Narrow Results

Given a principal $G$-bundle $P\to M$ and two $C^1$ curves in $M$ with coinciding endpoints, we say that the two curves are holonomically equivalent if the parallel transport along them is identical for any smooth connection on $P$. The main result in this paper is that if $G$ is semi-simple, then the two curves are holonomically equivalent if and only if there is a thin, i.e. of rank at most one, $C^1$ homotopy linking them. Additionally, it is also demonstrated that this is equivalent to the factorizability through a tree of the loop formed from the two curves and to the reducibility of a certain transfinite word associated to this loop. The curves are not assumed to be regular.

We give a topological interpretation of the free metabelian group, following the plan described in [11, 12]. Namely, we represent the free metabelian group with d-generators as an extension of the group of the first homology of the d-dimensional lattice (as Cayley graph of the group ℤ^d), with a canonical 2-cocycle. This construction opens a possibility to study metabelian groups from new points of view; in particular to give useful normal forms of the elements of the group, leading to applications to the random walks, and so on. We also describe the satellite groups which correspond to all 2-cocycles of cohomology group associated with the free solvable groups. The homology of the Cayley graph can be used for studying the wide class of groups which include the class of all solvable groups.

Leech’s (co)homology groups of finite cyclic monoids are computed.

We introduce a family of cyclic presentations of groups depending on a finite set of integers. This family contains many classes of cyclic presentations of groups, previously considered by several authors. We prove that, under certain conditions on the parameters, the groups defined by our presentations cannot be fundamental groups of closed connected hyperbolic 3–dimensional orbifolds (in particular, manifolds) of finite volume. We also study the split extensions and the natural HNN extensions of these groups, and determine conditions on the parameters for which they are groups of 3–orbifolds and high–dimensional knots, respectively.

This paper shows how, in principle, simplicial methods, including the well-known Dold–Kan construction can be applied to convert link homology theories into homotopy theories. The paper studies particularly the case of Khovanov homology and shows how simplicial structures are implicit in the construction of the Khovanov complex from a link diagram and how the homology of the Khovanov category, with coefficients in an appropriate Frobenius algebra, is related to Khovanov homology. This Khovanov category leads to simplicial groups satisfying the Kan condition that are relevant to a homotopy theory for Khovanov homology.

In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang–Baxter operators (we will call it the “algebraic” version in this paper). In 2012, Przytycki defined another homology theory for pre-Yang–Baxter operators which has a nice graphic visualization (we will call it the “graphic” version in this paper). We show that they are equivalent. The “graphic” homology is also defined for pre-Yang–Baxter operators, and we give some examples of its one-term and two-term homologies. In the two-term case, we have found torsion in homology of Yang–Baxter operator that yields the Jones polynomial.

Ng constructed an invariant of knots in ${ℝ}^3$, a combinatorial knot contact homology. Extending his study, we construct an invariant of surface-knots in ${ℝ}^4$ using diagrams in ${ℝ}^3$.

Feller et al. showed the homology and the intersection form of a closed trisected four-manifold are described in terms of trisection diagram. In this paper, it is confirmed that we are able to calculate those of a trisected four-manifold with boundary in a similar way. Moreover, we describe a representative of the second Stiefel–Whitney class by the relative trisection diagram.

C.F. Gauss gave a necessary condition for a word to be the intersection word of a closed normal planar curve and he gave an example which showed that his condition was not sufficient. M. Dehn provided a solution to the planarity problem [3] and subsequently, different solutions have been given by a number of authors (see [9]). However, all of these solutions are algorithmic in nature. As B. Grünbaum remarked in [7], “they are of the same aesthetically unpleasing character as MacLane’s [1937] criterion for planarity of graphs. A characterization of Gauss codes in the spirit of the Kuratowski criterion for planarity of graphs is still missing”. In this paper we use the work of J. Scott Carter [2] to give a necessary and sufficient condition for planarity of signed Gauss words which is analogous to Gauss’s original condition.

In this paper, relations between topological entropy, volume growth and the growth in topological complexity from homotopical and homological point of view are discussed for random dynamical systems. It is shown that, under certain conditions, the volume growth, the growth in fundamental group and the growth in homological group are all bounded from above by the topological entropy.

Starting with a numerically noncritical (at $p$) Hecke eigenclass $f$ in the homology of a congruence subgroup $\mathrm{\Gamma }$ of ${\mathrm{\text{SL}}}_3(\mathbb{Z})$ (where $p$ divides the level of $\mathrm{\Gamma }$) with classical coefficients, we first show how to compute to any desired degree of accuracy a lift of $f$ to a Hecke eigenclass $F$ with coefficients in a module of $p$-adic distributions. Then we show how to find to any desired degree of accuracy the germ of the projection $Z$ to weight space of the eigencurve around the point $z$ corresponding to the system of Hecke eigenvalues of $F$. We do this under the conjecturally mild hypothesis that $Z$ is smooth at $z$.

Uniform spanning trees on finite graphs and their analogues on infinite graphs are a well-studied area. On a Cayley graph of a group, we show that they are related to the first ℓ²-Betti number of the group. Our main aim, however, is to present the basic elements of a higher-dimensional analogue on finite and infinite CW-complexes, which relate to the higher ℓ²-Betti numbers. One consequence is a uniform isoperimetric inequality extending work of Lyons, Pichot, and Vassout. We also present an enumeration similar to recent work of Duval, Klivans and Martin.

We show that for every finite-volume hyperbolic $3$-manifold $M$ and every prime $p$ we have $dim{H}_1(M;F_p)<168.602\cdot \mathrm{\text{vol}}(M)$. There are slightly stronger estimates if $p=2$ or if $M$ is non-compact. This improves on a result proved by Agol, Leininger and Margalit, which gave the same inequality with a coefficient of $334.08$ in place of $168.602$. It also improves on the analogous result with a coefficient of about $260$, which could have been obtained by combining the arguments due to Agol, Leininger and Margalit with a result due to Böröczky. Our inequality involving homology rank is deduced from a result about the rank of the fundamental group: if $M$ is a finite-volume orientable hyperbolic $3$-manifold such that ${\pi }_1(M)$ is $2$-semifree, then $\mathrm{rank}{\pi }_1(M)<1+\lambda 0\cdot \mathrm{\text{vol}}M$, where $\lambda 0$ is a certain constant less than $167.79$.