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(H_t) denote the number mod 2 of quadruple points of a generic regular homotopy H_t:F → M. We are interested in defining an invariant Q : GI → ℤ/2 such that q(H_t) = Q(H₀) - Q(H₁) for any generic regular homotopy H_t:F → M. Such an invariant exists iff q = 0 for any closed generic regular homotopy (abbreviated CGRH). We prove that indeed q(H_t) = 0 for any CGRH H_t : 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.