arXiv research

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

This paper explains how to compute Khovanov homology of torus links using the Kauffman bracket polynomial.

The study examines the topology of complements of polytopal skeletons.

We establish a long exact sequence for Legendrian submanifolds L in P x R, where P is an exact symplectic manifold, which admit a Hamiltonian isotopy that displaces the projection of L off of itself. In this sequence, the singular homology H_* maps to linearized contact cohomology CH^* which maps to linearized contact …

In this paper we construct Cech cohomology groups that form a Gysin-type long exact sequence for principal torus bundles. This sequence is modeled on a de Rham cohomology sequence published in earlier work by Bouwknegt, Hannabuss and Mathai, which was developed to compute the global properties of T-duality in the prese…

In the paper [1] (arXiv:math/0408333) the authors discuss two possible definitions of the relative Cheeger-Simons characters, the second one fitting into a long exact sequence. Here we relate that picture to the one of the relative Deligne cohomology groups, defined via the mapping cone: we show that there are three me…

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q, Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifols, that essentially involves a complete Ehresmann connection.…

In this paper we analyse the topological group cohomology of finite-dimensional Lie groups. We introduce a technique for computing it (as abelian groups) for torus coefficients by the naturally associated long exact sequence. The upshot in there is that certain morphisms in this long exact coefficient sequence can be a…

We calculate the smooth structure set of $S^p \times S^q$, $S(p, q)$, for $p, q \geq 2$ and $p+q \geq 5$. As a consequence we show that in general $S(4j-1, 4k)$ cannot admit a group structure such that the smooth surgery exact sequence is a long exact sequence of groups. We also show that the image of forgetful map $F:…

In work of Higson-Roe the fundamental role of the signature as a homotopy and bordism invariant for oriented manifolds is made manifest in how it and related secondary invariants define a natural transformation between the (Browder-Novikov-Sullivan-Wall) surgery exact sequence and a long exact sequence of C*-algebra K-…

We partially solve the conjecture by A.Shumakovitch about torsion in the Khovanov homology of prime, non-split links in S^3. We give a size restriction on the Khovanov homology of almost alternating links. We relate the Khovanov homology of the connected sum of a link diagram and the Hopf link with the Khovanov homolog…

To a Legendrian knot, one can associate an $\mathcal{A}_{\infty}$ category, the augmentation category. An exact Lagrangian cobordism between two Legendrian knots gives a functor of the augmentation categories of the two knots. We study the functor and establish a long exact sequence relating the corresponding cohomolog…

We provide a suitable axiomatic framework for differential cohomology in the relative case and we deduce the corresponding long exact sequences. We also construct the relative version of the generalized Cheeger-Simons characters and we define the integration map when the fibre has a boundary.

We introduce the holonomy of a singular leaf $L$ of a singular foliation as a sequence of group morphisms from $π_n(L)$ to the $π_{n-1}$ of the universal Lie $\infty$-algebroid of the transverse foliation of $L$. We include these morphisms in a long exact sequence, thus relating them to the holonomy groupoid of Androul…

Let h be a rationally even cohomology theory and h^ the natural differential refinement, as defined by Hopkins and Singer. We consider the possible definitions of the relative differential cohomology groups, generalizing the analogous picture for the Deligne cohomology, and we show the corresponding long exact sequence…

A topological groupoid G is K-pointed, if it is equipped with a homomorphism from a topological group K to G. We describe the homotopy groups of such K-pointed topological groupoids and relate these groups to the ordinary homotopy groups in terms of a long exact sequence. As an application, we give an obstruction to pr…

In this paper, we generalize the notion of Serre fibration to the Morita category of topological groupoids and derive the associated long exact sequence of homotopy groups. We use this results for calculation of homotopy groups of various groupoids, such as the foliation groupoid of a Riemannian foliation.

Given two Lagrangian spheres in an exact symplectic manifold, we find conditions under which the Dehn twists about them generate a free non-abelian subgroup of the symplectic mapping class group. This extends a result of Ishida for Riemann surfaces. The proof generalises the categorical version of Seidel's long exact s…

This is the first paper in a series of two papers. In this paper we construct complexes of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. We call them $k$-Dirac complexes. More explicitly, we will show that each $k$-Dirac complex arises as…

The paper develops a Mayer-Vietoris sequence for groupoid homology.

