial for simply connected elliptic surfaces to completely classify them in terms of topological invariants and the order of their multiple fibers. (See also [e21], [e30], [e22].) His work with Kieran O’Grady [11] for the case of $ b_2^+ = 3 $ was done via calculations with moduli spaces of stable vector bundles, while the papers with Mrowka [9], [15] made use of cut and paste methods described below. The calculations via moduli spaces of stable vector bundles required an additional step. In brief, while Donaldson had identified the moduli spaces, for the purposes of calculating the Donaldson invariants in terms of stable bundles, one needed to know that their natural compactifications are compatible. This was carried out independently by John [10] and Jun Li [e19].

Our work with John

The foundational work of Uhlenbeck [e8], [e7] and Taubes [e5], [e10], introduced the yin and yang ideas of compactness and gluing. Uhlenbeck showed that sequences of ASD connections on a manifold without a convergent subsequence give rise to a limiting ASD connection on manifolds that themselves are limits of sequences of related metrics on the original manifold. Taubes laid down the paradigm to reverse the procedure: given ASD connections on the limiting manifolds, he produces sequences of ASD connections on the original manifold that limit to the given one (or finds obstructions to doing so). These ideas were already crucial to Donaldson’s original work [e9], and suggested that gauge-theoretic invariants for a manifold $ X $ could be computed by splitting $ X $ into pieces and understanding how solutions on $ X $ could be recovered in terms of solutions on the pieces. Donaldson’s connected sum theorem [e16] put this idea into practice, showing that the polynomial invariants of a connected sum of manifolds, each having positive $ b_2^+ $, must vanish. Donaldson also showed that the polynomial invariants of $ X $ can be recovered from the invariants of a blowup $ X \#\overline{\mathbb{C}P}^2 $. This distinction between the effects of adding $ \mathbb{C}P^2 $ or $ \overline{\mathbb{C}P}^2 $ was very much in line with a dictum of John’s (from an unpublished manuscript [7]): It has long been realized by those studying the topology of complex surfaces that there seems to be a polarization, or a preferred sign. John was not the only one advocating this point of view; the connection between gauge theory and complex geometry was central to Donaldson’s work from the very beginning.

A natural idea was to extend this method (referred to as cutting and pasting, a nonlinear analogue of the Mayer–Vietoris principle) to compute Donaldson’s invariants for manifolds split along more complicated 3-manifolds into pieces with common boundaries. Many different people took up this idea and in particular Floer’s insights led to the most generally useful point of view [e33]. Tom took a direct analytic approach and in his thesis began developing some of the analytic tools that eventually became incorporated in our book [13]. Tom was very fortunate on the one hand to learn the analysis from Taubes and on the other be able to discuss these ideas and learn topology and many other parts of mathematics from John as well as learn something of the tradecraft of writing mathematics.

The natural way to study gauge theory on a manifold $ X $ with boundary seemed to be to consider the associated manifold with a product end $ \partial X \times [0,\infty) $. A good deal of foundational work was necessary to get this off the ground, including the monograph [13] that we wrote with John. In the late 1980s, John was a frequent visitor to Harvard and MIT, doing lots of mathematics and (successfully!) courting Ellen Corenswet. During this time he gave a well-attended course on 4-manifold topology and the emerging Donaldson invariants. Richard Stong, with help from Tom, took notes for this course. John’s lectures were always crisp and clean, and became a great reference for the field in the early days. The lectures covered accounts of the basics of smooth 4-manifold topology, basics of connections and principal bundles, the algebraic topology of the moduli spaces of connections, analysis behind the construction of the moduli spaces of ASD-connections, some gluing theory, and the connection with complex geometry, culminating with the definition of Donaldson invariants. Many of the topics were done differently than in the published accounts. Among the many ideas in these lectures was a basic method for defining the Donaldson invariants even when moduli spaces (and their compactifications) could have nasty singularities. A few years later similar ideas were used in the theory of pseudo-holomorphic curves and became known as the virtual fundamental class.

