Arnold-type invariants of wave fronts on surfaces

Vladimir Tchernov

Recently Arnold's $\St$ and $J^{\pm}$ invariants of generic planar curves
have been generalized to the case of generic planar wave fronts.  Here these
invariants are generalized to the case of wave fronts on an arbitrary
surface.  All the invariants satisfying the axioms, which naturally
generalize the axioms used by Arnold, are explicitly described.  As a
byproduct result we calculate all the homotopy groups of the space of
Legendrian immersions of $S^1$ to the spherical cotangent bundle of a
surface.