The homotopy type of the space of symplectic balls in rational ruled 4-manifolds

Abstract

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_ω(B^{4}(c),M) be the space of symplectic embeddings of the standard ball B^4(c) \subset \R^4 of radius r and of capacity c:= πr^2 into (M,\om). By the work of Lalonde and Pinsonnault, we know that there exists a critical capacity \ccrit \in (0,w_{M}] such that, for all c\in(0,\ccrit), the embedding space Emb_ω(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_ω(B^{4}(c),M) changes when c reaches \ccrit and that it remains constant for all c \in [\ccrit,w_{M}). In this paper, we compute the rational homotopy type, the minimal model, and the cohomology with rational coefficients of \Emb_ω(B^{4}(c),M) in the remaining case c \in [\ccrit,w_{M}). In particular, we show that it does not have the homotopy type of a finite CW-complex.

38 pages; revised version

References in corpus (3)

• The topology of the space of symplectic balls in rational 4-manifolds
• Symplectomorphism groups and embeddings of balls into rational ruled 4-manifolds
• Compatible complex structures on symplectic rational ruled surfaces

Cited by in corpus (7)

• Lagrangian blow-ups, blow-downs, and applications to real packing
• ECH embedding obstructions for rational surfaces
• Stability of the symplectomorphism group of rational surfaces
• Topology of symplectomorphism groups and ball-swappings
• Continuous Covers on Symplectic Manifolds
• Essential tori in spaces of symplectic embeddings
• Symplectic isotopy on non-minimal ruled surfaces