Research on Splitting Isomorphism of Leibniz Algebra and Non-Abelian expansion

In this study, Leibniz algebras and the derivations and properties of Leibniz algebras were given, respectively. The stable automorphism group of explicit splitting extension was calculated via the stable automorphism group of Abelian extension of finite group splitting. Based on the stable automorphism group of the splitting extension studied, the non-Abelian extension and the second order non-Abelian co-homology group of Leibniz algebra were investigated in detail according to the stable automorphism group of the splitting extension.

Computing the fundamental group of a higher-rank graph

We compute a presentation of the fundamental group of a higher-rank graph using a coloured graph description of higher-rank graphs developed by the third author. We compute the fundamental groups of several examples from the literature. Our results fit naturally into the suite of known geometrical results about higher-rank graphs when we show that the abelianization of the fundamental group is the homology group. We end with a calculation which gives a non-standard presentation of the fundamental group of the Klein bottle to the one normally found in the literature.

On the Loop Homology of a Certain Complex of RNA Structures

In this paper, we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence–structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure and that the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

On the Loop Homology of a Certain Complex of RNA Structures

In this paper we establish a topological framework of τ-structures to quantify the evolutionary transitions between two RNA sequence-structure pairs. τ-structures developed here consist of a pair of RNA secondary structures together with a non-crossing partial matching between the two backbones. The loop complex of a τ-structure captures the intersections of loops in both secondary structures. We compute the loop homology of τ-structures. We show that only the zeroth, first and second homology groups are free. In particular, we prove that the rank of the second homology group equals the number γ of certain arc-components in a τ-structure, and the rank of the first homology is given by γ−χ+1, where χ is the Euler characteristic of the loop complex.

A note on super Koszul complex and the Berezinian

We construct the super Koszul complex of a free supercommutative A-module V of rank p|q and prove that its homology is concentrated in a single degree and it yields an exact resolution of A. We then study the dual of the super Koszul complex and show that its homology is concentrated in a single degree as well and isomorphic to $$\Pi ^{p+q} A$$ Π p + q A , with $$\Pi $$ Π the parity changing functor. Finally, we show that, given an automorphism of V, the induced transformation on the only non-trivial homology class of the dual of the super Koszul complex is given by the multiplication by the Berezinian of the automorphism, thus relating this homology group with the Berezinian module of V.

The Topological Entropy Conjecture

For a compact Hausdorff space X, let J be the ordered set associated with the set of all finite open covers of X such that there exists nJ, where nJ is the dimension of X associated with ∂. Therefore, we have Hˇp(X;Z), where 0≤p≤n=nJ. For a continuous self-map f on X, let α∈J be an open cover of X and Lf(α)={Lf(U)|U∈α}. Then, there exists an open fiber cover L˙f(α) of Xf induced by Lf(α). In this paper, we define a topological fiber entropy entL(f) as the supremum of ent(f,L˙f(α)) through all finite open covers of Xf={Lf(U);U⊂X}, where Lf(U) is the f-fiber of U, that is the set of images fn(U) and preimages f−n(U) for n∈N. Then, we prove the conjecture logρ≤entL(f) for f being a continuous self-map on a given compact Hausdorff space X, where ρ is the maximum absolute eigenvalue of f*, which is the linear transformation associated with f on the Čech homology group Hˇ*(X;Z)=⨁i=0nHˇi(X;Z).

One-relator groups and algebras related to polyhedral products

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$ , we specify a necessary and sufficient combinatorial condition for the commutator subgroup $RC_K'$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex $\mathcal {R}_K$ , to be a one-relator group; and for the Pontryagin algebra $H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For $RC_K'$ , it is given by a condition on the homology group $H_2(\mathcal {R}_K)$ , whereas for $H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of $\mathcal {Z}_K$ .

Semisymmetric Zp-covers of the C20 graph

A graph X is said to be G-semisymmetric if it is regular and there exists a subgroup G of A:=Aut(X) acting transitively on its edge set but not on its vertex set. In the case of G=A, we call X a semisymmetric graph. Finding elementary abelian covering projections can be grasped combinatorially via a linear representation of automorphisms acting on the first homology group of the graph. The method essentially reduces to finding invariant subspaces of matrix groups over prime fields. In this study, by applying concept linear algebra, we classify the connected semisymmetric zp-covers of the C20 graph.

Homology of polyomino tilings on flat surfaces

The homology group of a tiling introduced by M. Reid is studied for certain topological tilings. As in the planar case, for finite square grids on topological surfaces, the method of homology groups, namely the non-triviality of some specific element in the group allows a ?coloring proof? of impossibility of a tiling. Several results about the non-existence of polyomino tilings on certain square-tiled surfaces are proved in the paper.

Multivariate Alexander quandles, II. The involutory medial quandle of a link (corrected)

Joyce showed that for a classical knot [Formula: see text], the involutory medial quandle [Formula: see text] is isomorphic to the core quandle of the homology group [Formula: see text], where [Formula: see text] is the cyclic double cover of [Formula: see text], branched over [Formula: see text]. It follows that [Formula: see text]. In this paper, the extension of Joyce’s result to classical links is discussed. Among other things, we show that for a classical link [Formula: see text] of [Formula: see text] components, the order of the involutory medial quandle is bounded as follows: [Formula: see text] In particular, [Formula: see text] is infinite if and only if [Formula: see text]. We also show that in general, [Formula: see text] is a strictly stronger invariant than [Formula: see text]. That is, if [Formula: see text] and [Formula: see text] are links with [Formula: see text], then [Formula: see text]; but it is possible to have [Formula: see text] and [Formula: see text]. In fact, it is possible to have [Formula: see text] and [Formula: see text].