A Sheaf-theoretic Model for Sl(2,c) Floer Homology
نویسنده
چکیده
Given a Heegaard splitting of a three-manifold Y , we consider the SL(2,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 . The hypercohomology of this sheaf can be viewed as a model for (the dual of) SL(2,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.
منابع مشابه
The Superpolynomial for Knot Homologies
We propose a framework for unifying the sl(N) Khovanov-Rozansky homology (for all N) with the knot Floer homology. We argue that this unification should be accomplished by a triply graded homology theory which categorifies the HOMFLY polynomial. Moreover, this theory should have an additional formal structure of a family of differentials. Roughly speaking, the triply graded theory by itself cap...
متن کاملHolomorphic Disks and Genus Bounds
We prove that, like the Seiberg-Witten monopole homology, the Heegaard Floer homology for a three-manifold determines its Thurston norm. As a consequence, we show that knot Floer homology detects the genus of a knot. This leads to a purely Morse-theoretic interpretation of the genus of a knot. The method of proof also shows that the canonical element of Heegaard Floer homology associated to a w...
متن کاملSheaf-Theoretic Stratification Learning
In this paper, we investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Alexandroff (1937) and McCord (1978), we aim to redirect efforts in the computational topology of triangulated compact polyhedra to the much more computable realm of sheaves on partially ordered sets. Our main result is the construction of stratification learning algorithms framed...
متن کاملDiscrete Morse Theory for Computing Cellular Sheaf Cohomology
Sheaves and sheaf cohomology are powerful tools in computational topology, greatly generalizing persistent homology. We develop an algorithm for simplifying the computation of cellular sheaf cohomology via (discrete) Morse-theoretic techniques. As a consequence, we derive efficient techniques for distributed computation of (ordinary) cohomology of a cell complex.
متن کاملSeiberg–Witten–Floer Homology and Heegard splittings
The Seiberg-Witten gauge theory on four-manifolds has a dimensional reduction that leads to equations on a three-manifold. As in the case of Donaldson theory, these equations are the gradient flow of a functional defined on a Banach manifold. The functional was originally introduced by Kronheimer and Mrowka in the proof of the Thom conjecture [14]. As pointed out by Donaldson [9], it is possibl...
متن کامل