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In this paper we determine the integral quandle homology groups of Alexander quandles of prime order. As a special case, this settles the delayed Fibonacci conjecture by M. Niebrzydowski and J. H. Przytycki from their 2009 paper. Further, we determine the cohomology group of the Alexander quandle and obtain relatively simple presentations of all higher degree cocycles which generate the cohomology group. Finally, we prove that the integral quandle homology of a finite connected Alexander quandle is annihilated by the order of the quandle. [ABSTRACT FROM AUTHOR]

We define families of invariants for elements of the mapping class group of Σ, a compact-orientable surface. For any characteristic subgroup , let J(H) denote the subgroup of mapping classes that induce the identity on π1(Σ)/H. To any unitary representation ψ of π1(Σ)/H, we associate a higher-order ρψ-invariant and a signature 2-cocycle σψ. These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular, each ρψ is a quasimorphism and each σψ is a bounded 2-cocycle on J(H). In one of the simplest nontrivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ. [ABSTRACT FROM PUBLISHER]

We identify Čech cocycles in nonabelian (formal) group cohomology with Maurer–Cartan elements in a suitable L ∞-algebra. Applications to deformation theory are described. [ABSTRACT FROM PUBLISHER]

We compute the third cohomology group of a dihedral quandle of odd order with finite field coefficient, generalizing our previous work. In particular, we obtain many new examples of non-trivial 3-cocycles. We check non-triviality of the associated invariant for knots. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to determine the homology groups of trivial quandles with good involutions. We also show that the quandle cocycle invariants of surface-links obtained from trivial quandles with good involutions are equivalent to the triple linking numbers. [ABSTRACT FROM AUTHOR]