## Symplectic packings in cotangent bundles of tori.(English)Zbl 0972.52010

Summary: 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 $$\mathbb{Z}^n$$ and the embeddings induce isomorphisms of first homology. When the target and domains are $$\mathbb{T}^n\times V$$ and $$\mathbb{T}^n\times U$$ in the cotangent bundle of the torus, all such symplectic packings give rise to packings of $$V$$ by copies of $$U$$ under $$\text{GL}(n,\mathbb{Z})$$ 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.

### MSC:

 52C17 Packing and covering in $$n$$ dimensions (aspects of discrete geometry) 53D05 Symplectic manifolds (general theory)

### Keywords:

symplectic packings
Full Text:

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