4-manifold - Smooth 4-manifolds

Smooth 4-manifolds

For manifolds of dimension at most 6, any piecewise linear (PL) structure can be smoothed in an essentially unique way, so in particular the theory of 4 dimensional PL manifolds is much the same as the theory of 4 dimensional smooth manifolds.

A major open problem in the theory of smooth 4-manifolds is to classify the simply connected compact ones. As the topological ones are known, this breaks up into two parts:

1. Which topological manifolds are smoothable?
2. Classify the different smooth structures on a smoothable manifold.

There is an almost complete answer to the first problem of which simply connected compact 4-manifolds have smooth structures. First, the Kirby Siebenmann invariant must vanish.

• If the intersection form is definite Donaldson's theorem (Donaldson 1983) gives a complete answer: there is a smooth structure if and only if the form is diagonalizable.
• If the form is indefinite and odd there is a smooth structure.
• If the form is indefinite and even we may as well assume that it is of nonpositive signature by changing orientations if necessary, in which case it is isomorphic to a sum of m copies of II_1,1 and 2n copies of E₈(−1) for some m and n. If m ≥ 3n (so that the dimension is at least 11/8 times the |signature|) then there is a smooth structure, given by taking a connected sum of n K3 surfaces and m − 3n copies of S2×S2. If m ≤ 2n (so the dimension is at most 10/8 times the |signature|) then Furuta proved that no smooth structure exists (Furuta 2001). This leaves a small gap between 10/8 and 11/8 where the answer is mostly unknown. (The smallest case not covered above has n=2 and m=5, but this has also been ruled out, so the smallest lattice for which the answer is not currently known is the lattice II_7,55 of rank 62 with n=3 and m=7.) The "11/8 conjecture" states that smooth structures do not exist if the dimension is less than 11/8 times the |signature|.

In contrast, very little is known about the second question of classifying the smooth structures on a smoothable 4-manifold; in fact, there is not a single smoothable 4-manifold where the answer is known. Donaldson showed that there are some simply connected compact 4-manifolds, such as Dolgachev surfaces, with a countably infinite number of different smooth structures. There are an uncountable number of different smooth structures on R4; see exotic R4. Fintushel and Stern showed how to use surgery to construct large numbers of different smooth structures (indexed by arbitrary integral polynomials) on many different manifolds, using Seiberg-Witten invariants to show that the smooth structures are different. Their results suggest that any classification of simply connected smooth 4-manifolds will be very complicated. There are currently no plausible conjectures about what this classification might look like. (Some early conjectures that all simply connected smooth 4-manifolds might be connected sums of algebraic surfaces, or symplectic manifolds, possibly with orientations reversed, have been disproved.)