  We develop a noncompact version of the Hopf maps based on the split algebras.
The split algebras consist of three species: split-complex numbers, split
quaternions, and split octonions. They correspond to three noncompact Hopf maps
that represent topological maps between hyperboloids in different dimensions
with hyperboloid bundle. We realize such noncompact Hopf maps in two ways: one
is to utilize the split-imaginary unit, and the other is to utilize the
ordinary imaginary unit. Topological structures of the hyperboloid bundles are
explored, and the canonical connections are naturally regarded as noncompact
gauge field of monopoles.
