  We study a topological phase of interacting bosons in (3+1) dimensions which
is protected by charge conservation and time-reversal symmetry. We present an
explicit lattice model which realizes this phase and which can be studied in
sign-free Monte Carlo simulations. The idea behind our model is to bind bosons
to topological defects called hedgehogs. We determine the phase diagram of the
model and identify a phase where such bound states are proliferated. In this
phase we observe a Witten effect in the bulk whereby an external monopole binds
half of the elementary boson charge, which confirms that it is a bosonic
topological insulator. We also study the boundary between the topological
insulator and a trivial insulator. We find a surface phase diagram which
includes exotic superfluids, a topologically ordered phase, and a phase with a
Hall effect quantized to one-half of the value possible in a purely
two-dimensional system. We also present models that realize symmetry-enriched
topologically-ordered phases by binding multiple hedgehogs to each boson; these
phases show charge fractionalization and intrinsic topological order as well as
a fractional Witten effect.
