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We use Heegaard Floer homology to define an invariant of homology cobordism. This invariant is isomorphic to a summand of the reduced Heegaard Floer homology of a rational homology sphere equipped with a spin structure and is analogous to Stoffregen’s connected Seiberg–Witten Floer homology. We use this invariant to study the structure of the homology cobordism group and, along the way, compute the involutive correction terms $$\bar{d}$$ and $$\text{}\underline{d}$$ for certain families of three-manifolds.
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Families of metrics with positive scalar curvature on spectral sequence cobordisms
We study families of metrics on the cobordisms that underlie the differential maps in Bloom’s monopole Floer spectral sequence, a spectral sequence for links in [Formula: see text] whose [Formula: see text] page is the Khovanov homology of the link, and which abuts to the monopole Floer homology of the double branched cover of the link. The higher differentials in the spectral sequence count parametrized moduli spaces of solutions to Seiberg–Witten equations, parametrized over a family of metrics with asymptotic behavior corresponding to a configuration of unlinks with 1-handle attachments. For a class of configurations, we construct families of metrics with the prescribed behavior, such that each metric therein has positive scalar curvature. The positive scalar curvature implies that there are no irreducible solutions to the Seiberg–Witten equations and thus, when the spectral sequences are computed with these families of metrics, only reducible solutions must be counted. The class of configurations for which we construct these families of metrics includes all configurations that go into the spectral sequence for [Formula: see text] torus knots, and all configurations that involve exactly two 1-handle attachments.
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- PAR ID:
- 10612182
- Publisher / Repository:
- Journal of Topology and Analysis
- Date Published:
- Journal Name:
- Journal of Topology and Analysis
- Volume:
- 17
- Issue:
- 06
- ISSN:
- 1793-5253
- Page Range / eLocation ID:
- 1747 to 1772
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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