  In this article, we propose a one-sample test to check whether the support of
the unknown distribution generating the data is homologically equivalent to the
support of some specified distribution or not OR using the corresponding
two-sample test, one can test whether the supports of two unknown distributions
are homologically equivalent or not. In the course of this study, test
statistics based on the Betti numbers are formulated, and the consistency of
the tests is established under the critical and the supercritical regimes.
Moreover, some simulation studies are conducted and results are compared with
the existing methodologies such as Robinson's permutation test and test based
on mean persistent landscape functions. Furthermore, the practicability of the
tests is shown on two well-known real data sets also.
