Extending the Dehn quandle to shears and foliations on the torus
Tom 225 / 2014
Streszczenie
The Dehn quandle, $Q$, of a surface was defined via the action of Dehn twists about circles on the surface upon other circles. On the torus, $\mathbb {T}^2$, we generalize this to show the existence of a quandle $\hat Q$ extending $Q$ and whose elements are measured geodesic foliations. The quandle action in $\hat Q$ is given by applying a shear along such a foliation to another foliation. We extend some results which related Dehn quandle homology to the monodromy of Lefschetz fibrations. We apply certain quandle 2-cycles to yield factorizations of elements of $\mathop {\rm SL}_2(\mathbb {R})$ fixing specified vectors (circles, foliations) and give examples. Using these, we show the quandle homology of $\hat Q$ is nontrivial in all dimensions.
