# Essential tori in spaces of symplectic embeddings

## TL;DR

This paper investigates the topology of spaces of symplectic embeddings between ellipsoids, showing that under certain size conditions, the inclusion of a torus induces an injective map on homology, using advanced holomorphic curve techniques.

## Contribution

It introduces a novel method to analyze the homological properties of symplectic embedding spaces via parametrized moduli spaces of holomorphic cylinders.

## Key findings

- Injective map on homology induced by the torus in embedding spaces.
- Use of parametrized moduli spaces of J-holomorphic cylinders in the proof.
- Results depend on inequalities relating symplectic sizes of ellipsoids.

## Abstract

Given two $2n$--dimensional symplectic ellipsoids whose symplectic sizes satisfy certain inequalities, we show that a certain map from the $n$--torus to the space of symplectic embeddings from one ellipsoid to the other induces an injective map on singular homology with mod $2$ coefficients. The proof uses parametrized moduli spaces of $J$--holomorphic cylinders in completed symplectic cobordisms.