Given a Heegaard splitting of a three-manifold $Y$, we consider the SL(2,$\mathbb C$) character variety of the Heegaard surface, and two complex Lagrangians associated to the handlebodies. We focus on the smooth open subset corresponding to irreducible representations. On that subset, the intersection of the Lagrangians is an oriented d-critical locus in the sense of Joyce. Bussi associates to such an intersection a perverse sheaf of vanishing cycles. We prove that in our setting, the perverse sheaf is an invariant of $Y$, i.e., it is independent of the Heegaard splitting. The hypercohomology of the perverse sheaf can be viewed as a model for (the dual of) SL(2,$\mathbb C$) instanton Floer homology. We also present a framed version of this construction, which takes into account reducible representations.We give explicit computations for lens spaces and Brieskorn spheres, and discuss the connection to the Kapustin–Witten equations and Khovanov homology.

中文翻译：

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SL（2，$ \ mathbb C $）Floer同源性的捆理论模型

给定三流形$ Y $的Heegaard分裂，我们考虑了Heegaard表面的SL（2，$ \ mathbb C $）字符变体，以及两个与车体相关的复杂拉格朗日数。我们专注于与不可约表示相对应的光滑开放子集。在该子集上，在乔伊斯的意义上，拉格朗日派的交点是定向的d临界轨迹。Bussi将这样一连串消失的循环与这种交汇联系在一起。我们证明，在我们的环境中，错误捆是$ Y $的不变式，即，它独立于Heegaard分裂。正交束的超同调可以看作是SL（2，$ \ mathbb C $）瞬时Floer同源性的模型。我们还介绍了此构造的框架版本，其中考虑了可简化的表示形式。