Arnold-type invariants of wave fronts on surfaces

Vladimir V. Tchernov

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arXiv · math/9901133

Arnold-type invariants of wave fronts on surfaces

Vladimir V. Tchernov

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Abstract

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.

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Vladimir V. Tchernov. 1999-08-11. Arnold-type invariants of wave fronts on surfaces. https://arxiv.org/abs/math/9901133

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