[Paper Review] Relative family Gromov-Witten invariants and symplectomorphisms

Olguţa Buşe|ArXiv.org|Oct 29, 2001

Geometric and Algebraic Topology8 references4 citations

TL;DR

This paper introduces relative family Gromov-Witten invariants to detect nontrivial homotopy classes in symplectomorphism groups of manifolds equipped with 1-parameter families of symplectic forms. By analyzing almost complex structures on $S^2 \times S^2 \times X$ with deformed cohomology classes, it proves the persistence of fragile homotopy elements—such as those found by Abreu-McDuff—in higher homotopy groups of symplectomorphism groups, establishing new infinite families of nontrivial elements in $\pi_k(G_\lambda^X)$ for countably many $k$ and $\lambda$. The invariants are symplectic deformation invariants and detect nontrivial relative homotopy classes in spaces of almost complex structures.

ABSTRACT

We study the symplectomorphism groups $G_λ=Symp_0(M,ω_λ)$ of an arbitrary closed manifold M equipped with a 1-parameter family of symplectic forms $ω_λ$ with variable cohomology class. We show that the existence of nontrivial elements in $π_*({\cal A},{\cal A}')$, where $({\cal A},{\cal A}')$ is a suitable pair of spaces of almost complex structures, implies the exiarxiv.org stence of families of nontrivial elements in $π_{*-i}G_λ$, for $i=1$ or 2. Suitable parametric Gromov Witten invariants detect nontrivial elements in $π_*({\cal A},{\cal A}')$. By looking at certain resolutions of quotient singularities we investigate the situation $(M,ω_λ)= (S^2 imes S^2 imes X,σ_F \oplus λσ_B \oplus ω_{st})$, with $(X,ω_{st})$ an arbitrary symplectic manifold. We find families of nontrivial elements in $π_k(G_λ^X)$, for countably many $k$ and different values of $λ$. In particular we show that the fragile elements $w_{\ell}$ found by Abreu-McDuff in $π_{4 \ell}(G_{\ell+1}^{pt})$ do not disappear when we consider them in $S^2 imes S^2 imes X$.

Motivation & Objective

• To understand how the topology of symplectomorphism groups $\mathrm{Symp}_0(M,\omega_\lambda)$ changes as the cohomology class of $\omega_\lambda$ varies in a 1-parameter family.
• To detect nontrivial elements in the homotopy groups of symplectomorphism groups using parametric Gromov-Witten invariants.
• To show that fragile homotopy elements—previously found in $\pi_{4\ell}(G_{\ell+1}^{\mathrm{pt}})$—persist when extended to $S^2 \times S^2 \times X$.
• To establish a link between nontrivial relative homotopy classes in spaces of almost complex structures and nontrivial homotopy classes in symplectomorphism groups via a fibration sequence.

Proposed method

• Define relative family Gromov-Witten invariants $PGW^{M,(J_B,J_{\partial B})}_{D,0,k}$ that count $J_b$-holomorphic stable maps in class $D$ for $b \in B$, with $J_b$ avoiding such maps on $\partial B$, ensuring well-definedness.
• Construct a homomorphism $\Theta_{k,\alpha_1,\dots,\alpha_k}$ from $\pi_*(\mathcal{A}_I, \mathcal{A}_{I,D}^c)$ to $\mathbb{Q}$ using these invariants, which are symplectic deformation invariants.
• Use the fibration $\mathrm{Symp}_0(M,\omega_\lambda) \to \mathrm{Diff}_0(M) \to \mathcal{A}_\omega$ to relate homotopy of almost complex structures to symplectomorphism group topology.
• Analyze the resolution of quotient singularities $\mathbb{C}^2/C_{2\ell}$ to construct explicit families $(J_{B_\ell}, \partial J_{B_\ell})$ in $\mathcal{A}_{[\ell+\epsilon,\ell]}$ with nontrivial invariants.
• Apply the long exact sequence of homotopy groups to the fibration to lift nontrivial classes in $\pi_k(\mathcal{A}_{\ell^+}, \mathcal{A}_\ell)$ to nontrivial elements in $\pi_{k-1}(G_\lambda^X)$.
• Prove that persistent elements in $\pi_k(\mathcal{A}_{\ell^+}, \mathcal{A}_\ell)$ yield nontrivial, non-germ elements in $\pi_{k-1}(G_\lambda^X)$, showing they are new and not liftable from $\mathrm{Diff}_0(M)$.

Experimental results

Research questions

• RQ1 Do nontrivial homotopy classes in spaces of almost complex structures detect nontrivial elements in symplectomorphism groups?
• RQ2 Can relative family Gromov-Witten invariants detect persistent homotopy elements in symplectomorphism groups under deformation of symplectic forms?
• RQ3 Do fragile homotopy elements in $\pi_{4\ell}(G_{\ell+1}^{\mathrm{pt}})$ survive when extended to $S^2 \times S^2 \times X$?
• RQ4 Are the nontrivial elements in $\pi_k(G_\lambda^X)$ constructed via these invariants new, or do they arise as germs of elements in $\pi_k(G_\lambda)$ for $\lambda < \ell$?
• RQ5 Can the topology of symplectomorphism groups be detected through parametric invariants sensitive to relative homotopy classes in $\mathcal{A}_{\omega_\lambda}$?

Key findings

• The relative family Gromov-Witten invariants $PGW^{M,(J_B,J_{\partial B})}_{D,0,k}$ are symplectic deformation invariants and depend only on the relative homotopy class of the pair $(J_B, J_{\partial B})$.
• For $M = S^2 \times S^2 \times X$ with $\omega_\lambda = \sigma_F \oplus \lambda \sigma_B \oplus \omega_{\mathrm{st}}$, nontrivial invariants exist for $\lambda \in [\ell, \ell+\epsilon]$ and $\ell \in \mathbb{N}$, leading to nontrivial elements in $\pi_{4\ell-2}(G_\lambda^X)$.
• The fragile elements $w_\ell$ found by Abreu-McDuff in $\pi_{4\ell}(G_{\ell+1}^{\mathrm{pt}})$ do not disappear when lifted to $S^2 \times S^2 \times X$, remaining nontrivial in $\pi_{4\ell-2}(G_\lambda^X)$ for $\lambda > \ell$.
• Nontrivial elements in $\pi_k(\mathcal{A}_{\ell^+}, \mathcal{A}_\ell)$ yield nontrivial elements in $\pi_{k-1}(G_\lambda^X)$, and these are not liftable from $\mathrm{Diff}_0(M)$, proving they are new and not germs of elements in $\pi_{k-1}(G_\lambda)$ for $\lambda < \ell$.
• Persistent elements in $\pi_k(\mathcal{A}_{\ell^+}, \mathcal{A}_\ell)$—defined as those not lifting to $\pi_k(\mathcal{A}_\ell)$—produce nontrivial, non-germ elements in $\pi_{k-1}(G_\lambda^X)$, confirming their topological novelty.
• The construction via resolution of $\mathbb{C}^2/C_{2\ell}$ provides explicit families of almost complex structures with nontrivial invariants, enabling the detection of countably many nontrivial homotopy classes in $\pi_k(G_\lambda^X)$ for $k = 4\ell - 2$.