John had been rethinking Donaldson’s connected sum theorem and blowup formula. At Cliff Taubes’ invitation, he gave a short series of lectures which greatly clarified many details. A few days later, Danny asked John (at a party at Ellen’s house in Newton) whether there should be similar theorems if one was splitting the 4-manifold along a 3-manifold whose fundamental group had only reducible $ \operatorname{SU}(2) $ representations. This condition is equivalent to requiring all flat $ \operatorname{SU}(2) $ connections are reducible–an elementary, but crucial, property of the 3-sphere. An intended application was to show that if some Donaldson invariant of $ X $ was non-zero, then $ X $ contained no essential 2-sphere of non-negative self-intersection. John said, “Let’s talk on Monday.”

After some thought, we realized that the application to spheres could be easily deduced from the blowup formula, and turned our attention to embedded tori, where the natural 3-manifold to consider is the circle bundle over $ T^2 $ with Euler class equal to the self-intersection of the torus. When the Euler class is even, there are only reducible $ \operatorname{SU}(2) $ connections, but now (crucially) these appear in a higher-dimensional non-smooth family. We tried to understand what the gluing picture would look like. The answer we expected was that if $ X $ was split along a 3-manifold $ N $ into pieces $ X_1 $ and $ X_2 $, then there should be a “map at infinity” $ \partial_i $ sending an ASD connection on $ X_i $ to a flat connection on $ N $, and gluing theory would show that the moduli space on $ X $ is (at least locally) some sort of fiber product of the moduli spaces on the $ X_i $ over the flat connections on $ N $.

A simple test of this (assuming sufficient smoothness for these moduli spaces, and transversality of the maps $ \partial_i $) was to compute the dimensions of all the moduli spaces and see if they added up properly. In doing this, we implicitly made (as we understood after the fact) the assumption that an ASD connection on a manifold with a cylindrical end the norm of whose curvature was square-integrable would in fact decay exponentially to a flat connection. Some tentative calculations based on [e3], [e2] suggested that something was off and that the most reasonable explanation was that there were in fact $ L^2 $ connections without exponential decay.

We trooped up to Cliff’s office to ask if there was such a thing as an $ L^2 $ moduli space, and he said yes (but that he hadn’t told anyone about it). He spent some time trying to explain where our “extra dimensions” came from; see [e20] for his perspective. Eventually, Cliff suggested that Tom (who was already studying the general gluing problem in his thesis) would be an ideal collaborator in this effort.

We initially worked in person at Harvard (often, at John’s insistence, over burgers at Charlie’s Kitchen in Harvard Square) but as time went on, work was mostly accomplished on occasional cross-country visits and many phone calls. Eventually an understanding of the behavior of the Chern–Simons flow on a cylinder emerged, and we were able to give a general picture for manifolds with arbitrary cylindrical ends. Applied to the case of circle bundles, this did indeed give restrictions on embedded tori. A different approach to Donaldson theory in relation to embedded surfaces by Tom with Peter Kronheimer [e18], [e27] [e26] yielded far stronger results. Just over the horizon was a general adjunction formula, and resolution of the Thom conjecture [e24], [17] using the new Seiberg–Witten invariants (see below).

The writing of the book took several more years. One of John’s magical powers we learned during this collaboration was his ability to suspend some work for a long period of time and be ready to pick up where we left off without a hitch. This really kept us on our toes. John moved back to New York, Danny stayed in Boston and Tom moved to Stanford. John and Tom talked on the phone often for quite a while to keep the writing going, interspersed with some subset of the three of us meeting, often at John’s apartment in New York. It is worth recalling that back in those days phone calls actually cost real money. Unfortunately Tom was not super organized about this and the calls were often quite long. At some point the head administrator in the Stanford Math Department called Tom in and said that he’d run up something like a \$1,000 dollar phone bill and asked how he’d like to pay for it. Tom slunk away without much of an answer and fortunately for him was never asked again. John was also now married to Ellen and living in NYC and it was a pleasure to watch John’s sartorial taste finally catch up to his mathematical taste, no doubt due to Ellen’s influence.

