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

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Correction-term inequality from a rational-ball cobordism map

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 $H_{*} (Y; Q) ≅ H_{*} (S^{3}; Q)$ . Let $t$ be a boundary Spin $^{c}$ structure extending over $B$ . Removing an open four-ball from the interior gives a Spin $^{c}$ cobordism $W$ with boundary $- S^{3} ⊔ Y$ , and each boundary inclusion induces an isomorphism on rational homology. In both orientations, the induced Heegaard Floer infinity map is an isomorphism of degree zero and the corresponding plus-map inequality gives $d (Y, t) \geq 0$ and $d (Y, t) \leq 0$ .

Statement references

The cobordism degree convention

The generic Heegaard Floer package needed for correction terms

Ozsvath and Szabo, Absolutely Graded Floer Homologies and Intersection Forms for Four-Manifolds with Boundary, Theorem 9.6 and Proposition 9.9

Baranowski, Borodzik, and Serrano de Rodrigo, Heegaard Floer Homologies and Rational Cuspidal Curves, Section 4

Facts & Assumptions

[F1]

Puncturing a rational homology ball gives a rational homology cobordism gives the two rational homology inclusion isomorphisms, the boundary orientations, $χ (W) = 0$ and the zero rational intersection module.

[F2]

Absolute rational Heegaard Floer grading and fixed-Spin-c cobordism maps and Infinity standardness and fixed-Spin-c rational-cobordism isomorphisms construct, for each fixed extending Spin $^{c}$ structure, the compatible infinity and plus maps and prove the degree-zero infinity isomorphism.

[F3]

Heegaard invariance and well-definedness of the correction term supplies the grading-preserving tower and its bottom degree, with $d (S^{3}) = 0$ ; Commuting Laurent and plus maps compare their bottom degrees supplies the bottom-degree comparison for the commuting square.

Proof

Given: The stated rational homology ball, its oriented boundary, and an extending Spin $^{c}$ structure; let $W$ be obtained by removing an interior open ball.

1.1F1F2

The two boundary inclusions are rational homology isomorphisms and $H^{2} (W; Q) = 0$ . In particular the square of the Chern class is zero, and $χ (W) = σ (W) = 0$ . The same assertions hold for the reverse cobordism from $Y$ to $S^{3}$ . For each of the fixed induced Spin $^{c}$ structures, the constructed infinity map is an isomorphism with degree $(0 - 2 \cdot 0 - 3 \cdot 0) / 4 = 0$ , and the corresponding plus map commutes with the infinity-to-plus projection. This uses the individual fixed-structure conclusion, including torsion extensions.

2.1F3step 1.1∎

Apply the supplied commuting-square comparison first to $W : S^{3} \to Y$ and then to its reverse $Y \to S^{3}$ . The degree-zero isomorphisms give respectively $d (Y, t) \geq d (S^{3}) = 0$ and $d (S^{3}) = 0 \geq d (Y, t)$ . These are exactly the two stated inequalities in both orientations.

Depends on

Used by

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

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

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)