Gluing conjecture for relative Seiberg–Witten invariants

Let X1X_1 be a cobordism between three-manifolds Y1Y_1 and Y2Y_2, and let X2X_2 be a cobordism between Y2Y_2 and Y3Y_3. Denote by DX1\mathcal{D}_{X_1}, DX2\mathcal{D}_{X_2}, and DX1X2\mathcal{D}_{X_1 \cup X_2} the morphisms associated with these cobordisms in the Seiberg–Witten–Floer stable homotopy category. Gluing conjecture. The morphism associated with the composite cobordism satisfies

DX1X2DX2DX1.\mathcal{D}_{X_1 \cup X_2} \cong \mathcal{D}_{X_2} \circ \mathcal{D}_{X_1}.

This is the expected functoriality or gluing law for the relative Seiberg–Witten invariant under composition of cobordisms. The supplied text gives no evidence that the statement has been proved or disproved.

Sources & referencesView supporting material

Primary source

Ciprian Manolescu, “Seiberg-Witten-Floer stable homotopy type of three-manifolds with b_1=0”, arXiv:math/0104024 (2019).

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