A diffeomorphism f has a C1-robust homoclinic tangency if there is a C1-neighbourhood of f such that every diffeomorphism in g∈ has a hyperbolic set
Nearby in the stack
_g
, depending continuously on
g
, such that the stable and unstable manifolds of
_g
have some non-transverse intersection. For every manifold of dimension greater than or equal to three, we exhibit a local mechanism (blender-horseshoes) generating diffeomorphisms with
C1
-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of
C1
-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display