Seiberg–Witten theory

John was one of the first mathematicians to perceive — with spectacular consequences — the power of the Seiberg–Witten invariants. Guided by Taubes’ Harvard lecture explaining the ideas of [e25], [e23] to the mathematical community in the early fall of 1994, John, along with Zoltán Szabó and Taubes [17], produced a proof of the Thom conjecture. This was done at essentially the same time by Kronheimer and Mrowka [e24] (whose announcement, in Kronheimer’s words, preceded that of [17] by originating in a more favorable time zone.) Both [e24] and [17] contained a version of the adjunction inequality

\[ \chi(C)+C\cdot C \le -|\langle c_1(\mathfrak{s}),C\rangle|, \]

where $ C $ is a smooth embedded surface with $ C\cdot C\ge 0 $ and $ \mathfrak{s} $ is a $ \mathrm{Spin}^c $ structure with non-vanishing Seiberg–Witten invariants satisfying

\[ c_1^2(\mathfrak{s})=2\chi(X)+3\sigma(X) \]

(i.e., when the formal dimension of the moduli space for $ \mathfrak{s} $ is zero.) A similar inequality had been proved earlier using Donaldson invariants for many complex surfaces by Peter and Tom [e26] but the Seiberg–Witten version greatly expanded the scope while simultaneously simplifying the proof. In a sense the vision John suggested in the AMS Bulletin article [3] now came into sharp focus. The Seiberg–Witten basic classes, defined for any smooth oriented four manifold, generalize the canonical class of a complex surface; the adjunction formula for the genus of an algebraic curve becomes the adjunction inequality.

The rapid progress in 4-manifold topology resulted from the power of the new Seiberg–Witten theory, but also from the groundwork laid by the previous decade of work in Donaldson theory. Witten [e23] showed how to solve the Seiberg–Witten equations on a Kähler surface, leading to efficient proofs of the $ \mathcal{C}^\infty $ invariance of the canonical class and plurigenera [18], [21], and in general to the resolution of most of the conjectures and speculations in [3]. (A notable exception is the still-standing $ 11/8 $ conjecture, on which important progress has been made [e32].) Friedman and Morgan were not alone in applying Seiberg–Witten theory to complex surfaces; see for instance [e29], [e30], [e28], [e23]. For topological applications, it was important to adapt the gluing approach to Donaldson invariants to the setting of Seiberg–Witten theory. Partial results were already explored in [17], and Morgan–Mrowka–Szabó [19] (see also [e31]) resolved the crucial case of gluing along 3-tori.

John as expositor

Every field that John has entered has been deeply impacted by his expository talents (see, for instance, his contributions to the Smith Conjecture volume [1]), and so it was with gauge theory and its applications to low-dimensional topology. He wrote several influential accounts as the theory grew and changed. His book with Friedman [14] set out to solidly lay the foundations of applications of gauge theory to the study of 4-manifolds, focusing particularly on complex surfaces. The book starts with an introduction to the Kodaira classification of complex surfaces with, not surprisingly, an emphasis on elliptic surfaces. It is rather taken for granted these days that we know models of the smooth 4-manifolds underlying elliptic surfaces. However it is a quite non-trivial fact, due essentially to Kodaira, that fixing some essentially topological invariants (genus of the base, the Euler characteristic, the number of multiple fibers and their multiplicity), the moduli space of such surfaces is connected and in particular the underlying manifolds are all diffeomorphic. Kodaira’s work is spread over a number of different papers. Friedman and Morgan give a very clean, compact, and readable account of this result.

