Automorphism groups and embeddings of symplectic balls into rational manifolds (Q1565900)

As they state in their abstract, the authors study the homotopy type of the space Pl\((c,\lambda)\) of symplectic embeddings of the closed ball \(B^4(c)\subset\mathbb R^4\) of capacity \(c\) in the symplectic manifold \(M_\lambda=(M,\omega_\lambda)=(S^2\times S^2,(1+\lambda)\omega_{st}\otimes\omega_{st})\), \(\lambda\geq0\). When \(\lambda=0\), they show that the homotopy type of \(M_\lambda\) does not depend on \(c\). When \(\lambda\in(0,1]\), they show that for all \((c,c')\) such that \(0<c\leq c'<\lambda\) or \(\lambda\leq c\leq c'<1\), the restriction Pl\((c',\lambda)\to\text{Pl}(c,\lambda)\) is a homotopy equivalence, and when the values of \(c\) and \(c'\) are on opposite sides of \(\lambda\), this is not the case. The details are found in [\textit{F. Lalonde} and \textit{M. Pinsonnault}, Duke Math. J. 122, No. 2, 347--397 (2004; Zbl 1063.57023), Preprint math.SG/0207096], and there they actually show that the homotopy type changes as \(c\) crosses the value \(\lambda\). The case when \(\lambda>0\) and embeddings of balls in \(\mathbb{C} P^2\) are investigated in an article in preparation by the second author. Basic ideas in the proofs and related results can be found in [\textit{M. Abreu} and \textit{D. McDuff}, J. Am. Math. Soc. 13, No. 4, 971--1009 (2000; Zbl 0965.57031), \textit{M. Gromov}, Invent. Math. 82, 307--347 (1985; Zbl 0592.53025) and \textit{D. McDuff}, Topology 30, No. 3, 409--421 (1991; Zbl 0731.53035)].