Surgery obstructions for knots in integer homology spheres

Abstract: For knot surgery in $S<sup>3$, 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.

• The paper generalizes Floer theoretic surgery obstructions to arbitrary integer homology spheres, establishing explicit numerical bounds for both positive and negative surgeries.
• It employs a mapping cone formalism and d-invariant estimates to restrict surgery outcomes, with specific applications to Brieskorn and Seifert fibered examples.
• The study yields implications for 4-manifold cobordisms by setting a universal lower bound on the second Betti number in smooth cobordisms between integer homology spheres.

Surgery Obstructions for Knots in Integer Homology Spheres

Introduction and Problem Setting

The paper "Surgery obstructions for knots in integer homology spheres" (2607.02028) addresses the natural extension of the classical surgery problem in 3-manifold topology: for a fixed integer homology sphere $Y$, which integer homology spheres $Z$ arise as Dehn surgery on a knot in $Y$? This question generalizes the well-studied case of surgeries in $S^3$, where both algebraic and Heegaard Floer theoretic obstructions are already known. The work leverages and significantly extends the Heegaard Floer theoretic obstructions of Hom-Karakurt-Lidman, developing a systematic framework applicable to arbitrary integer homology spheres.

The main contributions are:

• Generalization of Floer-theoretic surgery obstructions to arbitrary integer homology spheres, for both positive and negative $1/m$-surgeries.
• Explicit numerical bounds for surgery obstructions in the context of Brieskorn spheres and Seifert fibered examples.
• Construction of infinitely many hyperbolic integer homology spheres not realizable as surgeries on knots in a fixed ambient homology sphere.
• Consequences for four-manifold cobordisms, yielding a universal lower bound on $b_2$ in smooth cobordisms between certain pairs of integer homology spheres.

Mapping Cone Formalism and Floer Homological Invariants

The technical base for the obstructions is the mapping cone formula for rational surgeries in Heegaard Floer theory, originally due to Ozsváth-Szabó. For a knot $K$ in an integer homology sphere $Y$, the knot Floer complex $CFK^\infty(Y,K)$ is equipped with a doubly-filtered structure, and the computations of the Floer homology of $Y_{p/q}(K)$ reduce to the homology of an explicit mapping cone constructed from truncations and filtrations of $Z$0.

The crucial invariants are sequences $Z$1 and $Z$2 derived from the knot Floer complex, satisfying key monotonicity and duality relations. The $Z$3-invariant (correction term) and the structure of the reduced Floer homology $Z$4, including the action of $Z$5, provide the main obstructions to the realization of a given $Z$6 as surgery on a knot in $Z$7.

In both positive and negative $Z$8-surgery, correction term estimates tightly constrain possible surgery outcomes:

• For $Z$9, $Y$0,
• For $Y$1, $Y$2, where $Y$3 captures the length of torsion summands in the odd-degree part of $Y$4.

Main Surgery Obstructions

Negative Surgeries

If $Y$5 is an integer homology sphere whose reduced Floer homology is concentrated in even $Y$6-grading, and $Y$7, then $Y$8 cannot be realized as $Y$9 unless $S^3$0. Moreover, if $S^3$1, the surgered manifold necessarily contains Floer homology elements in odd grading, which is a contradiction. This provides a sharp obstruction for negative surgeries, generalizing known results for $S^3$2.

Positive Surgeries

For positive $S^3$3-surgeries, the obstruction is subtler: if the difference $S^3$4 exceeds an explicit threshold $S^3$5, then $S^3$6 for some even grading $S^3$7. The constant $S^3$8 depends only on the ambient manifold and the grading. Since in many cases $S^3$9 acts trivially on $1/m$0 for the candidate $1/m$1, this gives powerful obstructions that often completely rule out the possibility that $1/m$2 for any $1/m$3.

A decisive numerical form appears in the case $1/m$4, where the threshold $1/m$5 is shown to be optimal: for all even $1/m$6, the Brieskorn sphere $1/m$7 cannot be realized as surgery on a knot in $1/m$8.

