Quadruple points of regular homotopies of surfaces in 3-manifolds

Let GI denote the space of all generic immersions of a surface F into a 3-manifold M. Let q(Ht) denote the number mod 2 of quadruple points of a generic regular homotopy Ht : F→M. We are interested in defining an invariant Q : GI→Z/2 such that q(Ht)=Q(H0)−Q(H1) for any generic regular homotopy Ht : F→M. Such an invariant exists iff q=0 for any closed generic regular homotopy (abbreviated CGRH). ve that indeed q(Ht)=0 for any CGRH Ht : F→R3 where F is any system of surfaces. We prove the same for general 3-manifolds in place of R3 under certain assumptions. We demonstrate the need for these assumptions with various counter-examples. We give an explicit formula for the invariant Q for embeddings of a system of tori in R3.