I am currently teaching myself the basics of persistent homology by reading the following set of notes by Gunnar Carlsson.
I first read the algebraic portion of the notes where he introduced the notion of persistent vector spaces before showing the equivalence between the isomorphism classes of finitely presented vector spaces and barcodes. Pg 316-321. I have also read the portion regarding the computation of the dimensions of the homology group of the abstract simplicial complexes. Pg 303-306
However, I am facing issues try to link the homology of the Vietoris Rips complexes to that of the finitely presented persistent vector spaces. Hence, would it be possible for someone to explain how the theory of finitely presented persistent vector spaces ties in to what is being said on Pg 315(Bottom half of page)?