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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)
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