G-invariant Persistent Homology on Quotient Spaces

Persistent homology, building its foundation on homology theory, has been successfully applied various fields containing computation of topology. Here, we derive and implement persistent homology theory to specefic topological spaces with G-invariant covering, whereby persistence over quotient space can be canonically formed. The implementation is validated by comparing against various transformation homomorphisms. Furthermore, it is proved that implementation over normal covering spaces can preserve persistence barcode diagram if charateristic of coefficient is coprime to |G|. These results shed light on potential application of persistent homology to clusters or crystallography in a view of hierarchy structure induced by symmetry.