  We prove John Hubbard's conjecture on the topological complexity of the
hyperbolic horseshoe locus of the complex H\'enon map. Indeed, we show that
there exist several non-trivial loops in the locus which generate infinitely
many mutually different monodromies. Our main tool is a rigorous computational
algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In
addition, we show that the dynamics of the real H\'enon map is completely
determined by the monodromy of a certain loop, providing the parameter of the
map is contained in the hyperbolic horseshoe locus of the complex H\'enon map.
