  Characteristic classes, which are abstract topological invariants associated
with vector bundles, have become an important notion in modern physics with
surprising real-world consequences. As a representative example, the incredible
properties of topological insulators, which are insulators in their bulk but
conductors on their surface, can be completely characterized by a specific
characteristic class associated with their electronic band structure, the first
Chern class. Given their importance to next generation computing and the
computational challenge of calculating them using first-principles approaches,
there is a need to develop machine learning approaches to predict the
characteristic classes associated with a material system. To aid in this
program we introduce the {\emph{Haldane bundle dataset}}, which consists of
synthetically generated complex line bundles on the $2$-torus. We envision this
dataset, which is not as challenging as noisy and sparsely measured real-world
datasets but (as we show) still difficult for off-the-shelf architectures, to
be a testing ground for architectures that incorporate the rich topological and
geometric priors underlying characteristic classes.
