be trivial) there is no algorithm to tell if two 4-manifolds have the same fundamental group. This is one reason why much of the work on 4-manifolds just considers the simply connected case: the general case of many problems is already known to be intractable. 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: Which topological manifolds are smoothable? 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 class must vanish. If the intersection form is definite Donaldson's theorem 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 II1,1 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 . 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 II7,55 of rank 62 with n=3 and m=7. See for recent (as of 2019) progress in this area.) 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