  The topology of photons in vacuum is interesting because there are no photons
with $\boldsymbol{k}=0$, creating a hole in momentum space. We show that while
the set of all photons forms a trivial vector bundle $\gamma$ over this
momentum space, the $R$- and $L$-photons form topologically nontrivial
subbundles $\gamma_\pm$ with first Chern numbers $\mp2$. In contrast, $\gamma$
has no linearly polarized subbundles, and there is no Chern number associated
with linear polarizations. It is a known difficulty that the standard version
of Wigner's little group method produces singular representations of the
Poincar\'{e} group for massless particles. By considering representations of
the Poincar\'{e} group on vector bundles we obtain a version of Wigner's little
group method for massless particles which avoids these singularities. We show
that any massless bundle representation of the Poincar\'{e} group can be
canonically decomposed into irreducible bundle representations labeled by
helicity, which in turn can be associated to smooth irreducible Hilbert space
representations. This proves that the $R$- and $L$-photons are globally
well-defined as particles and that the photon wave function can be uniquely
split into $R$- and $L$-components. This formalism offers a method of
quantizing the EM field without invoking discontinuous polarization vectors as
in the traditional scheme. We also demonstrate that the spin-Chern number of
photons is not a purely topological quantity. Lastly, there has been an
extended debate on whether photon angular momentum can be split into spin and
orbital parts. Our work explains the precise issues that prevent this
splitting. Photons do not admit a spin operator; instead, the angular momentum
associated with photons' internal degree of freedom is described by a
helicity-induced subalgebra corresponding to the translational symmetry of
$\gamma$.
