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Abstract We consider manifold-knot pairs$$(Y,K)$$, whereYis a homology 3-sphere that bounds a homology 4-ball. We show that the minimum genus of a PL surface$$\Sigma $$in a homology ballX, such that$$\partial (X, \Sigma ) = (Y, K)$$can be arbitrarily large. Equivalently, the minimum genus of a surface cobordism in a homology cobordism from$$(Y, K)$$to any knot in$$S^3$$can be arbitrarily large. The proof relies on Heegaard Floer homology.
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On the Frøyshov invariant and monopole Lefschetz number
Given an involution on a rational homology 3-sphere Y with quotient the 3-sphere, we prove a formula for the Lefschetz num- ber of the map induced by this involution in the reduced mono- pole Floer homology. This formula is motivated by a variant of Witten’s conjecture relating the Donaldson and Seiberg–Witten invariants of 4-manifolds. A key ingredient is a skein-theoretic ar- gument, making use of an exact triangle in monopole Floer homol- ogy, that computes the Lefschetz number in terms of the Murasugi signature of the branch set and the sum of Frøyshov invariants as- sociated to spin structures on Y . We discuss various applications of our formula in gauge theory, knot theory, contact geometry, and 4-dimensional topology.
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- Award ID(s):
- 1811111
- PAR ID:
- 10469627
- Publisher / Repository:
- Journal of Differential Geometry
- Date Published:
- Journal Name:
- Journal of Differential Geometry
- Volume:
- 123
- Issue:
- 3
- ISSN:
- 0022-040X
- Page Range / eLocation ID:
- 523-593
- Subject(s) / Keyword(s):
- Monopole homology, Floer homology, gauge theory, Froyshov invariant
- 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/10.4310/jdg/1683307008
