A Remark on the Topology of the Regular Loci of Some Complexified Hamiltonian Systems

Abstract

We study the topology of the regular loci of two complexified Hamiltonian integrable systems using the Zariski-van Kampen method. In particular, we show that the fundamental group of the regular locus for the complexified planar Kepler problem is ${\mathbf{F}}_2⋊\mathbb{Z}$ where the action of the generator of $\mathbb{Z}$ interchanges the generators of ${\mathbf{F}}_2$, whereas that for the complexified spherical pendulum is $A\left({\tilde{A}}_3\right)$, that is the affine Artin group of type ${\tilde{A}}_3$.

arXiv comment14 pages, 7 figures, some mistakes have been corrected