  Motivated by the recent Kitaev's K-theory analysis of topological insulators
and superconductors, we adopt the same framework to study the topological phase
structure of Jackiw-Rebbi model in 3+1 dimensions. According to the K-theory
analysis based on the properties of the charge conjugation and time reversal
symmetries, we classify the topological phases of the model. In particular, we
find that there exist $\mathbf{Z}$ Majorana zero-modes hosted by the
hedgehogs/t'Hooft-Polyakov monopoles, if the model has a $T^2=1$ time reversal
symmetry. Guided by the K-theory results, we then explicitly show that a single
Majorana zero mode solution exists for the SU(2) doublet fermions in some
co-dimensional one planes of the mass parameter space. It turns out we can see
the existence of none or a single zero mode when the fermion doublet is only
two. We then take a step further to consider four-fermion case and find there
can be zero, one or two normalizable zero mode in some particular choices of
mass matrices. Our results also indicate that a single normalizable Majorana
zero mode can be compatible with the cancellation of SU(2) Witten anomaly.
