On the Cohomology of Configuration Spaces on Surfaces

The integral cohomology rings of the configuration spaces ofn-tuples of distinct points on arbitrary surfaces (not necessarilyorientable, not necessarily compact and possibly with boundary)are studied. It is shown that for punctured surfaces the cohomologyrings stabilize as the number of points tends to infinity, similarlyto the case of configuration spaces on the plane studied byArnold, and the Goryunov splitting formula relating the cohomologygroups of configuration spaces on the plane and punctured planeto arbitrary punctured surfaces is generalized. Moreover, onthe basis of explicit cellular decompositions generalizing theconstruction of Fuchs and Vainshtein, the first cohomology groupsfor surfaces of low genus are given.