The book also contains an in-depth discussion of the definition of the Donaldson polynomial invariant. Key to the construction of the Donaldson invariant is finding a compactification carrying a fundamental class. A messy issue was handling the bottom stratum of the compactification where the limit was the trivial connection. (A trick to avoid this issue via the blowup formula was introduced by John and Tom [8].) When $ b^+ > 0 $ the trivial connection isn’t a smooth point of the moduli space and there is an obstruction to gluing (as discovered by Taubes, building on ideas of Kuranishi [e1]), making the construction of the compactification delicate. The book formalizes Taubes’ construction as the thickened moduli space. This is a tool for understanding the ends of moduli space: one finds open subsets of the moduli space a living inside spaces of connections parametrized by a finite dimensional manifold that carries a distinguished vector bundle and section. The actual solutions are zeros of the section but one might not have control over transversality of the section. This data provides enough structure to construct what is called a $ \delta $-approximation of the fundamental class. (This was covered also in John’s lectures at Harvard around 1988.) The book covers many other important topics culminating in building up computations of some Donaldson invariants of algebraic surfaces.

John’s notes on Seiberg–Witten invariants [16], derived from lectures he gave as the theory was just developing, were another expository gift to the community. The book lays out the basics of the theory, starting from $ \mathrm{Spin}^c $ structures and going through Witten’s solution to the Seiberg–Witten equations for Kähler manifolds. The writing is to some degree pitched towards those with some familiarity with gauge theory, yet even for novices provides an efficient and clear route to the key ideas that remains useful to this day. John gave lecture courses at a number of conferences and workshops, making gauge theory accessible to a new generation of students and researchers. See [22], [20], [25], [24], [23].

John’s Ph.D. students in gauge theory

While John was actively working on gauge theory, he had 11 Ph.D. students in the area, many of whom have gone on to distinguished careers. Some of their early work reflects John’s interests, but they have also emulated John by opening many new research directions. We list those students below, with degrees from Columbia University, unless otherwise noted.

Paolo Lisca (1991) On smoothly embedded tori in four-manifolds
Hongjie Yang (1992)
Transition functions and a blow-up formula for Donaldson polynomials
Thomas Leness (1994)
Blow-up formulae for SO(3)-Donaldson polynomials
Peter Steven Ozsváth (1994, Princeton University)
On blowup formulas For SU(2) Donaldson polynomials
András Stipsicz (1994, Rutgers University)
Computation of Donaldson invariants by cut and paste techniques
Zoltán Szabó (1994, Rutgers University)
On the smooth structures of elliptic surfaces and irreducible four-manifolds
Yuhan Lim (1995)
Computation of Donaldson invariants for elliptic surfaces of geometric genus one
Dosang Joe (1998)
Symplectic structures on connected sums with a ruled surface and product formulas for Seiberg–Witten invariants along a nilmanifold
Brendan Edward Owens (2000)
Instantons on cylindrical manifolds and stable bundles
Pedram Safari (2000)
A gluing theorem For Seiberg–Witten moduli spaces
Gregory Charles Langmead (2001)
A supersymmetric quantum field theory formulation of the Donaldson polynomial invariants

Tom Mrowka is a professor of mathematics at MIT. His completed his Ph.D. under the direction of Rob Kirby and Cliff Taubes, with John Morgan as a third advisor. He works mainly in low-dimensional topology, often jointly with Peter Kronheimer.

Daniel Ruberman is Professor Emeritus of Mathematics at Brandeis University. He wrote his Ph.D. at the University of California, Berkeley from 1977–82 under the supervision of Rob Kirby. He has worked on low-dimensional topology, especially on applications of gauge theory to 4-manifold topology.