Develops a spectral sequence for Lie group actions on manifolds.

For each graph we construct graded cohomology groups whose graded Euler characteristic is the chromatic polynomial of the graph. We show the cohomology groups satisfy a long exact sequence which corresponds to the well-known deletion-contraction rule. This work is motivated by Khovanov's work on categorification of the…

The paper improves collapsing Alexandrov spaces results using good coverings.

A new framework for efficient sequence maps using Bayesian filtering and covariance.

We prove that Manolescu and Woodward's Symplectic Instanton homology, and its twisted versions are natural, and define maps associated to four dimensional cobordisms within this theory. This allows one to define representations of the mapping class group and the fundamental group of a 3-manifold, and to have a geometri…

Study shows spherical embedding and immersion components are related to homotopy groups.

The search for generating compatibility conditions (CC) for a given operator is a very recent problem met in General Relativity in order to study the Killing operator for various standard useful metrics (Minkowski, Schwarschild and Kerr). In this paper, we prove that the link existing between the lack of formal exactne…

We provide an explicit section for a mapping class group sequence.

The paper studies splitting maps in link Floer homology using skein exact sequences.

Paper presents new cobordism sequences for singular maps.

We prove that the mod $2$ Betti numbers of double coverings of a complex hyperplane arrangement complement are combinatorially determined. The proof is based on a relation between the mod $2$ Aomoto complex and the transfer long exact sequence. Applying the above result to the icosidodecahedral arrangement ($16$ planes…

X.S. Lin and O. Dasbach proved that the sum of the absolute value of the second and penultimate coefficients of the Jones polynomial of an alternating knot is equal to the twist number of the knot. In this paper we give a new proof of their result using Khovanov homology. The proof is by induction on the number of cros…

Given a knot, we ask how its Khovanov and Khovanov-Rozansky homologies change under the operation of introducing twists in a pair of strands. We obtain long exact sequences in homology and further algebraic structure which is then used to derive topological and computational results. Two of our applications include giv…

This paper answers a basic question about the Birman exact sequence in the theory of mapping class groups. We prove that the Birman exact sequence does not admit a section over any subgroup $Γ$ contained in the Torelli group with finite index. A fortiori this proves that there is no section of the Birman exact sequence…

Corrects earlier work on surface orbifold pure braid groups.

RTRL optimizes long sequences without truncation, converging to loss minima.

Monopole Floer homology is used to prove that real projective three-space cannot be obtained from Dehn surgery on a non-trivial knot in the three-sphere. To obtain this result, we use a surgery long exact sequence for monopole Floer homology, together with a non-vanishing theorem, which shows that monopole Floer homolo…

Grid homology theory for spatial graphs extends skein sequence.

The paper studies mapping class groups of 3-manifolds fibered over surfaces.

New exact sequence links cohomology, automorphisms, and extensions of symmetric quandles.

We develop an analogue of the Birman exact sequence for the Torelli subgroup of Aut(F_n). This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group Aut(F_n). These results play an important role in the authors' recent work on the second homology group of the To…

A 2-manifold's group structure is deduced from orbit configuration spaces.

In this paper, we construct a category of short exact sequences of vector bundles and prove that it is equivalent to the category of double vector bundles. Moreover, operations on double vector bundles can be transferred to operations on the corresponding short exact sequences. In particular, we study the duality theor…

This paper defines a spectral sequence connecting knot homologies.

We present the mixture-of-parents maximum entropy Markov model (MoP-MEMM), a class of directed graphical models extending MEMMs. The MoP-MEMM allows tractable incorporation of long-range dependencies between nodes by restricting the conditional distribution of each node to be a mixture of distributions given the parent…

This is the third part of the work on the exact triangles. We construct chain homomorphisms and show exactness of the resulting sequence.

We show a connection between a surgery exact sequence in knot Floer homology and the sequence derived in [18]. As a consequence of this relationship we see that the exact sequence in [18] also works with coherent orientations and admits refinements with respect to spinc-structures. As an application of this discussion,…

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid gr…