Abundance of $C^1$-robust homoclinic tangencies

A diffeomorphism $f$ has a $C^1$-robust homoclinic tangency if there is a $C^1$-neighbourhood $\text{cU}$ of $f$ such that every diffeomorphism in $g\in \text{cU}$ has a hyperbolic set ${\text{La}}_g$, depending continuously on $g$, such that the stable and unstable manifolds of ${\text{La}}_g$ 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 $C^1$-robust homoclinic tangencies. Using blender-horseshoes, we prove that homoclinic classes of $C^1$-generic diffeomorphisms containing saddles with different indices and that do not admit dominated splittings (of appropriate dimensions) display $C^1$-robust homoclinic tangencies.