Abstract

Given two loops on a compact surfaces F, it is natural to ask: what is their minimal intersectionnumber during homotopy classes? This number is usually said to be the geometric intersectionnumber. In this talk, we shall explain an algorithmic treatment of such a problem: Determination ofgeometric intersection and selfintersection numbers of loops on surfaces. Some applications ingeometric topology will be illustrated. Our integration have two parts: Nielsen fixed point theory andGrbner-Shirshov basis.