
4. Embedding problems and configurations of curves

1. Problem 4.1.

[Libgober] Consider the collections of algebraic, symplectic, and smooth surface knot groups in $\mathbb{C}P^2$, i.e. the groups $\pi_1(\mathbb{C}P^2 \setminus \Sigma)$ for $\Sigma$ a surface in $\mathbb{C}P^2$.

1. (a) How do these collections of groups differ?

2. (b) How do the collections of Alexander polynomials for surface complements in these categories differ?

3. (c) What if we fix the topological genus of the surface?

4. (d) (Ruberman) Do we know anything special about $H_2(\pi_1)$ in the algebraic and symplectic categories?

1. Remark. [Meier?] In the smooth case, it is unknown which groups arise as surface knot groups in $S^4$, but the collection of such groups is strictly larger than the above collections.
• Problem 4.2.

[Casals] Describe Stein structures on complements of symplectic surfaces in $\mathbb{C}P^2$ using Kirby diagrams.
1. Remark. This may allow you to calculate the Fukaya category of the surface complement using the Legendrian DGA.
• Problem 4.3.

[Palka] [$\mathbb{Q}$-acyclic surfaces, plumbing trees]
• Problem 4.4.

[Borodzik] [HF criterion, Bezout, involutive HF, constraint from algebraic geometry]

Cite this as: AimPL: Symplectic four-manifolds through branched coverings, available at http://aimpl.org/symplecticfour.