Immersions of non-orientable surfaces

Abstract

Let F be a closed non-orientable surface. We classify all finite order invariants of immersions of F into R 3 , 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 → R 3 , 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 → R 3 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 H 1 ( F ; Z / 2 ) , h ∗ ∗ is the map induced by h on H 1 ( F ; Q ) , and for 0 ≠ r ∈ Q , ε ( r ) ∈ Z / 2 is 0 or 1 according to whether r is positive or negative, respectively.