On the topology of cooriented wave fronts in spaces of small dimensions

We consider Legendre singularities with respect to Legendre equivalence preserving a coorientation of the contact structure. In this case, we calculate the adjacency indices of multisingularities of generic Legendre mappings to smooth manifolds of the dimension $n\leq6$. As a corollary, we find new coexistence conditions on singularities of wave fronts. Namely, we find all linear relations with real coefficients between the Euler characteristics of manifolds of singularities of any generic compact cooriented wave front in any $n$-dimensional space.