  Over the past three decades, it has been shown that discrete and continuous
media can support topologically nontrivial waves. Recently, it was shown that
the same is true of the vacuum, namely, right (R) and left (L) circularly
polarized photons are topologically nontrivial. Here, we study the topology of
another class of massless particles, namely gravitons. We show that the
collection of all gravitons forms a topologically trivial vector bundle over
the lightcone, allowing us to construct a globally smooth basis for gravitons.
The graviton bundle also has a natural geometric splitting into two
topologically nontrivial subbundles, consisting of the R and L gravitons. The R
and L gravitons are unitary irreducible bundle representations of the
Poincar\'{e} group, and are thus elementary particles; their topology is
characterized by the Chern numbers $\mp 4$. This nontrivial topology obstructs
the splitting of graviton angular momentum into spin and orbital angular
momentum.
