Thurston has claimed (unpublished) that central extensions of word hyperbolic groups by finitely generated abelian groups are automatic. We show that they are in fact biautomatic. Further, we show that every 2-dimensional cohomology class on a word hyperbolic group can be represented by a bounded 2-cocycle. This lends weight to the claim of Gromov that for a word hyperbolic group, the cohomolog...

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed family of explicit easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees $(\sigma,\sigma^*)$-compatible orientations, where the orientations are representatives equivalence classes up to cycle-cocycle reversal which determined by cycle signature $\sigma$ cocycle $\sigma^*$. Their proof makes use zo...

We give two sufficient and necessary conditions for the Hochschild extension of a finite-dimensional algebra by its dual bimodule 2-cocycle to be symmetric algebra.

Abelian string and particle braiding in topological order: Modular SL(3,Z) representation and (3 + 1)-dimensional twisted gauge theory." Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use. The MIT Faculty has made this article openly available. Please share how this access benefits you...

We introduce the notion of an algebraic cocycle as the algebraic analogue of a map to an Eilenberg-MacLane space. Using these cocycles we develop a “cohomology theory” for complex algebraic varieties. The theory is bigraded, functorial, and admits Gysin maps. It carries a natural cup product and a pairing to L-homology. Chern classes of algebraic bundles are defined in the theory. There is a na...

We show that the integration of a 1-cocycle I(X) space long knots in R^3 over Fox-Hatcher 1-cycles gives rise to Vassiliev invariant order exactly three. This result can be seen as continuation previous work second named author, proving Gramain is Casson invariant, unique nontrivial two (up scalar multiplications). The present paper also analogous part Mortier's result. Our differs from, but mo...

We consider the standard contact structure on the supercircle, S1|1, and the supergroups E(1|1), Aff(1|1) and SpO(2|1) of contactomorphisms, defining the Euclidean, affine and projective geometry respectively. Using the new notion of p|q-transitivity, we construct in synthetic fashion even and odd invariants characterizing each geometry, and obtain an even and an odd super cross-ratios. Startin...

We develop a new framework for investigating linear equivalence of divisors on graphs using a generalization of Gioan’s cycle-cocycle reversal system for partial orientations. An oriented version of Dhar’s burning algorithm is introduced and employed in the study of acyclicity for partial orientations. We then show that the Baker-Norine rank of a partially orientable divisor is one less than th...

Infinite presentations are given for all of the higher Torelli groups of once-punctured surfaces. In the case of the classical Torelli group, a finite presentation of the corresponding groupoid is also given. Furthermore, the first Johnson homomorphism, which is defined from the classical Torelli group to the third exterior power of the homology of the surface, is shown to lift to an explicit c...

We construct an associative algebra with a decomposition into the direct sum of the underlying vector spaces of another associative algebra and its dual space such that both of them are subalgebras and the natural symmetric bilinear form is invariant or the natural antisymmetric bilinear form is a Connes 2-cocycle. The former is called a double construction of Frobenius algebra and the latter i...