Arnold-type invariants of wave fronts on surfaces

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves have been generalized to the case of generic planar wave fronts. We generalize these invariants to the case of wave fronts on an arbitrary surface $F$. All invariants satisfying the axioms which naturally generalize the axioms used by Arnold are explicitly described. We also give an explicit formula for the finest order one $J^+$-type invariant of fronts on an orientable surface $F\neq S^2$. We obtain necessary and sufficient conditions for an invariant of nongeneric fronts with one nongeneric singular point to be the Vassiliev-type derivative of an invariant of generic fronts. As a byproduct, we calculate all homotopy groups of the space of Legendrian immersions of $S^1$ into the spherical cotangent bundle of a surface.