## Symplectic structures with non-isomorphic primitive cohomology on open 4-manifolds

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- by Matthew Gibson, Li-Sheng Tseng and Stefano Vidussi PDF
- Trans. Amer. Math. Soc.
**375**(2022), 8399-8422 Request permission

## Abstract:

We analyze four-dimensional symplectic manifolds of type $X=S^1 \times M^3$ where $M^3$ is an open $3$-manifold admitting inequivalent fibrations leading to inequivalent symplectic structures on $X$. For the case where $M^3 \subset S^3$ is the complement of a $4$-component link constructed by McMullen-Taubes, we provide a general algorithm for computing the monodromy of the fibrations explicitly. We use this algorithm to show that certain inequivalent symplectic structures are distinguished by the dimensions of the primitive cohomologies of differential forms on $X$. We also calculate the primitive cohomologies on $X$ for a class of open $3$-manifolds that are complements of a family of fibered graph links in $S^3$. In this case, we show that there exist pairs of symplectic forms on $X$, arising from either equivalent or inequivalent pairs of fibrations on the link complement, that have different dimensions of the primitive cohomologies.## References

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## Additional Information

**Matthew Gibson**- Affiliation: Department of Mathematics, University of California, Irvine, California 92697
- MR Author ID: 1407020
- Email: gibsonmd@uci.edu
**Li-Sheng Tseng**- Affiliation: Department of Mathematics, University of California, Irvine, California 92697
- Email: lstseng@math.uci.edu
**Stefano Vidussi**- Affiliation: Department of Mathematics, University of California, Riverside, California 92521
- MR Author ID: 617938
- Email: svidussi@ucr.edu
- Received by editor(s): May 26, 2021
- Published electronically: October 3, 2022
- Additional Notes: The second author was supported by the Simons Collaboration Grant No. 636284 and the third author was supported by the Simons Collaboration Grant No. 524230.
- © Copyright 2022 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**375**(2022), 8399-8422 - MSC (2020): Primary 57R57, 57K43, 57K20
- DOI: https://doi.org/10.1090/tran/8747