Symplectic (−2)-spheres and the symplectomorphism group of small rational 4-manifolds

Abstract

For (CP 2 #5CP 2 , ω), let Nω be the number of (−2)-symplectic spherical homology classes. We completely determine the Torelli symplectic mapping class group (Torelli SMCG): Torelli SMCG is trivial if Nω > 8; it is π 0 (Diff + (S 2 , 5)) if Nω = 0 (by [1],[2]); it is π 0 (Diff + (S 2 , 4)) in the remaining case. Further, we completely determine the rank of π 1 (Symp(CP 2 #5CP 2 ) for any given symplectic form. Our results can be uniformly presented regarding Dynkin diagrams of type A and type D Lie algebras. We also provide a solution to the smooth isotopy problem of rational 4-manifolds.Compared to the monotone case when C 0 is contractible (where the form is of type D 5 ), we fall short of computing the homotopy type of it directly: indeed, the topology of the open strata of almost complex structure can be very complicated even in much simpler manifolds, see [6].We took a new approach here. Starting from a class of standard RP 2 packing symplectic forms (RP 2 forms for short, see Definition 3.16), we show that the map φ is indeed surjective when there is an RP 2 packing in X. This surjectivity is in turn related to another relative ball-packing problem and makes use of the ball-swapping symplectomorphism constructed in [20]. We then use a stability argument inspired by [21], paired with a Cremona equivalence computation, to relate a type A form with a RP 2 form.The forms of type D 4 is more complicated. We will construct a key commutative diagram (28) (compare [1]). The punchline is to remove those strata of almost complex structures which allows more than one (−2)-sphere, or spheres with self-intersection no greater than (−3) from the space of ω-compatible almost complex structure. This yields a 2-connected space. Such a space is not homeomorphic to C 0 , but captures π i (C 0 ) for i = 0, 1, 2, which suffices for the study of π 0 and π 1 of Symp(X, ω). An extensive study of diagram (28) enables one to compare the induced homotopy sequence in the lowest degrees with the strand-forgetting sequencewhich eventually deduces our main theorem for D 4 using the Hopfian property of braid groups.Remark 1.5. After the first draft of this manuscript was posted, Silvia Anjos informed us about her work with Sinan Eden ([22], [23] ), in which they independently obtain similar results in some toric cases for the 4-fold blow-up of CP 2 , including the generic case and the case where λ = 1 in the Table 1. Moreover, they have a result to show that the generators of π 1 (Ham(X, ω)) also generate the homotopy Lie algebra of Ham(X, ω), using similar ideas from [8]. K c J := {[ω] ∈ H 2 (M ; R)|ω is compatible with J}. K t J and K c J are both convex cones in the positive cone P = {c ∈ H 2 (M ; R)|c · c > 0}. Note that we have the tamed Nakai-Moishezon theorem for rational surfaces when Euler number is small: Theorem 2.19 (Theorem 1.6 in [33]). Suppose M = S 2 × S 2 or CP 2 #kCP 2 , k ≤ 9, and let C >0 J := {c ∈ H 2 (M ; R)|[Σ] = c for some some J-holomorphic subvariety Σ} be the curve cone of J. For an almost Kähler J on M , C ∨,>0...

Topology of symplectomorphism groups and ball-swappings

“…We can follow the route of Proposition 6.4 in [McD08] to give a proof of the following result, refining [McD08, Proposition 6.4, Corollary 6.9]. See also [LLnt,Proposition 4.13].…”

“…In general, we know SymppXq is always a countable CW-complex [MS98], but any concrete computations are highly nontrivial, see for example [Abr98,McD00,AM00,Anj02,LP04,Bus11]. This thread of study of the full homotopy type has been deeply intertwined with the stratification of almost complex structures, which again inspired a series of interesting works [McD00,AGK09,Bus05,LLnt], etc. If one narrows the attention to the connectedness of Symp h pXq, the symplectomorphism subgroup that acts homologically trivially on X, interesting phenomenon can already be observed.…”

“…As it turns out, this explicit construction has interesting applications to studying mapping classes and classifying Lagrangian submanifolds in A n -surface singularities [Wu14], see also Theorem 3.8. Later, in the authors' joint works [LLW15,LLWnt] with T.-J. Li, we use this construction to study π 0 pSymppX, ωqq for rational surfaces with χpXq " 8 (Theorem 3.3).…”

Symplectic isotopy on non-minimal ruled surfaces

“…Now let us focus on the case of non-minimal ruled surfaces. Following the same strategies employed by [15], one needs to find a sufficiently fine stratification of the spaces of almost complex structures and show that they only differ by the addition of said (finite codimension) strata when ω crosses the walls of the chambers of the arithmetic regions in ∆ n+1 .…”

Centralizers of Hamiltonian circle actions on rational ruled surfaces

“…Regarding the first question, most efforts as been devoted to the study of symplectomorphism groups of rational 4-manifolds. Following the seminal work of M. Gromov [Gro85] who showed that the group of compactly supported symplectomorphisms of R 4 is contractible, the homotopical properties of the group of symplectomorphisms of CP 2 , S 2 ˆS2 and of the k-fold symplectic blow-ups CP 2 #kCP 2 , k ď 5, were studied in several papers such as [Abr98], [AGK09], [AM00], [Pin08b], [AG04], [AP13], [AE19], and [LLW22]. In particular, for CP 2 , S 2 ˆS2 , and CP 2 #kCP 2 , k ď 3, the rational homotopy type of SymppM, ωq can be described precisely in terms of the cohomology class rωs.…”