Related papers: Skein Homology
This survey reviews recent advances connecting link homology theories to invariants of smooth 4-manifolds and extended topological quantum field theories. Starting from joint work with Morrison and Walker, I explain how functorial link…
If a (possibly finite) compact Lie group acts effectively, locally linearly, and homologically trivially on a closed, simply-connected four-manifold with second Betti number at least three, then it must be isomorphic to a subgroup of S^1 x…
Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we…
In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We…
Given a closed $n$-manifold, we consider the set of simple homotopy types of $n$-manifolds within its homotopy type, called its simple homotopy manifold set. We characterise it in terms of algebraic K-theory, the surgery obstruction map,…
The goal of this paper is to generalize some of the existing toolkit of combinatorial algebraic topology in order to study the homology of abstract chain complexes. We define shellability of chain complexes in a similar way as for cell…
The B-side of Kontsevich's Homological Mirror Symmetry Conjecture is discussed. We give first a self-contained study of derived categories and their homological algebra, and later restrict to the bounded derived category of schemes and…
We introduce a new spectral sequence for the study of $\mathcal{K}$-manifolds which arises by restricting the spectral sequence of a Riemannian foliation to forms invariant under the flows of $\{\xi_1,...,\xi_s\}$. We use this sequence to…
Scissors congruence groups have traditionally been expressed algebraically in terms of group homology. We give an alternate construction of these groups by producing them as the $0$-level in the algebraic $K$-theory of a Waldhausen…
Skein lasagna modules are a recent tool developed for the study of 4-manifolds. We provide general formula for 1-, 2-, and 3-handle attachments for skein modules defined with any functorial link theory in $S^3 \times I$ generalizing…
Compact K\"{a}hler manifolds satisfy several nice Hodge-theoretic properties such as the Hodge symmetry, the Hard Lefschetz property and the Hodge-Riemann bilinear relations, etc. In this note, we investigate when such nice properties hold…
We construct a new family of skein exact triangles for link Floer homology. The skein triples are described by a triple of rational tangles $(R_0,R_1,R_{2n+1})$, where $R_0$ is the trivial tangle and $R_k$ is obtained from it by applying…
We show certain symmetry of the dimensions of cohomologies of the funda- mental groups of compact Sasakian manifolds by using the Hodge theory of twisted basic cohomology. As applications, we show that the polycyclic fundamental groups of…
In this paper we introduce the notion of admissible skein modules associated to an ideal in a pivotal category. We explain how these modules are generalizations of the Kauffman skein algebra and how they relate to renormalized quantum…
Homology of braid groups and Artin groups can be related to the study of spaces of curves. We completely calculate the integral homology of the family of smooth curves of genus $g$ with one boundary component, that are double coverings of…
We use chain level genus zero Gromov-Witten theory to associate to any closed monotone symplectic manifold a formal group (loosely interpreted), whose Lie algebra is the odd degree cohomology of the manifold (with vanishing bracket). When…
As an appropriate generalisation of the features of the classical (Schein) theory of representations of inverse semigroups in $\mathscr{I}_{X}$, a theory of representations of inverse semigroups by homomorphisms into complete atomistic…
We study the homology concordance group of knots in integer homology three-spheres which bound integer homology four-balls. Using knot Floer homology, we construct an infinite number of $\mathbb{Z}$-valued, linearly independent homology…
We prove a generalised version of finiteness of skein modules for 3-manifolds by including boundary. We show that internal skein modules are holonomic modules over the internal skein algebra of the boundary - a property including finite…
The ${\ell}^1$-convolution algebra of a semilattice is known to have trivial cohom ology in degrees 1,2 and 3 whenever the coefficient bimodule is symmetric. We ex tend this result to all cohomology groups of degree $\geq 1$ with symmetric…