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Affine Winding Numbers and Their Applications to the Study of Front Propagation

Let M be an oriented manifold. We study the question how many times a wave front W on M has passed through a given point p ∈ M. Namely, we introduce the numerical topological invariant winp(W (t)) such that winp(W (t 1)) − winp(W (t 2)) is the algebraic number of times the wave front W passed through p between the time moments t 1 and t 2. This invariant essentially is the affine winding number and it can be easily calculated from the current picture of wave fronts on M without the knowledge of the time-dependent front propagation law. Moreover, in fact we even do not need to know the topology of M outside of a part M of M such that W (t 1), W (t 2), p ⊂ M and W (t 1) and W (t 2) are homotopic in M .