Quasi-abelian crossed modules and nonabelian cohomology

Crossed squares and 2-crossed modules

A. M. S. Classification: 18G30, 18G55.

Braids and crossed modules

For n ≥ 4, we describe Artin’s braid group on n strings as a crossed module over itself. In particular, we interpret the braid relations as crossed module structure relations. Subject classification: Primary: 20F36 57M05 57M20 57M25; Secondary: 18D50 18G55 20C08 55P48

Nonabelian Cohomology and Obstructions, following Wojtkowiak [2]

Let D be a small category. We define a variety of cohomology objects. Each starts with a functor to a category of " coefficents, " and produces a different sort of object as output. The coefficient categories are: sets Set, groups Gp, abelian groups Ab, and " bands " HGp, that is, the category whose objects are groups and whose morphisms are conjugacy classes of homomorphisms. Let S : D → Set b...

Nonabelian Algebraic Topology : filtered spaces , crossed complexes ,

The Abelian/nonabelian Correspondence and Frobenius Manifolds

We propose an approach via Frobenius manifolds to the study (began in [BCK2]) of the relation between rational Gromov-Witten invariants of nonabelian quotients X//G and those of the corresponding “abelianized” quotients X//T, for T a maximal torus in G. The ensuing conjecture expresses the Gromov-Witten potential of X//G in terms of the potential of X//T. We prove this conjecture when the nonab...