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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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This content will become publicly available on July 24, 2026
A classification of incompleteness statements
Abstract For which choices of$$X,Y,Z\in \{\Sigma ^1_1,\Pi ^1_1\}$$does no sufficiently strongX-sound andY-definable extension theory prove its ownZ-soundness? We give a complete answer, thereby delimiting the generalizations of Gödel’s second incompleteness theorem that hold within second-order arithmetic.
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- Award ID(s):
- 2054379
- PAR ID:
- 10634723
- Publisher / Repository:
- arxiv
- Date Published:
- Journal Name:
- Canadian Mathematical Bulletin
- ISSN:
- 0008-4395
- Page Range / eLocation ID:
- 1 to 8
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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