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[e29] R. Brussee: “The canonical class and the $ C^\infty $ properties of Kähler surfaces,” New York J. Math. 2 (1996), pp. 103–146. MR 1423304 article

[e30] R. Fintushel and R. J. Stern: “Rational blowdowns of smooth 4-manifolds,” J. Differential Geom. 46 : 2 (1997), pp. 181–235. MR 1484044 article

[e31] C. H. Taubes: “The Seiberg–Witten invariants and 4-manifolds with essential tori,” Geom. Topol. 5 (2001), pp. 441–519. MR 1833751 article

[e32] M. Furuta: “Monopole equation and the $ \frac{11}8 $-conjecture,” Math. Res. Lett. 8 : 3 (2001), pp. 279–291. MR 1839478 article

[e33]S. K. Donaldson: Floer homology groups in Yang–Mills theory. Cambridge Tracts in Mathematics 147. Cambridge University Press, 2002. With the assistance of M. Furuta and D. Kotschick. MR 1883043 Zbl 0998.53057

Works

[1] J. W. Morgan: “The Smith conjecture,” pp. 3–6 in The Smith conjecture (New York, 1979). Edited by J. W. Morgan and H. Bass. Pure Appl. Math. 112. Academic Press (Orlando, FL), 1984. MR 758460 incollection

[2] R. Friedman, B. Moishezon, and J. W. Morgan: “On the $ C^\infty $ invariance of the canonical classes of certain algebraic surfaces,” Bull. Amer. Math. Soc. (N.S.) 17 : 2 (1987), pp. 283–286. MR 903733 Zbl 0627.57014 article

The results announced in this article concern certain aspects of the diffeomorphism classification of algebraic surfaces, and in particular, the role of the canonical class. We establish our results by developing a general criterion under which the possibilities for Donaldson’s polynomial invariants for smooth 4-manifolds [2] are severely limited. We then use these limitations to conclude that in many cases the canonical class of an algebraic surface is a diffeomorphism invariant up to a multiple. Two classes of surfaces satisfying our general criterion are complete intersections and simply connected elliptic surfaces with $ pg \equiv 0\, (\mathrm{mod}\ 2) $ (see Corollary 8). A third class of such surfaces are certain abelian branched coverings of $ CP^1 \times CP^1 $ which are surfaces of general type (see §4). These latter surfaces provide infinitely many examples of pairs of homeomorphic, nondiffeomorphic, simply connected surfaces of general type.

§2 gives a brief review of the part of Donaldson’s theory needed for what we do here. The material described in §3 represents the work of the first and last authors and is some evidence for a general conjecture described in [5]. The material described in §4 represents the work of the second author and will be explained in detail in [6].

[3] R. Friedman and J. W. Morgan: “Algebraic surfaces and 4-manifolds: some conjectures and speculations,” Bull. Amer. Math. Soc. (N.S.) 18 : 1 (1988), pp. 1–19. MR 919651 Zbl 0662.57016 article

[4] R. Friedman and J. W. Morgan: “On the diffeomorphism types of certain algebraic surfaces, I,” J. Differential Geom. 27 : 2 (1988), pp. 297–369. MR 925124 Zbl 0669.57016 article

[5] R. Friedman and J. W. Morgan: “On the diffeomorphism types of certain algebraic surfaces, II,” J. Differential Geom. 27 : 3 (1988), pp. 371–398. MR 940111 Zbl 0669.57017 article

[6] R. Friedman and J. W. Morgan: “Complex versus differentiable classification of algebraic surfaces,” pp. 135–139 in Proceedings of the 1987 Georgia Topology Conference (Athens, GA, 1987), published as Topology Appl. 32 : 2. Issue edited by N. Habegger and C. McCrory. 1989. MR 1007985 Zbl 0694.14013 inproceedings

[7]J. W. Morgan, T. Mrowka, and D. Ruberman: Self-intersection numbers of embedded 2-spheres in algebraic surfaces, 1991. Unpublished manuscript. misc

[8] J. W. Morgan and T. S. Mrowka: “A note on Donaldson’s polynomial invariants,” Internat. Math. Res. Notices 10 (1992), pp. 223–230. MR 1191573 Zbl 0787.57011 article

