The theta function of a unimodular positive definite lattice is a modular form whose weight is one half the rank.
Due to the dimension bound on spaces of modular forms, the minimum norm of a nonzero vector of an even unimodular lattice is no greater than ⎣n/24⎦ + 1.
An even unimodular lattice that achieves this bound is called extremal.
Even positive definite unimodular lattice exist only in dimensions divisible by 8.
Unimodular lattices with no roots (vectors of norm 1 or 2) have been classified up to dimension 28.
The following table from (King 2003) 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.
The second cohomology group of a closed simply connected oriented topological 4-manifold is a unimodular lattice.
In particular if we take the lattice to be 0, this implies the Poincaré conjecture for 4-dimensional topological manifolds.