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A sheaf-theoretic model for SL \((2, \mathbb C)\) Floer homology. (English) Zbl 1455.57018

Given are a \(3\)-dimensional manifold \(Y\) with a fixed Heegaard splitting, the \(\operatorname{SL}(2, \mathbb C)\) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies in the Heegaard splitting. The authors of the paper under review consider the smooth open subset corresponding to irreducible representations, on that subset the intersection of the Lagrangians is an oriented \(d\)-critical locus (this notion was defined by D. Joyce [J. Differ. Geom. 101, No. 2, 289–367 (2015; Zbl 1368.14027)]). Following the work of V. Bussi [“Categorification of Lagrangian intersections on complex symplectic manifolds using perverse sheaves of vanishing cycles”, Preprint, arXiv:1404.1329] the authors associate to such an intersection a perverse sheaf of vanishing cycles. The main result of the paper states the following:
Theorem: Let \(Y\) be a closed, connected, oriented \(3\)-dimensional manifold. Then, the perverse sheaf of vanishing cycles \(P^{\bullet}(Y)\) (constructed from a Heegaard decomposition, as discussed above) is an invariant of the three-manifold \(Y\), up to canonical isomorphism in a category of perverse sheaves. As a consequence, its hypercohomology \[ HP^*(Y) := \mathbb H^*( P^{\bullet}(Y))\] is also an invariant of \(Y\), well-defined up to canonical isomorphism in the category of \(\mathbb Z\)-graded Abelian groups.
The hypercohomology of the perverse sheaf can be seen as a model for the dual of \(\operatorname{SL}(2, \mathbb C)\) instanton Floer homology. In addition, the authors of the paper under review develop a framed version of this construction, which takes into account reducible representations. Finally, the authors provide explicit computations for Brieskorn spheres and lens spaces, and discuss the connection to Khovanov homology and the Kapustin-Witten equations.

MSC:

57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
53D40 Symplectic aspects of Floer homology and cohomology
57R58 Floer homology

Citations:

Zbl 1368.14027
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References:

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