  Stable, holomorphic vector bundles are constructed on an torus fibered,
non-simply connected Calabi-Yau threefold using the method of bundle
extensions. Since the manifold is multiply connected, we work with equivariant
bundles on the elliptically fibered covering space. The cohomology groups of
the vector bundle, which yield the low energy spectrum, are computed using the
Leray spectral sequence and fit the requirements of particle phenomenology. The
physical properties of these vacua were discussed previously. In this paper, we
systematically compute all relevant cohomology groups and explicitly prove the
existence of the necessary vector bundle extensions. All mathematical details
are explained in a pedagogical way, providing the technical framework for
constructing heterotic standard model vacua.
