2002/07/31 by Francois Lalonde, Martin Pinsonnault · 2 citations
Mathematics · #math.SG #math.DG #msc:57R17 #msc:57S05 #msc:53D35 #msc:55R20
paper · pdf · doi:10.1215/s0012-7094-04-12223-7
published as Duke Math. J. Volume 122, Number 2 (2004), 347-397 · Typos corrected, 2 minor corrections in the text. Numbering consistant with the published version
arxiv created 2004/06/04 · arxiv updated 2011/04/26
We study in this paper the rational homotopy type of the space of symplectic embeddings of the standard ball B4(c) ⊂ \R4 into 4-dimensional rational symplectic manifolds. We compute the rational homotopy groups of that space when the 4-manifold has the form Mλ= (S2 × S2, μω0 ⊕ ω0) where ω0 is the area form on the sphere with total area 1 and μ belongs to the interval [1,2]. We show that, when μ is 1, this space retracts to the space of symplectic frames, for any value of c. However, for any given 1 < μ< 2, the rational homotopy type of that space changes as c crosses the critical parameter ccrit = μ- 1, which is the difference of areas between the two S2 factors. We prove moreover that the full homotopy type of that space changes only at that value, i.e the restriction map between these spaces is a homotopy equivalence as long as these values of c remain either below or above that critical value.