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Naturality and movie invariance of the decorated knot Floer map

Statement

Let $(W, Σ, A)$ be a decorated knot cobordism between doubly pointed knots in $S^{3}$ , and let $s \in {Spin}^{c} (W)$ . The representative maps from any two parametrized Kirby decompositions and admissible regular Heegaard choices are filtered $F_{2} [U, V]$ -equivariantly chain homotopic. If the decorated cobordism is a composite of $(W_{1}, Σ_{1}, A_{1})$ followed by $(W_{2}, Σ_{2}, A_{2})$ , then for fixed Spin-c structures $s_{i}$ on $W_{i}$ whose restrictions agree on the identified intermediate boundary $Y_{m i d}$ , so that $s_{1} ∣_{Y_{m i d}} = s_{2} ∣_{Y_{m i d}}$ ,

{\tilde{F}}_{W_{2}, Σ_{2}, A_{2}, s_{2}} \circ {\tilde{F}}_{W_{1}, Σ_{1}, A_{1}, s_{1}} ≃ \sum_{\begin{array}{c} s \in {Spin}^{c} (W) \\ s ∣_{W_{i}} = s_{i}, i = 1, 2 \end{array}} {\tilde{F}}_{W, Σ, A, s} .

The sum is over Spin-c structures on the glued cobordism restricting to the specified structures on the two pieces. For $W = S^{3} \times [0, 1]$ , ${Spin}^{c} (W)$ has one element; the sum has one term if the specified structures on the pieces are its restrictions, and is empty otherwise. In the compatible case the composition law is ordinary composition up to filtered equivariant chain homotopy. If the restrictions on $Y_{m i d}$ differ, the sector maps are not composable; the formula makes no assertion for that pair.

Statement references

The decorated surface and filtered two-color target

Maps for fixed decomposition data and one Spin-c structure

Triangle analysis required by the map construction

Zemke, Link cobordisms and functoriality in link Floer homology, Theorems A and B, §§10–13

Juhasz, Cobordisms of sutured manifolds and the functoriality of link Floer homology, §§8 and 11.3

Facts & Assumptions

Given: The decorated cobordism, fixed Spin-c structures, representative maps and, for composition, actual gluing specified in the Statement, with the construction assumptions in [F3].

[F1]

External full naturality theorem. Zemke, arXiv:1610.05207v3, Theorem A (p.3), gives a fixed-Spin-c $Z^{P}$ -filtered $F_{2} [X_{p} : p \in P]$ -equivariant chain map, invariant under diffeomorphism up to filtered equivariant chain homotopy. Diagram changes are compared by the transitive system of Definition 2.15 and Proposition 3.5 (§§2.2 and 3.3, pp.11–14). Theorem 11.5 and its proof (pp.74 and 84–85), followed by Theorem A' and its fourteen-move proof (§12.4, pp.91–95), supply changes of parametrized Kirby decomposition. Definition 12.8, Lemma 12.9 and the final proof of Theorem A (§12.5, pp.95–97) supply empty-end puncturing and its independence. These are imported results; their move checks are not claimed as locally reconstructed. (Source)

[F2]

External Spin-c composition theorem. Zemke, Theorem B (pp.3 and 97–98), gives the exact extension-sum formula in the Statement. Its proof stacks decompositions, reorders the elementary maps, applies the two-handle composition law (Proposition 5.10, p.43), and handles empty ends by puncturing. The sum and its chain-homotopy interpretation are part of this imported theorem; no unique global extension is assumed in a general gluing. (Source)

[F3]

The map Definition binds these imports to the specified full dividing-set category and to the two-color specialization, with actual representatives of the required auxiliary homology classes and natural endpoint transition maps. Its explicit finite/countable choice specialization supplies the stated CC scope; the publication is not assumed to have made that foundations analysis itself. The existing local analytic triangle theorem is preserved at its stated scope and is not used as a substitute for [F1] or [F2]. (The decorated knot Floer cobordism map)

Proof

1.1F1F3given

Send the w-regions and z-regions of the given surface to the two colors $U, V$ . The source ring is now exactly $F_{2} [U, V]$ and its variable-power filtration is exactly the specified two-coordinate filtration. The curvature vanishes by the two-color identity. Hence [F1] applies to the actual maps of [F3] as genuine filtered equivariant chain maps, not merely as hat homology maps or uncolored counts.

2.1F1step 1.1

For different endpoint diagrams first use [F1]'s canonical transition chain-homotopy equivalences to compare maps on the same chosen endpoint representatives. Theorem A then gives a filtered equivariant chain homotopy for any two admitted decompositions and regular auxiliary choices. The source's fourteen Cerf moves include the band, basepoint and quasi-stabilization relations; §12.5 additionally handles components missing one or both ends. Boundary-fixed ambient isotopy preserves the given decorations and induces precisely the required diffeomorphism of decorated cobordisms, so it is covered by the same theorem.

2.2F2F3step 1.1

The matching boundary restrictions make the target sector of the first map the source sector of the second, after the natural intermediate-diagram identification. Apply [F2] to this typed composition and the two specified structures $s_{1}, s_{2}$ . Restriction of each summand to the two pieces is exactly the condition displayed in the Statement; no unmarked or total-Spin-c map has replaced these maps. The two-color specialization respects the composition and homotopies, giving the displayed formula over $F_{2} [U, V]$ .

3.1givenstep 2.1step 2.2∎

Finally $H^{2} (S^{3} \times [0, 1]; Z) = 0$ and the product is spin. Its nonempty Spin-c torsor therefore consists of one element. If the chosen structures on the pieces restrict from this unique product structure, the extension set has that one element; otherwise it is empty. In the compatible case the law reduces to ordinary composition up to filtered equivariant chain homotopy. The empty-sum convention does not define a composition between different intermediate sectors. The argument imports the established naturality/composition theorems, not an unproved assertion that the local triangle identity checks all movies.

Depends on

Used by

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Sources

Ian Zemke, Link cobordisms and functoriality in link Floer homology, Theorems A and B, §§10–13 (standard reference, not scraped)
Andras Juhasz, Cobordisms of sutured manifolds and the functoriality of link Floer homology, §§8 and 11.3 (standard reference, not scraped)