Related papers: The rational classification of links of codimensio…

Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…

If L is an oriented link with $n$ components, then the rank of its Khovanov homology is at least $2^n$. We classify all the links whose Khovanov homology with Z/2-coefficients achieves this lower bound, and show that such links can be…

Let A be a Hopf algebra and H a coalgebra. We shall describe and classify up to an isomorphism all Hopf algebras E that factorize through A and H: that is E is a Hopf algebra such that A is a Hopf subalgebra of E, H is a subcoalgebra in E…

We prove that for any prime $p$ the finite $p$-groups of fixed coclass have only finitely many different mod-$p$ cohomology rings between them. This was conjectured by Carlson; we prove it by first proving a stronger version for groups of…

We establish a bridge between homotopy groups of spheres and commutator calculus in groups, and solve in this manner the "dimension problem" by providing a converse to Sjogren's theorem: every abelian group of bounded exponent can be…

We propose definitions of complex manifolds $\mathcal{P}_M(X,m,n)$ that could potentially be used to construct the symplectic Khovanov homology of $n$-stranded links in lens spaces. The manifolds $\mathcal{P}_M(X,m,n)$ are defined as moduli…

We consider arrangements of n connected codimensional one submanifolds in closed d-dimensional manifold M. Let f be the number of connected components of the complement in M to the union of submanifolds. We prove the sharp lower bound for f…

We show there exists a linear embedding of $K_{3,3,1}$ with n nontrivial 2-component links if and only if n = 1, 2, 3, 4, or 5.

Let Y be a closed oriented 3-manifold with a contact form such that all Reeb orbits are nondegenerate. The embedded contact homology (ECH) index associates an integer to each relative 2-dimensional homology class of surfaces whose boundary…

Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L)…

Given a manifold N and a number m, we study the following question: is the set of isotopy classes of embeddings N->S^m finite? In case when the manifold N is a sphere the answer was given by A. Haefliger in 1966. In case when the manifold N…

In order to classify concordance classes of codimension 2 embeddings in a manifold M, we need to determine the complement of such an embedding. These complements are spaces over M well defined up to some homology equivalence. We construct a…

Let $R$ be a commutative ring that is free of rank $k$ as an abelian group, $p$ a prime, and $SL(n,R)$ the special linear group. We show that the Lie algebra associated to the filtration of $SL(n,R)$ by $p$-congruence subgroups is…

We consider Lie groups ${\rm SU}(n,1)$ and ${\rm Sp}(n,1)$ that act as the isometries of the complex and quaternionic hyperbolic spaces respectively. We classify pairs of semisimple elements in ${\rm Sp}(n,1)$ and ${\rm SU}(n,1)$ up to…

We prove a general dichotomy theorem for groups of finite Morley rank with solvable local subgroups and of Pr\"ufer p-rank at least 2, leading either to some p-strong embedding, or to the Pr\"ufer p-rank being exactly 2.

Finite $p$-groups of nilpotency class 2 are treated from the perspective of central extensions. Given finite abelian groups $G,A$, we derive an explicit formula for cocycles representing elements of $H^2(G,A)$, compute $H^2(G,A)$, and…

The main theorem shows that if M is an irreducible compact connected orientable 3-manifold with non-empty boundary, then the classifying space BDiff(M rel dM) of the space of diffeomorphisms of M which restrict to the identity map on…

Habegger and Lin gave a classification of the link-hmotopy classes of links as the link-homotopy classes of string links modulo the actions of conjugations and partial conjugations for string links. In this paper, we calculated the actions…

Let M_n be the class of all multiloop algebras of finite dimensional simple Lie algebras relative to n-tuples of commuting finite order automorphisms. It is a classical result that M_1 is the class of all derived algebras modulo their…

Przytycki and Sokolov proved that a three-manifold admits a semi-free action of the finite cyclic group of order $p$ with a circle as the set of fixed points if and only if $M$ is obtained from the three-sphere by surgery along a strongly…