Surgery obstructions and Heegaard Floer homology

Hom, Jennifer; Karakurt, Çağrı; Lidman, Tye

doi:10.2140/gt.2016.20.2219

Cited by 27 publications

(50 citation statements)

References 19 publications

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“…The graded roots of Seifert spaces were studied in . Our brief introduction to graded roots will follow [, § 4] extremely closely.…”

Section: Graded Rootsmentioning

confidence: 99%

“…Here

S^{2} false(p, q, r false)

is the orbifold with underlying space

S^{2}

and cone singularities modeled on the actions of

Z / p

Z / q

, and

Z / r

. This convention for

Σ (p, q, r)

agrees with the notation of , but is opposite the notation of .…”

Section: Graded Rootsmentioning

confidence: 99%

“…Theorem is proved using the technology of graded roots, introduced by Némethi , and refinements of the method of graded roots for Seifert spaces in . The proof is essentially borrowed from the partial calculation of

{italicHF}^{+} false(Y_{p} false)

for even

p

by Hom, Karakurt, and Lidman .…”

Section: Introductionmentioning

confidence: 99%

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Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

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Abstract. We give inequalities for the Manolescu invariants α, β, γ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a Z 8 subgroup of the homology cobordism group. To our knowledge, this is the first proof of Furuta's Theorem using monopoles. We also provide information about Manolescu invariants of the connected sum of n copies of a three-manifold Y , for large n.

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“…The graded roots of Seifert spaces were studied in . Our brief introduction to graded roots will follow [, § 4] extremely closely.…”

Section: Graded Rootsmentioning

confidence: 99%

“…Here

S^{2} false(p, q, r false)

is the orbifold with underlying space

S^{2}

and cone singularities modeled on the actions of

Z / p

Z / q

, and

Z / r

. This convention for

Σ (p, q, r)

agrees with the notation of , but is opposite the notation of .…”

Section: Graded Rootsmentioning

confidence: 99%

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Manolescu invariants of connected sums

Stoffregen

2017

Proc. London Math. Soc.

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“…By Proposition 3, d(Y j,n ) ≤ −10 and by Proposition 2, U 2 · HF red (Y j,n ) = 0. The same arguments used to prove Theorem 1.2 of [HKL14] imply that if Y j,n is surgery on a knot in S 3 with d(Y j,n ) ≤ −10, then U 2 · HF red (Y j,n ) = 0. Therefore, we obtain a contradiction.…”

mentioning

confidence: 93%

A note on surgery obstructions and hyperbolic integer homology spheres

Hom

Lidman

2017

Proc. Amer. Math. Soc.

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Abstract. Auckly gave two examples of irreducible integer homology spheres (one toroidal and one hyperbolic) which are not surgery on a knot in the three-sphere. Using Heegaard Floer homology, the authors and Karakurt provided infinitely many small Seifert fibered examples. In this note, we extend those results to give infinitely many hyperbolic examples, as well as infinitely many examples with arbitrary JSJ decomposition.Lickorish [Lic62] and Wallace [Wal60] proved that any closed, oriented three-manifold can be obtained by surgery on a link in the three-sphere. Thus, a natural question to ask is which manifolds can be described via the simplest possible surgery description, i.e., as surgery on a knot. Irreducible integer homology spheres are a particularly interesting family to consider, since the simplest obstructions (e.g., [BL90]) to being surgery on a knot all vanish. Note that Gordon and Luecke [GL89] showed that a reducible integer homology sphere can never be surgery on a knot. Auckly [Auc97] provided the first two examples (one toroidal and one hyperbolic) of irreducible integer homology spheres which are not surgery on a knot, answering [Kir95, Problem 3.6(C)] in the affirmative. For over 15 years, these two manifolds were the only known examples, until the authors and Karakurt [HKL14] provided an infinite family of small Seifert fibered integer homology spheres which are not surgery on a knot. In this note, we refine that result to give an infinite family of hyperbolic examples as well. Theorem 1. There exist infinitely many hyperbolic integer homology spheres which are not surgery on a knot in S 3 . Similarly, one can construct infinitely many examples with arbitrarily complicated JSJ decomposition. Finally, one can arrange that none of these examples are rationally homology cobordant.Theorem 1 is a consequence of the following results about the behavior of Heegaard Floer homology under Dehn surgery. The following results were originally proved for knots in S 3 , but hold more generally for knots in arbitrary integer homology sphere L-spaces.Proposition 2 ([Gai14, Theorem 3]). Let Y be an integer homology sphere L-space and K ⊂ Y a genus one knot. Then U 2 · HF red (Y 1/n (K)) = 0 for any n.

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“…Based on this result, one can ask which manifolds have surgery representations with some restrictions. For example, using Heegaard-Floer homology, [10] provides necessary conditions on manifolds having surgery representation along a knot. In this context, Theorem 3.1.1 can be viewed also as a criterion for a manifold having surgery representation of form S 3 −p/q (K) with K a connected sum of algebraic knots.…”

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confidence: 99%

Seiberg–Witten Invariant of the Universal Abelian Cover of $${S_{-p/q}^{3}(K)}$$

Bodnár

Némethi

2017

Singularities and Computer Algebra

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Abstract. We prove an additivity property for the normalized Seiberg-Witten invariants with respect to the universal abelian cover of those 3-manifolds, which are obtained via negative rational Dehn surgeries along connected sum of algebraic knots. Although the statement is purely topological, we use the theory of complex singularities in several steps of the proof. This topological covering additivity property can be compared with certain analytic properties of normal surface singularities, especially with functorial behaviour of the (equivariant) geometric genus of singularities. We present several examples in order to find the validity limits of the proved property, one of them shows that the covering additivity property is not true for negative definite plumbed 3-manifolds in general.

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Surgery obstructions and Heegaard Floer homology

Cited by 27 publications

References 19 publications

Manolescu invariants of connected sums

Manolescu invariants of connected sums

A note on surgery obstructions and hyperbolic integer homology spheres

Seiberg–Witten Invariant of the Universal Abelian Cover of $${S_{-p/q}^{3}(K)}$$

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