  Data analysis often concerns not only the space where data come from, but
also various types of maps attached to data. In recent years, several related
structures have been used to study maps on data, including Reeb spaces, mappers
and multiscale mappers. The construction of these structures also relies on the
so-called \emph{nerve} of a cover of the domain.
  In this paper, we aim to analyze the topological information encoded in these
structures in order to provide better understanding of these structures and
facilitate their practical usage.
  More specifically, we show that the one-dimensional homology of the nerve
complex $N(\mathcal{U})$ of a path-connected cover $\mathcal{U}$ of a domain
$X$ cannot be richer than that of the domain $X$ itself. Intuitively, this
result means that no new $H_1$-homology class can be "created" under a natural
map from $X$ to the nerve complex $N(\mathcal{U})$. Equipping $X$ with a
pseudometric $d$, we further refine this result and characterize the classes of
$H_1(X)$ that may survive in the nerve complex using the notion of \emph{size}
of the covering elements in $\mathcal{U}$. These fundamental results about
nerve complexes then lead to an analysis of the $H_1$-homology of Reeb spaces,
mappers and multiscale mappers.
  The analysis of $H_1$-homology groups unfortunately does not extend to higher
dimensions. Nevertheless, by using a map-induced metric, establishing a
Gromov-Hausdorff convergence result between mappers and the domain, and
interleaving relevant modules, we can still analyze the persistent homology
groups of (multiscale) mappers to establish a connection to Reeb spaces.
