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OP-11676Status unassessedGeometry and topology

Branched covers and braided surfaces · Question 3.45

Symplectic four-manifolds through branched coverings · AIMPL workshop record

MSC 53D · Symplectic geometry, contact geometry

Editorial mapping of AIMPL list topic: Symplectic four-manifolds through branched coverings. http://aimpl.org/symplecticfour/ Not an author-supplied MSC code.

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Problem statement

By work of Rudolph and Boileau-Orevkov, quasipositive knots in S3S^3 are precisely those knots which can be realized as positive transverse knots in (S3,ξst⁡)(S^3,\xi_{\operatorname{st}}) bounding properly embedded symplectic surfaces in (B4,ωst⁡)(B^4,\omega_{\operatorname{st}}). Let C\mathcal{C} denote the concordance group of knots in S3S^3 and Q\mathcal{Q} the subgroup generated by quasipositive knots.

  • [(a)] What is the quotient C/Q\mathcal{C}/\mathcal{Q}? Does it have infinite codimension?

  • [(b)] Is there a relation between this group and the relation of symplectic cobordism in S3×[0,1]S^3 \times [0,1]?

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Background

Section context

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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References
  1. [1]Symplectic four-manifolds through branched coverings · Original question link
  2. [2]AIMPL source bibliography link
  3. [3]AIMPL text license · CC BY-SA 3.0 linkAIM Problem Lists (AIMPL), original contributors; CC BY-SA 3.0. Adaptation: structural TeX converted to Markdown, documented source macros expanded, and source bibliography links added. Mathematical wording retained. These adapted source texts remain under CC BY-SA 3.0.

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