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3. Branched covers and braided surfaces

In what follows, all branched covering maps between symplectic four-manifolds are assumed to be of the form described by Denis Auroux in his talk on Monday. [A rigorous definition should be inserted here.]


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      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?


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          Problem 3.1.

          [Casals] Is there a “universal” singular symplectic curve in $\mathbb{C}P^2$ over which all symplectic four-manifolds can be realized as branched covers?
            1. Remark. [Auroux] Probably not, cf. Chisini’s conjecture.


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                  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 four-manifold?


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                      Problem 3.2.

                      [Auroux] Let $(X_1,\omega_1)$ and $(X_2,\omega_2)$ be symplectic four-manifolds with the same $c_1^2$, $c_1 \cdot \omega$, $c_2$, and $\omega^2$.
                      1. (a) Are $X_1$ and $X_2$ related by Luttinger surgery?
                      2. (b) Are $X_1$ and $X_2$ related by partial conjugation of their braid monodromy?


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                          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 $X \to \mathbb{C}P^2$?


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                              Problem 3.3.

                              [?] Let $X \to \mathbb{C}P^2$ be a symplectic branched covering (with smooth or singular branch locus). Can the Seiberg-Witten invariants of $X$ be recovered from the topological data of the branch locus?
                                1. Remark. [?] There are finite covers that kill $SW$.


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                                      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 $\pi \circ \iota =f$?


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                                          Problem 3.4.

                                          [Meier and Lambert-Cole] Efficient trisections of four-manifolds are well-behaved under branched covers. Motivated by this, we ask:

                                          1. (a) Does every simply-connected (smooth/symplectic/complex) four-manifold $X$ admit and efficient trisection?

                                          2. (b) Assuming the above, does every symplectic curve $C \subset X$ with $\pi_1(X \setminus C)$ finite admit an efficient bridge trisection?
                                            1. Remark. This would imply that there are no 1-handles, hence no exotic symplectic $\mathbb{C}P^2$, $S^2 \times S^2$, or $S^2 \tilde{\times} S^2$.


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                                                  Problem 3.45.

                                                  [Borodzik] By work of Rudolph and Boileau-Orevkov, 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.

                                                  1. (a) What is the quotient $\mathcal{C}/\mathcal{Q}$? Does it have infinite codimension?

                                                  2. (b) Is there a relation between this group and the relation of symplectic cobordism in $S^3 \times [0,1]$?
                                                    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 Ozsváth-Szabó, respectively.


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                                                          Problem 3.5.

                                                          [?] Let $(Y,\xi)$ be a contact three-manifold obtained as a cyclic branched cover of the standard contact $S^3$ along a quasipositive knot $K$. It is known that $(Y,\xi)$ is Stein-fillable. Can every Stein filling of $(Y,\xi)$ be obtained as branched cover of $B^4$ along a complex curve $\Sigma \subset B^4$ with $\partial \Sigma = K$?


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                                                              Problem 3.55.

                                                              [Hayden] Suppose that $Y$ be a three-manifold 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?


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                                                                  Problem 3.6.

                                                                  [Perutz] Consider the relation of domination among symplectic 4-manifolds, 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.).
                                                                  1. (a) Can you say anything about all symplectic $(X',\omega')$ above/below a given $(X,\omega)$?
                                                                  2. (b) [Starkston] What invariants respect this relation?
                                                                    1. 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 four-manifolds through branched coverings, available at http://aimpl.org/symplecticfour.