Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R³, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T ⊕ P ⊕ Q, where T is a Z valued invariant reflecting the number of triple points of the immersion, and P, Q are Z / 2 valued invariants characterized by the property that for any regularly homotopic immersions i, j : F → R³, P (i) - P (j) ∈ Z / 2 (respectively, Q (i) - Q (j) ∈ Z / 2) is the number mod 2 of tangency points (respectively, quadruple points) occurring in any generic regular homotopy between i and j. For immersion i : F → R³ and diffeomorphism h : F → F such that i and i ○ h are regularly homotopic we show:P (i ○ h) - P (i) = Q (i ○ h) - Q (i) = (rank (h_* - Id) + ε (det h_{* *})) mod 2 where h_* is the map induced by h on H₁ (F ; Z / 2), h_{* *} is the map induced by h on H₁ (F ; Q), and for 0 ≠ r ∈ Q, ε (r) ∈ Z / 2 is 0 or 1 according to whether r is positive or negative, respectively.