Finite order q-invariants of immersions of surfaces into 3-space

Abstract.

Given a surface F, we are interested in ${{\Bbb Z}/2}$ valued invariants of immersions of F into ${{\Bbb R}^3}$, which are constant on each connected component of the complement of the quadruple point discriminant in ${Imm(F,\E)}{{\Bbb R}^3}$. Such invariants will be called “q-invariants.” Given a regular homotopy class $A \subseteq {Imm(F,\E)}{{\Bbb R}^3}$, we denote by $V_n(A)$ the space of all q-invariants on A of order $\leq n$. We show that ifF is orientable, then for each regular homotopy class A and each n, $\dim (V_n (A) / V_{n-1}(A) ) \leq 1$.