Gauge Theory and Low-Dimensional Topology (Part II: Smooth Four-Manifolds)

In the last post, I attempted to give an overview of the state of affairs in four-manifold topology leading up to the introduction of gauge theory. In particular, we discussed the correspondence between (topological) four-manifolds and their intersection forms afforded by Freedman’s theorem, and briefly touched on the relevance of this relationship to the difference between continuous and smooth topology in dimension four. Today, we will elaborate on this a bit by describing what happens to Freedman’s theorem in the smooth category and will try to give a vague idea (to be expanded upon next time) of why gauge theory might have something to say about smooth manifolds.

TO SMOOTH OR NOT TO SMOOTH

In order to make sure that we are all on the same page, let’s briefly review the difference between topological and smooth manifolds. Recall (see for example previous posts on this blog!) that an $n$-dimensional topological manifold $M$ is defined using the data of local charts, each of which may be identified with an open subset of $\mathbb{R}^n$, together with continuous transition functions between them. If these transition functions are in addition smooth (in the usual sense of smooth maps on $\mathbb{R}^n$), then we say that $M$ has been given a smooth structure and is a smooth manifold. In order to talk about a diffeomorphism between two manifolds, we of course require that the manifolds themselves are smooth.

To those who have not thought about low-dimensional topology (and even for those who have), it is often difficult to get a feel for the difference between topological and smooth manifolds. (I certainly do not have any such intuition.) This is partly due to the fact that in low dimensions (less than or equal to three), every topological manifold admits a unique smooth structure. Thus, in order to think of a topological manifold that (for example) has no smooth structure, or two different smooth structures, we are already forced to consider examples in more dimensions than most of us are comfortable visualizing. Even setting this aside, it is difficult to see how one would go about distinguishing two smooth structures anyway, or how to prove from the definitions that a given topological manifold does or does not admit a smooth structure. Indeed, the first construction of a pair of distinct smooth structures on the same topological manifold (given for $S^7$, by John Milnor in 1956) came as a shock to many mathematicians.

Part of the difficulty in studying smooth structures on manifolds is that many of the introductory invariants in algebraic topology are either formulated purely in terms of the continuous structure (as is the case for homotopy or homology), or turn out to depend only on the continuous structure (as is the case for de Rham cohomology). Thus, slightly fancier tools are needed if one wants to systematically study smooth manifolds. We shall see later that, in dimension four, the primary (and in many cases, the only) strategy for studying smooth topology turns out to be afforded by gauge theory.

INTERSECTION FORMS (AGAIN)

Before we continue, let’s recall our discussion of Freedman’s theorem. Associated to any simply-connected, topological four-manifold $M$, we described a unimodular bilinear pairing, called the intersection form, which could be viewed as a symmetric integer matrix with determinant $\pm 1$. Freedman showed that for any unimodular pairing $Q$, one could find a simply-connected, topological four-manifold $M$ with intersection form $Q$. Even more surprisingly, he proved that if $Q$ was even, such an $M$ was unique up to homeomorphism (among simply-connected, topological four-manifolds), while for odd $Q$ there were exactly two possible $M$. We might thus hope for a similar relationship to hold in the smooth category; or, failing this, to understand how the behavior of smooth and topological manifolds differ.

The first question to ask is whether given a unimodular pairing $Q$, it is always possible to construct a smooth simply-connected manifold with intersection form $Q$. It turns out that the answer is a resounding no. The first result towards this end was established by Simon Donaldson in 1983 using gauge-theoretic methods:

Let $M$ be a smooth, simply-connected four-manifold. Suppose that the intersection form $Q$ of $M$ is positive (or negative) definite. Then $Q$ must be diagonalizable.

Here, an intersection form $Q$ is said to be positive (or negative) definite if $Q(x, x) > 0$ (or $Q(x, x) < 0$) for all $x$, and is said to be diagonalizable if it is equivalent to a diagonal matrix over $\mathbb{Z}$ (which in this case must be plus or minus the identity). Although being definite is certainly a restriction on the lattice, it turns out that most unimodular lattices in high dimensions are in fact either positive or negative definite—for example, there are over a billion distinct definite lattices of rank 32. According to Freedman’s theorem, each one of these arises as the intersection form of a simply-connected topological four-manifold. But by Donaldson’s theorem, almost all of these are not smooth manifolds—the only definite unimodular pairings that arise as the intersection forms of smooth four-manifolds are (equivalent to) plus or minus the identity!

