  (abridged) The scale of the acoustic oscillation of baryons at the
baryon-photon decoupling is imprinted on the spatial distribution of galaxies
in the Universe, known as the baryon acoustic oscillation (BAO). The
correlation functions and power spectrum are used as a central tool for the
studies on the BAO analysis. In this work, we analyzed the spatial distribution
of galaxies with a method from the topological data analysis (TDA), in order to
detect and examine the BAO signal in the galaxy distribution. The TDA provides
a method to treat various types of "holes" in point set data, by constructing
the persistent homology (PH) group from the geometric structure of data points
and handling the topological information of the dataset. We can obtain the
information on the size, position, and statistical significance of the holes in
the data. A particularly strong point of the persistent homology is that it can
classify the holes by their spatial dimension, i.e., a 0-dim separation, 1-dim
loop, 2-dim shell, etc. We first analyzed the simulation datasets with and
without the baryon physics to examine the performance of the PH method. We
found that the PH is indeed able to detect the BAO signal: simulation data with
baryon physics present a prominent signal from the BAO, while data without
baryon physics does not show this signal. Then, we applied the PH to a quasar
sample at $z <1.0$ from extended Baryon Oscillation Spectroscopic Survey in
Sloan Digital Sky Survey Data Release 14. We discovered a characteristic hole
(a hollow shell) at a scaler $\sim150\ [{\rm Mpc}]$. This exactly corresponds
to the BAO signature imprinted in the galaxy/quasar distribution. We performed
this analysis on a small subsample of 2000 quasars. This clearly demonstrates
that the PH analysis is very efficient in finding this type of topological
structures even if the sampling is very sparse.
