arXiv research

New algorithm calculates Steenrod squares in Khovanov cohomology.

We extend the definition of Bockstein basis $σ(G)$ to nilpotent groups $G$. A metrizable space $X$ is called a {\it Bockstein space} if $\dim_G(X) = \sup\{\dim_H(X) | H\in σ(G)\}$ for all Abelian groups $G$. Bockstein First Theorem says that all compact spaces are Bockstein spaces. Here are the main results of the pape…

We consider a simple and natural coboundary operator, on the Lie algebra valued differential forms on a manifold, which in the abelian case reduces to usual exterior derivative of such forms. Using the corresponding de Rham cohomology Lie superalgebra H*(M,G) we obtain numerical smooth invariants--as opposed to homotop…

We prove torsion constraints in Khovanov homology for certain links.

We prove a generalization of the Edwards-Walsh Resolution Theorem: Theorem: Let G be an abelian group for which $P_G$ equals the set of all primes $\mathbb{P}$, where $P_G=\{p \in \mathbb{P}: \Z_{(p)}\in$ Bockstein Basis $σ(G)\}$. Let n in N and let K be a connected CW-complex with $π_n(K)\cong G$, $π_k(K)\cong 0$ for…

Spectral sequence connects knot homologies via algebraic geometry.

For a fixed closed manifold $P$, we construct a cobordism category of embedded manifolds with a single Baas-Sullivan singularity of type $P$. Our main theorem identifies the homotopy type of the classifying space of this cobordism category with that of the infinite loop-space of a certain spectrum related to the spectr…

In the paper titled "Bockstein basis and resolution theorems in extension theory" (arXiv:0907.0491v2), we stated a theorem that we claimed to be a generalization of the Edwards-Walsh resolution theorem. The goal of this note is to show that the main theorem from (arXiv:0907.0491v2) is in fact equivalent to the Edwards-…

We answer a weaker version of the classification problem for the homotopy types of $(n-2)$-connected closed orientable $(2n-1)$-manifolds. Let $n\geq 6$ be an even integer, and $X$ be a $(n-2)$-connected finite orientable Poincaré $(2n-1)$-complex such that $H^{n-1}(X;\mathbb{Q})=0$ and $H^{n-1}(X;\mathbb{Z}_2)=0$. The…

Characterizes a general range decreasing group homomorphism.

We construct an algebra of non-trivial homological operations on Khovanov homology with coefficients in $\mathbb Z_2$ generated by two Bockstein operations. We use the unified Khovanov homology theory developed by the first author to lift this algebra to integral Khovanov homology. We conjecture that these two algebras…

We extend certain homomorphisms defined on the higher Torelli subgroups of the mapping class group to crossed homomorphisms defined on the entire mapping class group. In particular, for every $k\geq 2$, we construct a crossed homomorphism $ε_k$ which extends Morita's homomorphism $\tilde τ_k$ to the entire mapping clas…

Two crossing homomorphisms on braid groups are shown to be equivalent.

The study classifies homomorphisms from mapping class groups using finite subgroups.

Study of Chebyshev-Frobenius homomorphism in 3-manifold skein modules.

Study on singularities of bundle homomorphisms induced by Morin maps.

Graph homomorphism numbers embed graphs for classification.

New conditions for weighted composition operators in group homomorphisms.

Study homomorphisms from groups to 3-manifold fundamental groups.

New homomorphisms from knot Floer homology help classify knots.

New homomorphism from Khovanov homology for knot concordance.

Stability of Lie group homomorphisms and subgroups via Moser type argument.

Classifies homomorphisms from braid groups, proving their extensions to automorphisms.

Classifies homomorphisms between specific braid groups.

We introduce the notion of tight homomorphism into a locally compact group with nonvanishing bounded cohomology and study these homomorphisms in detail when the target is a Lie group of Hermitian type. Tight homomorphisms between Lie groups of Hermitian type give rise to tight totally geodesic maps of Hermitian symmetr…

Paper surveys Johnson homomorphisms and related tools.

A chord index homomorphism for knots in thickened surfaces is constructed.

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

Study on Euler class and flux homomorphisms for non-orientable surfaces.

We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,…

We extend each higher Johnson homomorphism to a crossed homomorphism from the automorphism group of a finite-rank free group to a finite-rank abelian group. We also extend each Morita homomorphism to a crossed homomorphism from the mapping class group of once-bounded surface to a finite-rank abelian group. This improve…

A new homomorphism connects group actions on circles to Euler classes.

The paper disproves a conjecture about satellite maps not inducing homomorphisms.

The paper extends Johnson homomorphisms to groups with extended N-series.

We propose an approach to study non-Abelian Iwasawa theory, using the idea of Johnson homomorphisms in low dimensional topology. We introduce arithmetic analogues of Johnson homomorphisms/maps, called the p-Johnson homomorphisms/maps, associated to the Zassenhaus filtration of a pro-p Galois group over a Z_p-extension …

Satellite operations with winding number ≠ 1 are not homomorphisms.

Study of handlebody group intersections with Torelli and Johnson kernels.

Homomorphism from braid groups to Steinberg groups defined.

Johnson and Livingston have characterized peripheral structures in homomorphs of knot groups. We extend their approach to the case of links. The main result is an algebraic characterization of all possible peripheral structures in certain homomorphic images of link groups.

The paper studies symmetries in quandles and their relative versions.

The paper studies new filtrations and homomorphisms related to mapping class groups and 3-manifold invariants.

If phi: G-->G' is a surjective homomorphism, we prove that the twisted Alexander polynomial of G is divisible by the twisted Alexander polynomial of G'. As an application, we show non-existence of surjective homomorphism between certain knot groups.

Study proves equality of LS-category and cohomological dimension for specific group homomorphisms.

The paper shows how surjective homomorphisms between surface braid groups factor and computes their automorphism groups.

Study de Rham homomorphism for Lipschitz cohomologies on metric simplicial complexes.

New homomorphism from Khovanov homology gives slice genus bounds.

Let K be the kernel of an epimorphism G -> Z, where G is a finitely presented group. If K has infinitely many subgroups of index 2, 3, or 4, then it has uncountably many. Moreover, if K is the commutator subgroup of a classical knot group G, then any homomorphism from K onto the symmetric group S_2 lifts to a homomorph…

Johnson has defined a surjective homomorphism from the Torelli subgroup of the mapping class group of the surface of genus $g$ with one boundary component to $\wedge^3 H$, the third exterior product of the homology of the surface. Morita then extended Johnson's homomorphism to a homomorphism from the entire mapping cla…