3. Branched covers and braided surfaces
In what follows, all branched covering maps between symplectic fourmanifolds are assumed to be of the form described by Denis Auroux in his talk on Monday. [A rigorous definition should be inserted here.]
Problem 3.05.
Let $(X,\omega)$ be symplectic but not complex (e.g. minimal but violating the Noether inequality), and let $f: X \to \mathbb{C}P^2$ be a branched covering with braided branch surface“in the sense of Denis”. What can be said about the branch surface? Can negative nodes be removed? 
Problem 3.1.
[Casals] Is there a “universal” singular symplectic curve in $\mathbb{C}P^2$ over which all symplectic fourmanifolds can be realized as branched covers?
Remark. [Auroux] Probably not, cf. Chisini’s conjecture.


Problem 3.15.
[Casals] Let $f: X \to \mathbb{C}P^2$ be a branched covering in the sense described above. In what ways can we modify the branch locus and preserve the symplectic fourmanifold? 
Problem 3.2.
[Auroux] Let $(X_1,\omega_1)$ and $(X_2,\omega_2)$ be symplectic fourmanifolds with the same $c_1^2$, $c_1 \cdot \omega$, $c_2$, and $\omega^2$. (a) Are $X_1$ and $X_2$ related by Luttinger surgery?
 (b) Are $X_1$ and $X_2$ related by partial conjugation of their braid monodromy?

Problem 3.25.
[Perutz] Suppose that $(X,\omega)$ has vanishing canonical class $K=0$. Conjecturally, $X$ is diffeomorphic to $K3$ or a $T^2$bundle over $T^2$. What can we understand about $X$ from the structure of branched covers $X \to \mathbb{C}P^2$? 
Problem 3.3.
[?] Let $X \to \mathbb{C}P^2$ be a symplectic branched covering (with smooth or singular branch locus). Can the SeibergWitten invariants of $X$ be recovered from the topological data of the branch locus?
Remark. [?] There are finite covers that kill $SW$.


Problem 3.35.
[Zuddas] Let $f: X^4 \to Y^4$ be a symplectic branched cover as above. Does there exist a bundle $\pi: E \to Y$ and an embedding $\iota: X \hookrightarrow E$ such that $\pi \circ \iota =f$? 
Problem 3.4.
[Meier and LambertCole] Efficient trisections of fourmanifolds are wellbehaved under branched covers. Motivated by this, we ask: (a) Does every simplyconnected (smooth/symplectic/complex) fourmanifold $X$ admit and efficient trisection?
 (b) Assuming the above, does every symplectic curve $C \subset X$ with $\pi_1(X \setminus C)$ finite admit an efficient bridge trisection?

Remark. This would imply that there are no 1handles, hence no exotic symplectic $\mathbb{C}P^2$, $S^2 \times S^2$, or $S^2 \tilde{\times} S^2$.
 (a) Does every simplyconnected (smooth/symplectic/complex) fourmanifold $X$ admit and efficient trisection?

Problem 3.45.
[Borodzik] By work of Rudolph and BoileauOrevkov, quasipositive knots in $S^3$ are precisely those knots which can be realized as positive transverse knots in $(S^3,\xi_{\operatorname{st}})$ bounding properly embedded symplectic surfaces in $(B^4,\omega_{\operatorname{st}})$. Let $\mathcal{C}$ denote the concordance group of knots in $S^3$ and $\mathcal{Q}$ the subgroup generated by quasipositive knots. (a) What is the quotient $\mathcal{C}/\mathcal{Q}$? Does it have infinite codimension?
 (b) Is there a relation between this group and the relation of symplectic cobordism in $S^3 \times [0,1]$?

Remark. The quotient $\mathcal{C}/\mathcal{Q}$ has rank at least three (over $\mathbb{Z}$). This follows from Hedden and Ording’s proof of $s \neq 2 \tau$, where $s$ and $\tau$ are the concordance invariants defined by Rasmussen and OzsváthSzabó, respectively.
 (a) What is the quotient $\mathcal{C}/\mathcal{Q}$? Does it have infinite codimension?

Problem 3.5.
[?] Let $(Y,\xi)$ be a contact threemanifold obtained as a cyclic branched cover of the standard contact $S^3$ along a quasipositive knot $K$. It is known that $(Y,\xi)$ is Steinfillable. Can every Stein filling of $(Y,\xi)$ be obtained as branched cover of $B^4$ along a complex curve $\Sigma \subset B^4$ with $\partial \Sigma = K$? 
Problem 3.55.
[Hayden] Suppose that $Y$ be a threemanifold containing a rationally fibered knot $K$ whose associated contact structure $\xi$ is (Stein or weakly/strongly symplectically) fillable. Let $(Y',\xi')$ be a cyclic branched cover of $(Y,\xi)$ along $K$. Is $(Y',\xi')$ fillable? 
Problem 3.6.
[Perutz] Consider the relation of domination among symplectic 4manifolds, writing $(X,\omega) \leq (X',\omega')$ if there is a symplectic branched covering $X' \to X$ in the sense of Auroux (symplectic branch loci, cusps, and $\pm$ double points, etc.). (a) Can you say anything about all symplectic $(X',\omega')$ above/below a given $(X,\omega)$?
 (b) [Starkston] What invariants respect this relation?

Remark. [?] As an example of an answer to (a), we have that $X \leq \mathbb{C}P^2$ implies $X \approx \mathbb{C}P^2$.
Cite this as: AimPL: Symplectic fourmanifolds through branched coverings, available at http://aimpl.org/symplecticfour.