  Quantum discord is an important measure of quantum correlations that can
serve as a resource for certain types of quantum information processing. \ Like
entanglement, discord is subject to destruction by external noise. The routes
by which this destruction can take place depends on the shape of the
hypersurface of zero discord $\mathcal{C}$ in the space of generalized Bloch
vectors. For 2 qubits, we show that with a few points subtracted, this
hypersurface is a simply-connected 9-dimensional manifold embedded in a
15-dimensional background space. \ We do this by constructing an explicit\
homeomorphism from a known manifold to the subtracted version of $\mathcal{C}
$. We also construct a coordinate map on $\mathcal{C}$ that can be used for
integration or other purposes. This topological characterization of $%
\mathcal{C}$ has important implications for the classification of the possible
time evolutions of discord in physical models. The classification for discord
contrasts sharply with the possible evolutions of entanglement.
  Using topological methods, we classify the possible joint evolutions of
entanglement and discord. There are 9 allowed categories: 6 categories for a
Markovian process and 3 categories for a non-Markovian process, respectively.
We illustrate these conclusions with an anisotropic XY spin model. All 9
categories can be obtained by adjusting parameters.
