We rigorously define the Liouville action functional for the finitely generated, purely loxodromic quasi-Fuchsian group using homology and cohomology double complexes naturally associated with the group action. We prove that classical action – the critical value of the Liouville action functional, considered as a function on the quasiFuchsian deformation space, is an antiderivative of a 1-form ...

It has long been known that every quasi-homogeneous normal complex surface singularity with Q–homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called “splice type singularities,” which generalize Brieskorn complete intersections....

1. Review of Thurston’s hyperbolization theorem and quasi-Fuchsian representations 2. Review of Thurston’s Surgery Theorem 3. Review of Freedman’s surfaces and Cooper-Long-Reid extension of Freedman’s work 4. Proof of the Surgery Theorem 5. A fundamental trihotomy for π1injective surfaces 6. Proof of the double coset theorem 7. Quasi-Fuchsian knots and analytic proof of Freedman-Cooper-Long the...

Extending ideas of Hedden-Ni, we show that the module structure on Khovanov homology detects split links. We also prove an analogue for untwisted Heegaard Floer branched double cover. Technical results proved along way include two interpretations in terms twisted and fact reduced complex a link is well defined up to quasi-isomorphism.

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a bas...

In this paper, we introduce the notion of Reidemeister torsion for quasi-isomorphisms of based chain complexes over a field. We call a chain map a quasi-isomorphism if its induced homomorphism between homology is an isomorphism. Our notion of torsion generalizes the torsion of acyclic based chain complexes, and is a chain homotopy invariant on the collection of all quasi-isomorphisms from a bas...

For a map f : X → Y of quasi-compact quasi-separated schemes, we discuss quasi-perfection, i.e., the right adjoint f of Rf∗ respects small direct sums. This is equivalent to the existence of a functorial isomorphism f×OY ⊗ L Lf(− ) −→ ∼ f(−); to quasi-properness (preservation by Rf∗ of pseudo-coherence, or just properness in the noetherian case) plus boundedness of Lf (finite tor-dimensionality...

A quasi-regular cell complex is defined and shown to have a reasonable barycentric subdivision. In this setting, Whitney's theorem that the ^-skeleton of the barycentric subdivision of a triangulated n-manifold is dual to the (n /c)th Stiefel-Whitney cohomology class is proven, and applied to projective spaces, lens spaces and surfaces. 1. QR complexes. A (finite) cell structure on a space X is...

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T, φ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of every T -bundle over S with first Betti number equal to one, and we compare our results with those of Lebow on the embedded contact homology of such torus bu...