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

TL;DR

This paper determines the rational homotopy type and cohomology of the space of symplectic embeddings of balls into rational ruled 4-manifolds for certain capacities, revealing it is not homotopy equivalent to a finite CW-complex.

Contribution

It computes the rational homotopy type, minimal model, and cohomology of the embedding space for capacities in the critical range, extending previous results.

Findings

The embedding space's rational homotopy type is explicitly computed.

The embedding space does not have the homotopy type of a finite CW-complex.

The results apply to capacities in the range [ ext{ccrit}, w_M).

Abstract

Let M:=(M^{4},\om) be a 4-dimensional rational ruled symplectic manifold and denote by w_{M} its Gromov width. Let Emb_{\omega}(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:= \pi r^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_{\omega}(B^{4}(c),M) is homotopy equivalent to the space of symplectic frames \SFr(M). We also know that the homotopy type of Emb_{\omega}(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_{\omega}(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.