Elliptic arrangements, elliptic Artin group and Elliptic Artin monoid
A talk in the Oberseminar Gruppen und Geometrie series by
Kyoji Saito
| Abstract: | Associated with a marked alliptic root system, we
introduce the elliptic period domain, on which elliptic Weyl
group is acting as a reflection group. The system of reflection
hyperplanes form, so calledm an elliptic arrangement. The
elliptic Weyl group acts regular and proper discontinuously
on the complement of the elliptic arrangement, and the regular
orbit space of the elliptic Weyl group is described as a
complement of elliptic discriminant loci in the invariant space. Similar to the classical finite Coxeter arrangement case, we ask the topology (homotopy type) of the elliptic discriminant complement. We have partial answers 1. and 2. to the question: 1. The fundamental group, called an elliptic Artin group, is presented by homogeneous relations defined on the elliptic diagram. Then, we introduce an elliptic Artin monoid using the same homogeneous relations. 2. Using the non-cancellative property of the elliptic Artin monoid, we construct second homotopy classes of the elliptic discriminant complement, and conjecture that the classes are non-vanishing and they generate the full second homotopy group. In particular, 2. should imply that an expectation that "the discriminant complement may be an Eilenberg-MacLane space" may not hold. |