Invariants of types \(j^+,j^-\), and st for smooth curves on two-dimensional manifolds (Q1974243)

In [Adv. Sov. Math. 21, 33-91 (1994; Zbl 0864.57027)], \textit{V. I. Arnold} defines three basic invariants of general plane curves, \(j^+\), \(j^-\) and \(st\), which are first-order Vassiliev invariants corresponding to certain strata; the same invariants are studied and partially extended by \textit{F. Aicardi} [J. Knot Theory Ramifications 5, No. 6, 743-778 (1996; Zbl 0874.57005)] and \textit{V. Tchernov} [ibid. 8, No. 1, 71-97 (1999; Zbl 0937.57013); Homotopy groups of the space of curves on a surface, Math. Scand. 86, No. 1, 36-44 (2000)]. The present paper takes into account the whole set \(\Omega\) of curves immersed on a 2-dimensional real manifold \(M^2\) (without assumptions about compactness, orientability and boundary properties) and performs a complete analysis of the irreducible components of suitable strata, denoted by \(J^+\), \(J^-\) and \(St\). Moreover, in case \(M^2\) being different from the Klein bottle \(Kl\), the possibility of introducing first-order Vassiliev invariants corresponding to these components is investigated, by means of necessary and sufficient conditions for the Poincaré dual elements (to the component) in the cohomology group \(H^1 (\Omega,\mathbb{Z})\) to vanish.