Central extensions of groups and adjoint groups of quandles

This paper develops an approach for describing centrally extended groups, as determining the adjoint groups associated with quandles. Furthermore, we explicitly describe such groups of some quandles. As a corollary, we determine some second quandle homologies.Comment: 15 pages, and no figure. This paper is an algebraic part of the previous paper, arXiv:1210.6528, into which I divided as two parts. Furthermore, I rewrote the introduction in the first version, and revised some statements on Symplectic quandles and Coxeter quandle

Finite presentations of centrally extended mapping class groups

We describe a finite presentation of $\mathcal{T}_{g,r}$ for $g \geq 3$. % or $(g,r)=(2,0)$. Here $\mathcal{T}_{g,r}$ is the universal central extension of the mapping class group of the surface of genus $g$ with $r$-boundaries. We also investigate the case $g=2$,Comment: 11 Pages, 5 figures. I revised English errors and some sentences. This paper will be published in Kyushu Journal of Mathematic

Twisted cohomology pairings of knots I; diagrammatic computation

We provide a diagrammatic computation for the bilinear form, which is defined as the pairing between the (relative) cup products with every local coefficients and every integral homology 2-class of every links in the 3-sphere. As a corollary, we construct bilinear forms on the twisted Alexander modules of links.Comment: Any comments are welcome. 23 pages and 5 figures. I divided a certain preprint into some parts. This paper is the first one. Furthermore, in this second version, I rewrote Sections 2,3,4 to make them more understabl

Cocycles of nilpotent quotients of free groups

We focus on the cohomology of the $k$-th nilpotent quotient of the free group, $F/F_k$. This paper describes all the group 2-, 3-cocycles in terms of Massey products, and gives expressions for some of the 3-cocycles. We also give simple proofs of some of the results on Milnor invariants and the Johnson-Morita homomorphisms.Comment: 15 pages. Any comment is welcom

Milnor-Orr invariants from the Kontsevich invariant

As nilpotent studies in knot theory, we focus on invariants of Milnor, Orr, and Kontsevich. We show that the Orr invariant of degree $k$ is equivalent to the tree reduction of the Kontsevich invariant of degree $< 2k$. Furthermore, we will see a close relation between the Orr invariant and the Milnor invariant, and discuss a method of computing these invariantsComment: 8 figures, 16 pages. I revised minor mistakes, and improve some proof

On third homologies of groups and of quandles via the Dijkgraaf-Witten invariant and Inoue-Kabaya map

We propose a simple method to produce quandle cocycles from group cocycles, as a modification of Inoue-Kabaya chain map. We further show that, in respect to "universal central extended quandles", the chain map induces an isomorphism between their third homologies. For example, all Mochizuki's quandle 3-cocycles are shown to be derived from group cocycles of some non-abelian group. As an application, we calculate some $\Z$-equivariant parts of the Dijkgraaf-Witten invariants of some cyclic branched covering spaces, via some cocycle invariant of links.Comment: 27 pages, 6 figures. This paper is a generalization and modification of the previous paper arXiv:1103.3839 [math.GT

Instanton effects in ABJM theory with general R-charge assignments

We study the large N expansion of the partition function of the quiver superconformal Chern-Simons theories deformed by two continuous parameters which correspond to the general R-charge assignment to the matter fields. Though the deformation breaks the conformal symmetry, we find that the partition function shares various structures with the superconformal cases, such as the Airy function expression of the perturbative expansion in 1/N with the overall constant A(k) related to the constant map in the ABJM case through a simple rescaling of k. We also identify five kinds of the non-perturbative effects in 1/N which correspond to the membrane instantons. The instanton exponents and the singular structure of the coefficients depend on the continuous deformation parameters, in contrast to the superconformal case where all the parameters are integers associated with the orbifold action on the moduli space. This implies that the singularity of the instanton effects would be observable also in the gravity side.Comment: 24 pages, two figures; v2: typos corrected and footnote adde

Homotopical interpretation of link invariants from finite quandles

This paper demonstrates a topological meaning of quandle cocycle invariants of links with respect to finite connected quandles $X$, from a perspective of homotopy theory: Specifically, for any prime $\ell$ which does not divide the type of $X$, the $\ell$-torsion of this invariants is equal to a sum of the colouring polynomial and a $\Z$-equivariant part of the Dijkgraaf-Witten invariant of a cyclic branched covering space. Moreover, our homotopical approach involves application of computing some third homology groups and second homotopy groups of the classifying spaces of quandles, from results of group cohomology.Comment: 34 pages, several figures. The previous version was be divided into two papers, as a topological paper and an algebraic one. This revision is the former, and the latter will be put in the arxiv soo

Twisted cohomology pairings of knots III; triple cup products

Given a representation of a link group, we introduce a trilinear form, as a topological invariant. We show that, if the link is either hyperbolic or a knot with malnormality, then the trilinear form equals the pairing of the (twisted) triple cup product and the fundamental relative 3-class. Further, we give some examples of the computation.Comment: 14 page

de Rham theory and cocycles of cubical sets from smooth quandles

We show a de Rham theory for cubical manifolds, and study rational homotopy type of the classifying spaces of smooth quandles. We also show that secondary characteristic classes in \cite{Dup2,DK} produce cocycles of quandles.Comment: 15 pages, 1 figur