Symplectic isotopy classes of ellipsoids and polydisks in dimension greater than four

TL;DR

This paper investigates the topology of symplectic embedding spaces in higher dimensions, revealing non-path-connectedness and limitations on extending embeddings to ellipsoids, thus advancing understanding of symplectic isotopy classes.

Contribution

It demonstrates that certain symplectic embedding spaces in dimensions greater than four are disconnected and that nonisotopic embeddings cannot be extended to the same ellipsoid.

Findings

Spaces of symplectic embeddings are not path connected in higher dimensions.

Nonisotopic embeddings cannot be extended to the same ellipsoid.

Provides new insights into symplectic isotopy classes in dimensions ≥ 6.

Abstract

In any dimension $2n \ge 6$ we show that certain spaces of symplectic embeddings of a polydisk into a product $B^4 \times \Bbb R^{2(n-2)}$ of a $4$-ball and Euclidean space, are not path connected. We also show that any pair of such nonisotopic embeddings can never be extended to the same ellipsoid.