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.
Bibliographical noteFunding Information:
E-mail address: email@example.com. URL: http://www.math.biu.ac.il/~tahl. 1 Partially supported by the Minerva Foundation.
- Finite order invariants
- Immersions of surfaces