  Neural networks can be thought of as applying a transformation to an input
dataset. The way in which they change the topology of such a dataset often
holds practical significance for many tasks, particularly those demanding
non-homeomorphic mappings for optimal solutions, such as classification
problems. In this work, we leverage the fact that neural networks are
equivalent to continuous piecewise-affine maps, whose rank can be used to
pinpoint regions in the input space that undergo non-homeomorphic
transformations, leading to alterations in the topological structure of the
input dataset. Our approach enables us to make use of the relative homology
sequence, with which one can study the homology groups of the quotient of a
manifold $\mathcal{M}$ and a subset $A$, assuming some minimal properties on
these spaces.
  As a proof of principle, we empirically investigate the presence of low-rank
(topology-changing) affine maps as a function of network width and mean weight.
We show that in randomly initialized narrow networks, there will be regions in
which the (co)homology groups of a data manifold can change. As the width
increases, the homology groups of the input manifold become more likely to be
preserved. We end this part of our work by constructing highly non-random wide
networks that do not have this property and relating this non-random regime to
Dale's principle, which is a defining characteristic of biological neural
networks.
  Finally, we study simple feedforward networks trained on MNIST, as well as on
toy classification and regression tasks, and show that networks manipulate the
topology of data differently depending on the continuity of the task they are
trained on.
