Traceless Character Varieties, the Link Surgeries Spectral Sequence, and Khovanov Homology
Abstract
In arXiv:1611.09927, we constructed a well-defined Lagrangian Floer invariant for any closed, oriented 3-manifold Y via the symplectic geometry of so-called traceless SU(2)-character varieties. This invariant, SI(Y), which we refer to as the symplectic instanton homology of Y, was also shown to satisfy an exact triangle for Dehn surgeries on knots which is typical of Floer-theoretic invariants of 3-manifolds. In this article, we demonstrate further structural properties of this symplectic instanton homology. For example, Floer theories are expected to roughly satisfy the axioms of a topological quantum field theory (TQFT), so that in particular they should be functorial with respect to cobordisms. Following a strategy used by Ozsv\'ath and Szab\'o in the context of Heegaard Floer homology, we prove that our theory is functorial with respect to connected 4-dimensional cobordisms, so that cobordisms induce homomorphisms between symplectic instanton homologies. We also generalize the surgery exact triangle by proving that Dehn surgeries on a link L in a 3-manifold Y induce a spectral sequence of symplectic instanton homologies -- the E2-page is isomorphic to a direct sum of symplectic instanton homologies of all possible combinations of 0- and 1-surgeries on the components of L, and the spectral sequence converges to SI(Y). For the branched double cover (L) of a link L ⊂ S3, we show there is a link surgery spectral sequence whose E2-page is isomorphic to the reduced Khovanov homology of L and which converges to the symplectic instanton homology of (L).