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

Given a surface F, we are interested in ℤ/2 valued invariants of immersions of F into ℝ³, which are constant on each connected component of the complement of the quadruple point discriminant in Imm(F, ℝ³). Such invariants will be called "g-invariants." Given a regular homotopy class A ⊆ Imm(F, ℝ³), we denote by V_n(A) the space of all q-invariants on A of order ≤ n. We show that if F is orientable, then for each regular homotopy class A and each n, dim( V_n(A)/V_n-1(A)) ≤ 1.