Donaldson’s theorem immediately implies the existence of a vast class of topological four-manifolds without any smooth structure. Moreover, it showed that the relationship between smooth four-manifolds and their intersection forms is rather more subtle than in the topological category. Since Donaldson’s theorem, much work has been done on investigating exactly which pairings $Q$ can arise as the intersection forms of smooth four-manifolds. So far, what is known is the following:

Let $M$ be a simply-connected, smooth four-manifold with intersection form $Q$. Then:

1) If $Q$ is definite, then $Q$ must be diagonalizable by Donaldson’s theorem. Conversely, all definite, diagonalizable $Q$ indeed arise as the intersection forms of smooth manifolds (namely, $m\mathbb{C}P^2$ or $m\overline{\mathbb{C}P}$$^2$).

2a) If $Q$ is indefinite and odd, then it is an algebraic fact (due to the classification of unimodular lattices) that $Q$ is necessarily equivalent to a diagonal matrix with $\pm 1$ on the diagonal. All such $Q$ indeed arise as the intersection forms of smooth manifolds (namely, $m\mathbb{C}P^2 \# n\overline{\mathbb{C}P}$$^2$).

2b) If $Q$ is indefinite and even, then it is an algebraic fact (due to the classification of unimodular lattices) that $Q$ is necessarily equivalent to a direct sum $aH \oplus b E_8$, where

\[H = \begin{bmatrix}
0 & 1 \\
1 & 0
\end{bmatrix}\]

and

\[E_8 = \begin{bmatrix}
2 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\
1 & 2 & 1 & 0 & 0 & 0 & 0 & 0\\
0 & 1 & 2 & 1 & 0 & 0 & 0 & 0 \\
0 & 0 & 1 & 2 & 1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 2 & 1 & 0 & 1 \\
0 & 0 & 0 & 0 & 1 & 2 & 1 & 0 \\
0 & 0 & 0 & 0 & 0 & 1 & 2 & 0 \\
0 & 0 & 0 & 0 & 1 & 0 & 0 & 2
\end{bmatrix}\]

If $M$ is smooth, then it is known that $b$ must be even and $|a| > |b|$. If $|a| \geq \frac{3}{2}|b|$, then one can explicitly realize $Q$ as the intersection form of a connected sum $mK3 \# nS^2\times S^2$. It is conjectured that for a smooth four-manifold the inequality $|a| \geq \frac{3}{2}|b|$ must hold in general; the strengthening of the condition $|a| > |b|$ to the condition $|a| \geq \frac{3}{2}|b|$ is referred to as the “11/8-conjecture”.

For those unfamiliar with the classification of unimodular lattices, the exact casework above is unimportant—the point is that unlike in the case of topological manifolds, the question of which lattices arise as the intersection forms of smooth four-manifolds is rather more complicated and involves some peculiar numerology. These results indicate that the theory of smooth four-dimensional manifolds is radically different from the study of topological four-manifolds, a divide that has colored the field to the present day. We should note also that the uniqueness analogue of Freedman’s theorem is significantly less well-understood than the existence part—there is not even a single four-manifold for which an exhaustive list of smooth structures has been proven, and in many examples there are an infinite number of known distinct smooth structures on the same topological four-manifold!

PRELUDE TO SMOOTH INVARIANTS

We have now come to the end of our brief history of the divide between continuous and smooth topology in dimension four. In the next post, we will begin introducing some basic ideas from gauge theory itself, but I will give an (extremely vague) overview here. Earlier, we alluded to the difficulty of studying smooth structures on manifolds using classical invariants due to their dependence only on the continuous structure. What was needed was thus a new set of tools which were formulated in such a way so as to explicitly see the smooth structure present on a manifold. The idea of mathematical gauge theory was to take certain partial differential equations from physics and study the moduli space of solutions to these PDEs when defined over the target manifold. These moduli spaces see both the topology of the manifold (in constraining the global solutions) and also, implicitly, the smooth structure (in defining the PDEs). The miracle of this approach is that, for the right PDEs, not only does the moduli space succeed in capturing the smooth structure, but also that it is tractable enough to condense into an invariant with good functorial properties!