Symplectic fillings of unit cotangent bundles of hyperbolic surfaces

We consider strong symplectic fillings of the unit cotangent bundle of a hyperbolic surface, equipped with its canonical contact structure. We show that every finitely presentable group can be realised as the fundamental group of such a filling.

💡 Research Summary

The paper investigates strong symplectic fillings of the unit cotangent bundle (ST^*\Sigma) of a closed hyperbolic surface (\Sigma) (genus (g\ge 2)) equipped with its canonical contact structure (\xi_{\mathrm{can}}). While many previous works focus on uniqueness results for symplectic fillings, the authors demonstrate a striking opposite phenomenon: there exist infinitely many non‑diffeomorphic (and non‑symplectomorphic) strong fillings, and moreover the fundamental group of a filling can be prescribed arbitrarily among all finitely presentable groups.

The construction begins with the standard filling (DT^\Sigma) (the disc bundle) equipped with the canonical symplectic form (\omega_{\mathrm{can}}=d\tilde\lambda). By adding a “magnetic” perturbation (\omega_{\mathrm{mag}}=\omega_{\mathrm{can}}+\varepsilon d(f\alpha)), where (\tilde\lambda=r\lambda) is the Liouville form, (\alpha) is the angular connection 1‑form on the sphere bundle, and (f(r)) is a smooth cut‑off equal to 1 near the zero section and 0 near the boundary, the authors obtain a new symplectic form that coincides with (\omega_{\mathrm{can}}) near the boundary but makes the zero section (\Sigma\subset DT^\Sigma) symplectic. This magnetic filling is still a strong filling of ((ST^*\Sigma,\xi_{\mathrm{can}})) but is not symplectomorphic to the standard one.

Next, the authors perform a Gompf symplectic sum between this magnetic filling and the product (\Sigma\times\Sigma) equipped with the split symplectic form (\sigma\oplus\sigma) (where (\sigma) is the hyperbolic area form). The diagonal (\Delta_\Sigma\subset\Sigma\times\Sigma) is a Lagrangian submanifold; after a sign change in the symplectic form on one factor, it becomes symplectic and its normal bundle has Euler class (\chi(\Sigma)). The zero section of the magnetic filling has normal bundle with Euler class (-\chi(\Sigma)). Gluing along these submanifolds yields a new symplectic 4‑manifold (W_0) diffeomorphic to ((\Sigma\times\Sigma)\setminus\Delta_\Sigma). This manifold is aspherical (π₂=0) and serves as a base filling whose fundamental group is a free product of surface groups.

To realize an arbitrary finitely presented group (G), the authors invoke Gompf’s construction of a symplectic 4‑manifold (W_G) whose fundamental group is exactly (G). (W_G) is built from (\Sigma_k\times T^2) (where (k) is the number of generators in a presentation of (G)) and a number of copies of the elliptic surface (V=\mathbb{CP}^2#9\overline{\mathbb{CP}}^2), glued along carefully chosen Lagrangian tori that encode the relations of (G).

The next step is to glue (W_G) to the previously constructed (W_0) via a Gompf sum along a torus (\lambda_1\times\lambda’1) (chosen from the product (\Sigma\times\Sigma)). This yields a manifold (W_1) whose fundamental group is still (G), but which still contains many extraneous generators coming from the (\Sigma\times\Sigma) factor. The authors then identify a collection of (2g-1) disjoint Lagrangian tori in (W_1) (derived from the standard symplectic basis of (H_1(\Sigma))) and perturb the symplectic form so that these tori become symplectic with equal area. Performing Gompf sums with additional copies of the elliptic surface (V) along each of these tori kills the unwanted generators, while preserving the relations that define (G). The final manifold (W_2) is a strong symplectic filling of ((ST^*\Sigma,\xi{\mathrm{can}})) with (\pi_1(W_2)\cong G).

Finally, the authors verify minimality: the magnetic perturbation makes the zero section symplectic, preventing the existence of any symplectically embedded ((-1))-spheres, and each Gompf sum is performed along tori with trivial normal bundle, ensuring no blow‑ups are introduced. Hence the resulting fillings are minimal in the sense of symplectic topology.

In summary, the paper provides a flexible construction that produces, for any finitely presentable group (G), a minimal strong symplectic filling of the unit cotangent bundle of a hyperbolic surface whose fundamental group is exactly (G). This demonstrates a high degree of non‑uniqueness for symplectic fillings of (ST^*\Sigma) and opens new avenues for exploring the interplay between contact geometry, symplectic topology, and group theory.