We show that all versions of Heegaard Floer homology, link Floer homology, and sutured Floer homology are natural. That is, they assign concrete groups to each based 3-manifold, based link, and balanced sutured manifold, respectively. Furthermore, we functorially assign isomorphisms to (based) diffeomorphisms, and show that this assignment is isotopy invariant. The proof relies on finding a simple generating set for the fundamental group of the “space of Heegaard diagrams,” and then showing that Heegaard Floer homology has no monodromy around these generators. In fact, this allows us to give sufficient conditions for an arbitrary invariant of multi-pointed Heegaard diagrams to descend to a natural invariant of 3-manifolds, links, or sutured manifolds.
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Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space
Abstract We show that the Seifert–Weber dodecahedral space $\textsf{SW}$ is a monopole Floer homology $L$-space. The proof relies on our approach to study Floer homology using hyperbolic geometry. Although $\textsf{SW}$ is significantly larger than previous manifolds studied with this technique, we overcome computational complexity issues inherent to our method by exploiting the many symmetries of $\textsf{SW}$. In particular, we prove that small eigenvalues on coexact $1$-forms on $\textsf{SW}$ have large multiplicity.
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- Award ID(s):
- 1948820
- NSF-PAR ID:
- 10287487
- Date Published:
- Journal Name:
- International Mathematics Research Notices
- ISSN:
- 1073-7928
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1093/imrn/rnaa310
