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Register a new userIntersection homology duality and pairings: singular, PL, and sheaf-theoretic
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We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential step in proving duality between the Goresky-MacPherson intersection homology product and the intersection cohomology cup product pairing recently defined by the author and McClure. We also provide a correction to the Goresky-MacPherson proof of a version of Poincare duality on pseudomanifolds that is used in the construction of the intersection product.
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, i.e., it is independent of the Heegaard splitting. 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.
Torsion sensitive intersection homology was introduced to unify severa
We provide a generalization of the Deligne sheaf construction of intersection homology theory, and a corresponding generalization of Poincare duality on pseudomanifolds, such that the Goresky-MacPherson, Goresky-Siegel, and Cappell-Shaneson duality theorems all arise as special cases. Unlike classical intersection homology theory, our duality theorem holds with ground coefficients in an arbitrary PID and with no local cohomology conditions on the underlying space. Self-duality does require local conditions, but our perspective leads to a new class of spaces more general than the Goresky-Siegel IP spaces on which upper-middle perversity intersection homology is self dual. We also examine categories of perverse sheaves that contain our torsion-sensitive Deligne sheaves as intermediate extensions.
This note corrects an error in the char(K)=2 case of the authors computation of the bordism groups of K-Witt spaces for the field K.
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