  Topological Data Analysis (TDA) is a modern approach to Data Analysis
focusing on the topological features of data; it has been widely studied in
recent years and used extensively in Biology, Physics, and many other areas.
However, financial markets have been studied slightly through TDA. Here we
present a quick review of some recent applications of TDA on financial markets,
including applications in the early detection of turbulence periods in
financial markets and how TDA can help to get new insights while investing.
Also, we propose a new turbulence index based on persistent homology -- the
fundamental tool for TDA -- that seems to capture critical transitions in
financial data; we tested our index with different financial time series
(S&P500, Russel 2000, S&P/BMV IPC and Nikkei 225) and crash events (Black
Monday crash, dot-com crash, 2007-08 crash and COVID-19 crash). Furthermore, we
include an introduction to persistent homology so the reader can understand
this paper without knowing TDA.
