Defining and computing persistent Z-homology in the general case

Ana Romero, Jónathan Heras, Julio Rubio, Francis Sergeraert

By general case we mean methods able to process simplicial sets and chain complexes not of finite type. A filtration of the object to be studied is the heart of both subjects persistent homology and spectral sequences. In this paper we present the complete relation between them, both from theoretical and computational points of view. One of the main contributions of this paper is the observation that a slight modification of our previous programs computing spectral sequences is enough to compute also persistent homology. By inheritance from our spectral sequence programs, we obtain for free persistent homology programs applicable to spaces not of finite type (provided they are spaces with effective homology) and with Z-coefficients (significantly generalizing the usual presentation of persistent homology over a field). As an illustration, we compute some persistent homology groups (and the corresponding integer barcodes) in the case of a Postnikov tower.

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