Applications to Seifert Fibered and Hyperbolic Homology Spheres

The results are explicitly realized in the context of Brieskorn spheres $1/m$9, for $b_2$0 even and large. The $b_2$1-invariant here is $b_2$2, and $b_2$3 acts trivially on $b_2$4. For any fixed integer homology sphere $b_2$5, there exists a threshold $b_2$6 such that for all $b_2$7, $b_2$8 cannot be obtained by $b_2$9 surgery on a knot in $K$0 (for any $K$1), regardless of the genus, Alexander polynomial, or fundamental group properties. This demonstrates that certain Seifert fibered integer homology spheres are universally obstructed from being surgeries in any fixed ambient $K$2.

Furthermore, the construction of Hom-Lidman yields infinitely many hyperbolic integer homology spheres $K$3, arising via surgery on hyperbolic knots in connected sums of Poincaré spheres, such that $K$4 with $K$5. For each fixed $K$6, all such $K$7 with sufficiently large $K$8 fail to be surgeries on a knot in $K$9, thereby showing the ubiquity of the obstruction amongst both geometric and JSJ types.

Consequences for 4-Manifold Cobordism

An immediate application of these obstructions is topological: any smooth, compact, oriented, simply connected 4-manifold $Y$0 with boundary $Y$1, where $Y$2 as above, must have $Y$3. This follows from the observation that if $Y$4, $Y$5 would have to be realizable as surgery on a knot in $Y$6 (the trace construction); the obstructions then imply this is impossible for the families constructed. This provides a universal lower bound for the second Betti number of cobordisms between broad classes of integer homology spheres, independent of their fundamental group or geometric structure.

Theoretical and Practical Implications

The paper provides a comprehensive, computable obstruction framework, grounded in Heegaard Floer homological invariants, for distinguishing which integer homology spheres can or cannot be related via knot surgery in a fixed host manifold. The results demonstrate that:

• Even within the class of irreducible, weight-one-manifold, integer homology spheres, surgery realization is highly constrained.
• Heegaard Floer theory provides not just group-theoretic or Casson-theoretic invariants, but also refined obstructions manifesting in the $Y$7-action and grading structure of $Y$8.
• The techniques generalize to arbitrary JSJ decompositions and provide new restrictions in the study of 4-manifold topology and smooth structures on homology cobordisms.

On the practical side, these obstructions are computable: given $Y$9, one can determine a numerical bound $CFK^\infty(Y,K)$0 and a family $CFK^\infty(Y,K)$1 to check for all large $CFK^\infty(Y,K)$2. The explicit thresholds facilitate computational searches and counterexample construction in low-dimensional topology.

Future Directions

Several avenues follow from the methods and results of the paper:

• Further refinement of the mapping cone techniques to account for more general surgeries (e.g., non-integral slopes, links).
• Investigations of the sharpness of the numerical bounds and possible uniformity across families.
• Analysis of the interaction between Floer-theoretic obstructions and torsion or symmetry phenomena in the fundamental group or $CFK^\infty(Y,K)$3-representation spaces.
• Applications to the study of homology cobordism group structure, and connections to questions on knot genus realization and exotic smooth structures.
• Extensions to involutive and equivariant Heegaard Floer theories, where new obstructions might arise, potentially distinguishing homology spheres that are not distinguished by the ordinary $CFK^\infty(Y,K)$4-invariant and $CFK^\infty(Y,K)$5 structures.

Conclusion

This work establishes a robust and general framework for analyzing knot surgery problems in arbitrary integer homology spheres, demonstrating the power and flexibility of Heegaard Floer homological invariants. The obstructions provided go substantially beyond previously known constraints, covering an extensive range of geometric, topological, and group-theoretic settings. The results underline the effectiveness of Floer-theoretic heuristics in low-dimensional topology and open broad new directions for further investigation in 3- and 4-manifold theory.