Immersions of non-orientable surfaces

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F   into R3R3, with values in any Abelian group. We show they are all functions of a universal order 1 invariant which we construct as T⊕P⊕QT⊕P⊕Q, where T   is a ZZ valued invariant reflecting the number of triple points of the immersion, and P,QP,Q are Z/2Z/2 valued invariants characterized by the property that for any regularly homotopic immersions i,j:F→R3, P(i)−P(j)∈Z/2P(i)−P(j)∈Z/2 (respectively, Q(i)−Q(j)∈Z/2Q(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○hi○h are regularly homotopic we show:P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h∗−Id)+ε(deth∗∗))mod2P(i○h)−P(i)=Q(i○h)−Q(i)=(rank(h∗−Id)+ε(deth∗∗))mod2 where h∗h∗ is the map induced by h   on H1(F;Z/2)H1(F;Z/2), h∗∗h∗∗ is the map induced by h   on H1(F;Q)H1(F;Q), and for 0≠r∈Q0≠r∈Q, ε(r)∈Z/2ε(r)∈Z/2 is 0 or 1 according to whether r is positive or negative, respectively.