TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)verified 2026-10-22 (gpt-6.1-sol)

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Correction terms vanish on extending Spin-c structures over rational homology balls

Statement

Let $B$ be a smooth compact connected oriented four-manifold with $H_{*} (B; Q) ≅ H_{*} (p o i n t; Q)$ and oriented boundary the rational homology sphere $Y$ , where $H_{*} (Y; Q) ≅ H_{*} (S^{3}; Q)$ . If a boundary Spin $^{c}$ structure $t$ extends over $B$ , then its Heegaard Floer correction term vanishes: $d (Y, t) = 0$ . The extension hypothesis is essential; no vanishing is asserted for Spin $^{c}$ structures that do not extend.

Statement references

The rational-ball cobordism-map inequalities

Ozsvath and Szabo, Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, Proposition 9.9

Baranowski, Borodzik, and Serrano de Rodrigo, Heegaard Floer Homologies and Rational Cuspidal Curves, Corollary 4.8

Facts & Assumptions

Given: A smooth rational homology four-ball $B$ , its oriented rational homology sphere boundary $Y$ , and an extending boundary Spin $^{c}$ structure $t$ .

[F1]

For this extending structure, puncturing the ball yields both inequalities $d (Y, t) \geq 0$ and $d (Y, t) \leq 0$ (Correction-term inequality from a rational-ball cobordism map).

Proof

technique · combine the two orientation-sensitive inequalities

Given: The data above.

1.1F1given

The forward punctured-cobordism map gives $d (Y, t) \geq 0$ by [F1].

2.1F1step 1.1algebra∎

The reversed punctured cobordism gives $d (Y, t) \leq 0$ by [F1]. A rational number satisfying both inequalities is zero, so $d (Y, t) = 0$ . The argument applies only to the Spin $^{c}$ structure that extends.

Depends on

Used by

Double-cover correction terms obstruct smooth sliceness Corollary
A nonextendible Spin-c structure with nonzero d on a rational-ball boundary Counterexample

Dependency tree · two levels

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Sources

Peter S. Ozsvath and Zoltan Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary (standard reference, not scraped)
Adam Baranowski, Maciej Borodzik, and Juan Serrano de Rodrigo, Heegaard Floer Homologies and Rational Cuspidal Curves (standard reference, not scraped)