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Surgery obstructions for knots in integer homology spheres

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Yuhui Chen

Geometric Topology · Submitted 2 Jul 2026 · 24 references

Pith's verdict: Extends the Hom-Karakurt-Lidman obstruction from S^3 to knots in arbitrary integer homology spheres Y and derives a cobordism bound from it.

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arXiv ID2607.02028 (version 1)
Pith ID7BLJCRND, Pith's permanent record of this paper and its review
CitationYuhui Chen. "Surgery obstructions for knots in integer homology spheres." arXiv:2607.02028 (2026). https://pith.science/paper/7BLJCRND
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The paper

The paper claims

The Hom-Karakurt-Lidman obstruction to 1/m-surgery on knots extends from S^3 to every integer homology sphere.

The reading

The paper establishes that the surgery obstruction previously known only for knots in the 3-sphere carries over directly to knots in an arbitrary integer homology sphere Y. The extension covers both positive and negative surgery coefficients of the form 1/m. Once available, the obstruction is applied to rule out surgery descriptions for infinitely many small Seifert fibered spaces and for hyperbolic examples. The same circle of ideas produces a lower bound on the second Betti number of any smooth cobordism connecting two integer homology spheres.

What carries the argument

The Hom-Karakurt-Lidman obstruction, now stated for knots inside any integer homology sphere Y and used to obstruct 1/m-surgery.

Extended reading notes

Core claim

We extend this obstruction to all integer homology spheres Y, for both positive and negative 1/m surgeries. This is used to test infinitely many small Seifert fibered examples and hyperbolic examples. Moreover, we deduce a lower bound on the b_2(W) of smooth cobordism between a pair of integer homology spheres.

Load-bearing premise

The original obstruction for knots in S^3 admits a direct generalization to knots in an arbitrary integer homology sphere without further restrictions on Y or the knot.

Editorial extensions

If this is right

  • The obstruction now applies to knots in every integer homology sphere rather than only to knots in S^3.
  • Infinitely many small Seifert fibered spaces and hyperbolic spaces can be shown not to arise by 1/m-surgery on any knot in an integer homology sphere.
  • Any smooth cobordism W between two integer homology spheres must satisfy a lower bound on its second Betti number that depends on the surgery data of the knots involved.
  • Both positive and negative 1/m coefficients are covered by the same extended obstruction.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same generalization technique might apply to other Floer-theoretic surgery obstructions that were originally proved only in S^3.
  • One could test whether the new bound on b_2(W) is sharp by constructing explicit cobordisms realizing the predicted minimal value.
  • The result supplies a new tool for deciding which integer homology spheres can appear as the result of surgery on a knot inside another integer homology sphere.
Original abstract

For knot surgery in $S^3$, Heegaard Floer homology provides an obstruction due to Hom--Karakurt--Lidman. We extend this obstruction to all integer homology spheres $Y$, for both positive and negative 1/m surgeries. This is used to test infinitely many small Seifert fibered examples and hyperbolic examples. Moreover, we deduce a lower bound on the $b_2(W)$ of smooth cobordism between a pair of integer homology spheres.

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