Symplectic packings in cotangent bundles of tori

Abstract

Finding optimal packings of a symplectic manifold with symplectic embeddings of balls is a well known problem. In the following, an alternate symplectic packing problem is explored where the target and domains are 2n-dimensional manifolds which have first homology group equal to $\funnyZ^n$ and the embeddings induce isomorphisms of first homology. When the target and domains are $\funnyT^n \times V$ and $\funnyT^n \times U$ in the cotangent bundle of the torus, all such symplectic packings give rise to packings of $V$ by copies of $U$ under $\GL(n,\funnyZ)$ and translations. For arbitrary dimensions, symplectic packing invariants are computed when packing a small number of objects. In dimensions 4 and 6, computer algorithms are used to calculate the invariants associated to packing a larger number of objects. These alternate and classic symplectic packing invariants have interesting similarities and differences.