and 24 in dimension 24, called the Niemeier lattices (examples: the Leech lattice, II24,0, II16,0 + II8,0, II8,03). Beyond 24 dimensions the number increases very rapidly; in 32 dimensions there are more than a billion of them. Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28. There are none of dimension less than 23 (other than the zero lattice!). There is one in dimension 23 (called the short Leech lattice), two in dimension 24 (the Leech lattice and the odd Leech lattice), and showed that there are 0, 1, 3, 38 in dimensions 25, 26, 27, 28, respectively. Beyond this the number increases very rapidly; there are at least 8000 in dimension 29. In sufficiently high dimensions most unimodular lattices have no roots. The only non-zero example of even positive definite unimodular lattices with no roots in dimension less than 32 is the Leech lattice in dimension 24. In dimension 32 there are more than ten million examples, and above dimension 32 the number increases very rapidly. The following table from gives the numbers of (or lower bounds for) even or odd unimodular lattices in various dimensions, and shows the very rapid growth starting shortly after dimension 24. Beyond 32 dimensions, the numbers increase even more rapidly. Applications The second cohomology group of a closed simply connected oriented topological 4-manifold is a unimodular lattice. Michael Freedman showed that this lattice almost determines the manifold: there is a unique such manifold for each even unimodular lattice, and exactly two for each odd unimodular lattice. In particular if we take the lattice to be 0, this implies the Poincaré conjecture for 4-dimensional topological manifolds. Donaldson's theorem states that if the manifold is smooth and the lattice is positive definite, then it must be a sum of copies of Z, so most of these manifolds have no smooth structure. One such example is the E8 manifold. References External links Gabriele Nebe and Neil Sloane's catalogue of unimodular lattices. Quadratic forms Lattice points