Seidel–Smith cohomology for tangles

Abstract

Abstract. We generalize the"symplectic Khovanov cohomology" of Seidel and Smith [17] to tangles using the notion of symplectic valued topological field theory introduced by Wehrheim and Woodward [19].

Quilted Floer cohomology

“…Quilted Floer homology was originally designed to construct symplectic versions of gauge theoretic invariants, in particular symplectic versions of Donaldson invariants, which we develop in later papers [42; 43], Seiberg-Witten invariants as in Perutz [25] and Lekili [16], and Khovanov invariants as in Rezazadegan [26]. Applications to symplectic topology are given for moduli spaces of flat bundles in [43], and to classification of Lagrangians in tori in Abouzaid-Smith [1].…”

“…Conjecturally this provides a symplectic construction of Donaldson type gauge theoretic invariants: SO.3/-instanton Floer homology, its higher rank version (though not strictly defined in the gauge theoretic setting), and the SU.n/-tangle invariants defined by KronheimerMrowka [14] from singular instantons. The same setup is used to give alternative constructions of Heegaard Floer homology [16] and Seidel-Smith invariants [26].…”

Floer Field Philosophy

“…Since its first announcement in [85], this Floer field philosophy has been applied to obtain various proposals for 2 + 1 field theories, which are inspired from various gauge theories. Unfortunately, these are still preprints [85,86], work in progress [40], or published [60,46,38,5] but hinging on generalizations of the crucial isomorphism in Floer homology under geometric compositions beyond the (compact monotone) setting in which it was proven in [83]; see Remarks 3.5.4-3.5.8. Instead of discussing the technicalities and possible obstructions, this section focusses on the motivations and thus presents both intuitive and naive reasonings why theories along these lines are to be expected.…”

“…The isomorphism L 12 #L 23 ∼ L 12 • L 23 should generalize directly to exact noncompact settings as long as the Lagrangians have a conical structure near infinity that allows one to use maximum principles to guarantee compactness. An application to the construction of a Floer field theory that extends the link invariants [68] was proposed in [60] but unfortunately seems to be lacking this conical structure.…”

“…The value of Floer field theory to 3-manifold topology is mostly of philosophical nature -giving a conceptual understanding for invariance proofs and a general construction principle for 3-manifold invariants (and similarly for knots and links), which has since been applied in a variety of contexts [5,38,46,60,85,86]. One main purpose of this paper and the content of §2 is to explain this philosophy and cast the construction principle into rigorous mathematical terms.…”