The purpose of this note is to exhibit two uses of the idea of stabilizing Donaldson polynomial invariants via the connected sum with $ \overline{CP}^2 $. The theme that we develop is that this operation allows one to avoid the anomalous behavior associated with flat connections. In some cases this simplifies arguments, and in other cases it actually permits one to extend definitions and computations which are not possible (at least directly) without the stabilization. Our first application concerns generalizing the usual $ SU(2) $-Donaldson polynomial invariants. Consider the Donaldson polynomial invariants of $ M \# \overline{CP}^2 $ associated to $ U(2) $-bundles with $ c_1 $ equal to the exceptional class $ [\overline{CP}^1] $. We show that these invariants are defined for any closed oriented smooth 4-manifold with $ b^+_2(M) > 1 $. These invariants are polynomials in the exceptional class with coefficients which are themselves polynomial invariants of $ M $. We show that, in the case when $ M $ is simply connected and the charge $ p_1 $ is sufficiently negative, the coefficient of the first power of the exceptional class agrees, up to a factor of $ (- 1)/2 $, with the usual $ SU(2) $-Donaldson polynomial invariant of $ M $. In this way we are led (after inverting 2) to a generalization of the Donaldson polynomial invariants to invariants for any closed, oriented 4-manifold with $ b > 1 $. (See Corollary 2.2 and Proposition 3.2.) Our second application is to give a simpler proof of Donaldson’s connected-sum theorem. (See Theorem 3.1 and Corollary 3.6.) Our proof uses dimension counts exactly as in Donaldson’s original argument [3] for the case when the limit connections on the sides are not flat, but, by the use of $ U(2) $-connections with nontrivial $ c_1 $, avoids the more subtle case when the connection on one of the sides is flat.

[9] J. W. Morgan and T. S. Mrowka: “On the diffeomorphism classification of regular elliptic surfaces,” Internat. Math. Res. Notices 6 (1993), pp. 183–184. MR 1224116 Zbl 0807.57015 article

[10] J. W. Morgan: “Comparison of the Donaldson polynomial invariants with their algebro-geometric analogues,” Topology 32 : 3 (1993), pp. 449–488. MR 1231956 Zbl 0801.57014 article

[11] J. W. Morgan and K. G. O’Grady: Differential topology of complex surfaces: Elliptic surfaces with $ p_g=1 $: smooth classification. Lecture Notes in Mathematics 1545. Springer, 1993. With the collaboration of Millie Niss. MR 1312610 Zbl 0789.14037 book

This book is about the smooth classification of a certain class of algebraic surfaces, namely regular elliptic surfaces of geometric genus one, i.e., elliptic surfaces with $ b_1 = 0 $ and $ b_2^+ = 3 $. The authors give a complete classification of these surfaces up to diffeomorphism. They achieve this result by partially computing one of Donaldson’s polynomial invariants. The computation is carried out using techniques from algebraic geometry. In these computations both the basic facts about the Donaldson invariants and the relationship of the moduli space of ASD connections with the moduli space of stable bundles are assumed known. Some familiarity with the basic facts of the theory of moduli of sheaves and bundles on a surface is also assumed. This work gives a good and fairly comprehensive indication of how the methods of algebraic geometry can be used to compute Donaldson invariants.

[12] D. Kotschick and J. W. Morgan: “$ \mathrm{ SO}(3) $-invariants for 4-manifolds with $ b^+_2=1 $, II,” J. Differential Geom. 39 : 2 (1994), pp. 433–456. MR 1267898 Zbl 0828.57013 article

In this paper we extend the definition of Donaldson polynomial invariants to cover the case of manifolds with $ b_1 = 0 $ and $ b^+_2 = 1 $. We shall consider $ SO(3) $-bundles with $ w_2 $ which lifts to an integral class. This paper generalizes [2] where the case of $ SU(2) $-bundles with $ c_2 = 1 $ was treated. It should be viewed as the continuation of [5], where $ SO(3) $-bundles with arbitrary $ p_1 $ were considered. It extends [5] in two ways. First of all, it completes the proof of the fact that the values of these invariants depend only on the chamber containing the self-dual harmonic 2-form for the metric used to define the anti-self-dual (ASD) equation. Secondly, it establishes more of the general properties of the differences of the values of the invariants as the self-dual 2-form crosses a wall. It follows from the properties that we establish here that, as conjectured in [5], the value of an invariant on every chamber is determined by its value on any one chamber; and in particular, the invariant is defined for all chambers regardless of whether they contain forms which are self-dual harmonic for some metric.

