Quadruple Points of Regular Homotopies of Surfaces in 3-Manifolds

Abstract

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(H_0) - Q(H_1) for any generic regular homotopy H_t : F -> M. Such an invariant exists iff q=0 for any "closed" generic regular homotopy (abbreviated CGRH.) Max and Banchoff proved that for any CGRH H_t: S^2 -> R^3 indeed q(H_t)=0. We generalize their result as follows: Theorem 3.9: Let F be a system of closed surfaces, and let H_t : F -> R^3 be any CGRH, then q(H_t)=0. Theorem 3.15: Let M be an orientable irreducible 3-manifold with pi_3(M)=0. Let F be a system of closed orientable surfaces. If H_t : F -> M is any CGRH in the regular homotopy class of an embedding, then q(H_t)=0. We demonstrate the need for the assumptions of Theorem 3.15 in various counter-examples. We give an explicit formula for the above mentioned invariant Q for embeddings of a system of tori in R^3.