  We present in this paper a framework which leverages the underlying topology
of a data set, in order to produce appropriate coordinate representations. In
particular, we show how to construct maps to real and complex projective
spaces, given appropriate persistent cohomology classes. An initial map is
obtained in two steps: First, the persistent cohomology of a sparse filtration
is used to compute systems of transition functions for (real and complex) line
bundles over neighborhoods of the data. Next, the transition functions are used
to produce explicit classifying maps for the induced bundles. A framework for
dimensionality reduction in projective space (Principal Projective Components)
is also developed, aimed at decreasing the target dimension of the original
map. Several examples are provided as well as theorems addressing choices in
the construction.