[13] J. W. Morgan, T. Mrowka, and D. Ruberman: The $ L^2 $-moduli space and a vanishing theorem for Donaldson polynomial invariants. Monographs in Geometry and Topology 2. International Press (Somerville, MA), 1994. MR 1287851 Zbl 0830.58005 book

[14] R. Friedman and J. W. Morgan: Smooth four-manifolds and complex surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)] 27. Springer, 1994. MR 1288304 Zbl 0817.14017 book

The heart of this work is a continuation of the study, via algebro-geometric techniques, of the Donaldson polynomial invariants of algebraic surfaces, and in particular, elliptic surfaces. The main result, which is proved by combining the classical algebro-geometric classification with these computations, is that there is only a finite discrepancy between the differentiable classification of complex algebraic surfaces and the complex analytic classification of algebraic surfaces up to deformation of their complex structure. (Two surfaces are of the same deformation type if there is a connected analytic family of surfaces that contains the two given surfaces. Roughly speaking this means that the two surfaces are diffeomorphic and that, using an appropriate diffeomorphism to identify them, we can find a continuous path of complex structures joining the two given ones.) More precisely, we shall prove the following two theorems:

Let $ M^4 $ be a smooth, oriented 4-manifold. Then the number of deformation equivalence classes of algebraic surfaces which are diffeomorphic to M is finite.

Theorem. Let $ S_1 $ and $ S_2 $ be two complex surfaces. If $ S_1 $ is diffeomorphic to $ S_2 $ , then either $ S_1 $ and $ S_2 $ have the same Kodaira dimension or one of the surfaces $ S_1 $ , $ S_2 $ is a rational surface and the other is a surface of general type.

[15] J. W. Morgan and T. S. Mrowka: “The smooth classification of elliptic surfaces,” pp. 246–292 in Geometry, topology, & physics for Raoul Bott. Edited by S.-T. Yau. Conf. Proc. Lecture Notes Geom. Topology. International Press (Somerville, MA), 1995. MR 1358620 Zbl 0874.57020 incollection

[16] J. W. Morgan: The Seiberg–Witten equations and applications to the topology of smooth four-manifolds. Mathematical Notes 44. Princeton University Press (Princeton, NJ), 1996. MR 1367507 Zbl 0846.57001 book

[17] J. W. Morgan, Z. Szabó, and C. H. Taubes: “A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture,” J. Differential Geom. 44 : 4 (1996), pp. 706–788. MR 1438191 Zbl 0974.53063 article

The Thom Conjecture asserts that any compact, embedded surface in $ CP^2 $ of degree $ d > 0 $ must have genus at least as large as the smooth algebraic curve of the same degree, namely $ (d - l)(d - 2)/2 $. More generally, one can ask whether in any algebraic surface a smooth algebraic curve is of minimal genus in its homology class. There was one significant result in this direction. Using $ SU(2) $-Donaldson invariants, Kronheimer showed in [3] that this result is true for curves of positive self-intersection in a large class of simply connected surfaces with $ b^{+}_2 > 1 $. Unfortunately, for technical reasons, this argument does not extend to cover the case of $ CP^2 $. It is the purpose of this paper to prove the general result that a smooth holomorphic curve of non-negative self-intersection in a compact Kähler manifold is genus minimizing.

