  (abridged) We present the Significant Cosmic Holes in Universe (SCHU) method
for identifying cosmic voids and loops of filaments in cosmological datasets
and assigning their statistical significance using techniques from topological
data analysis. Persistent homology is used to find different dimensional holes.
For dark matter halo catalogs and galaxy surveys, the 0-, 1-, and 2-dimensional
holes can be identified with clusters, loops of filaments, and voids. The
procedure overlays halos/galaxies on a 3D grid, and a distance-to-measure (DTM)
function is calculated at each point of the grid. A filtration is generated
over the lower-level sets of the DTM across increasing threshold values. The
filtered simplicial complex can be used to summarize the birth/death times of
the different dimension homology group generators (i.e., the holes).
Persistence diagrams are produced from the dimension and birth/death times of
each homology group generator. Using the persistence diagrams and bootstrap
sampling, we explain how $p$-values can be assigned to each homology group
generator. The homology group generators on a persistence diagram are not, in
general, uniquely located back in the original dataset volume so we propose a
method for finding a representation of the homology group generators. This
method provides a novel, statistically rigorous approach for locating
informative generators in cosmological datasets, which may be useful for
providing complementary cosmological constraints on the effects of, for
example, the sum of the neutrino masses. The method is tested on a Voronoi foam
simulation, and then applied to a subset of the SDSS galaxy survey and a
cosmological simulation. Lastly, we calculate Betti functions for two of the
MassiveNuS simulations and discuss implications for using the persistent
homology of the density field to help break degeneracy in the cosmological
parameters.