[18] R. Friedman and J. W. Morgan: “Algebraic surfaces and Seiberg–Witten invariants,” J. Algebraic Geom. 6 : 3 (1997), pp. 445–479. MR 1487223 article

[19] J. W. Morgan, T. S. Mrowka, and Z. Szabó: “Product formulas along $ T^3 $ for Seiberg–Witten invariants,” Math. Res. Lett. 4 : 6 (1997), pp. 915–929. MR 1492130 article

[20] J. W. Morgan: “An introduction to gauge theory,” pp. 51–143 in Gauge theory and the topology of four-manifolds (Park City, UT, 1994). Edited by R. Friedman and J. W. Morgan. IAS/Park City Math. Ser. 4. American Mathematical Society (Providence, RI), 1998. MR 1612968 Zbl 0911.57024 incollection

Gauge theory is the study of principal bundles, connections on them, and the curvatures of these connections. Much of it is an analytic study of certain especially important types of connections — called (Anti)-Self-Dual connections. These objects were introduced by physicists and have proved to be a central ingredient in modern particle physics, being the mechanism that allows for unifying various of the four fundamental forces. The subject thus has many sides — the physical, differential geometric, analytic, and the topological being the most important. We are interested in the application of these techniques to the classification of four-dimensional smooth manifolds. Before turning our attention to these applications, however, we need to understand the other sides of the subject. In fact, most of this course will consist of a basic introduction to the analytic, topological, and differential geometric aspects of gauge theory. Only toward the end will we turn to its applications. Even then we will only be able to scratch the surface — a much more detailed study than we have time for is necessary before one can attempt more serious applications.

[21] R. Friedman and J. W. Morgan: “Obstruction bundles, semiregularity, and Seiberg–Witten invariants,” Comm. Anal. Geom. 7 : 3 (1999), pp. 451–495. MR 1698386 Zbl 0946.14034 article

[22] J. W. Morgan: “Smooth invariants of 4-manifolds,” pp. 95–189 in Low dimensional topology. Edited by J. Böröczky, Károly, W. Neumann, and A. Stipsicz. Bolyai Soc. Math. Stud. 8. János Bolyai Math. Soc. (Budapest), 1999. Proceedings of five lecture series held during the Summer School on Low Dimensional Topology, 2–14 August 14, 1998 in Budapest, Hungary, and at the EMS Summer Schools No. 1, Algebraic Geometry, in Eger, Hungary in 1996. With discussion sessions by András I. Stipsicz. MR 1747269 Zbl 0946.57022 incollection

[23] M. Audin, J. W. Morgan, P. Vogel, and D. Bennequin: Nouveaux invariants en géométrie et en topologie. Edited by F. Dumas, J.-Y. Le Dimet, and S. Paycha. Panoramas et Synthèses [Panoramas and Syntheses] 11. Société Mathématique de France (Paris), 2001. With an afterword by Daniel Bennequin. MR 1882443 Zbl 1007.53066 book

[24] J. W. Morgan: “Seiberg–Witten invariants,” pp. 61–98 in Nouveaux invariants en géométrie et en topologie. Edited by F. Dumas, J.-Y. Le Dimet, and S. Paycha. Panoramas et Synthèses [Panoramas and Syntheses] 11. Société Mathématique de France (Paris), 2001. MR 1882445 Zbl 0994.57028 incollection

Recent years have seen many new invariants of manifolds, invariants defined using moduli spaces of solutions to certain partial differential equations. These include the anti-self-dual equations and Donaldson invariants, the pseudo-holomorphic curve equations of Gromov and the resulting Gromov and Gromov–Witten invariants, quantum cohomology, various Chern–Simon’s invariants, Floer homology, and, the subject of this article, the Seiberg–Witten invariants. We cover the background necessary to define these invariants, give the definition, and then several applications to the topology of smooth four-manifolds including to symplectic four-manifolds and complex algebraic surfaces.

[25] J. W. Morgan: “Definition of the Seiberg–Witten (SW) invariants of 4-manifolds,” pp. 1–11 in Low dimensional topology. Edited by B. Li, S. Wang, and X. Zhao. New Stud. Adv. Math. 3. International Press (Somerville, MA), 2003. MR 2052242 Zbl 1044.57012 incollection