In this section, we introduce the necessary background in persistent homology [80, 34]. In order to underline the special character of persistent homology with respect to its multiparameter counterpart treated in Section 1.6, we call it one-parameter persistent homology, or simply one-persistent homology. Whereas, by persistent homology, we mean both one-parameter and multiparameter persistent homology. We warn the reader that the content of Section1.6includes the content of this section as a special case. In this section, we refer to the notion of homology of a Lefschetz complex as introduced in Section1.3. In Section1.5.1, we introduce the notion of a one-parameter filtration. This defines an intermediate step towards multiparameter filtrations introduced in Section1.6.1. Moreover, this defines the object where to apply the construction of one-persistent homology in Section1.5.2that leads to the definition of the persistence module in the one-parameter case. The analogous construction for multiparameter persistent homology is given in Section1.6.2.
1.5.1
One-parameter filtrations
In this section, we introduce the notion of a one-parameter filtration of a Lefschetz complex L (Definition1.3.1). Lefschetz complexes are taken into account to include both cases of regular cell complexes (Definition1.1.11) and Morse complexes (Definition1.4.8). Afterwards, we introduce our own notion of a discrete gradient and a Morse complex compatible with a one-parameter filtration. These latter is generalized in Section1.6.1.
1.5.1 Definition (One-parameter filtration). A one-parameter filtration, or simply a one-filtration of Lis a map F taking each u P Z to a closed subcomplex Luof L in such a way that
• L0“ H;
• there exists a z P Z such that Lz“ L;
• u ď v in Z implies that Luis a closed subcomplex of Lv.
Each element u P Z is called a filtration grade, or simply a grade. Each subcomplex Luis called a filtration stepor simply a step. We call the image L“ FpZq a filtered (Lefschetz) complex. We say that L is filtered by F or, indifferently, by L.
We introduce the following property of a discrete gradient over a regular cell complex with respect to a one-filtration.
1.5.2 Definition (Discrete gradient compatible with a one-filtration). Given a regular cell complex X filtered by X , a discrete gradient V over X is said to be compatible with X if
@pa, bq P V, @u P Z, a P Xuô b P Xu.
A Morse complex M obtained from a discrete gradient compatible with a filtered complex X is said to be compatible with X .
1.5.3 Definition (Morse filtered complex). Given a filtered complex X relative to X with grades in Z and a Morse complex M over X , we define the Morse filtered complex M relative to M and compatible with X by setting in each filtration step pMu, κu
Mq, for u P Z,
Mu:“ ta P M | a P Xuu,
(a) (b)
Figure 1.5: a one-filtration indicated by numbers over cells. A discrete gradient which is non-compatible with the one-filtration (a), and another discrete gradient which is compatible with the same one-filtration (b). Arrows denote discrete vectors, red arrows denote non-compatible discrete vectors, colored cells are critical cells. The first component in discrete vectors correspond to the tail and second component to the head of the arrow.
1.5.4 Definition (One-parameter filtering function). A one-parameter filtering function over X is a function φ : X ´Ñ Z such that, for all cells a, b P X
a ă b ñ φpaq ď φ pbq.
A filtering function φ of pX , κSq induces a sublevel filtration Fφ by setting Fφpuq “ pXu, κSuq with
Xu:“ ta P X | φ paq ď uu, κSu:“ κSrestricted to Xuˆ Xu.
Indeed, each set Xu is closed in X and then in each Xv with u ď v. The filtered complex X obtained through sublevels under φ is denoted by Lφ.
1.5.5 Remark. A one-parameter filtered complex X can always be thought of as a Xφ for some
one-parameter filtering function φ . Indeed, it suffices to take, for each cell a in X , φ paq equal to the smallest grade u such that a P Xu.
1.5.2
One-parameter persistence module
Given a filtered complex L, the homology construction Hkp¨; Rq can be applied to each filtration
step. Since all definitions in this section does not depend on R, we skip to indicate it. Each filtration step has its associated chain complex and, by following Remark1.2.4, each inclusion between successive filtration steps induces a linear map of corresponding homology spaces
As already mentioned, maps ιu,vare not necessarily injective or surjective. In this sense, a class is persistent from grade u to grade vif it is not trivial in HkpLuq and still non-trivial in HkpLvq. This
motivates the following definition.
1.5.6 Definition (One-persistent homology). Given a Lefschetz complex L filtered by L with filtration grades in Z, the one-persistent kth-homology from grade u to grade v, with u ď v, is the image of the inclusion-induced map ιu,vof HkpLvq.
Let us consider a Lefschetz complex L filtered L. The global information of one-persistent homology for all possible filtration grades u ď v is captured by the following definition.
1.5.7 Definition (One-parameter kth-persistence module). The one-parameter kth-persistence module, or simply, one-persistence module HkpLq of a filtered complex L is the family of homology spaces HkpLuq with u P Z along with all linear maps ιu,vwith u ď v induced by the
inclusion of complexes. (a) F2x ¯ay ˆ1 0 ˙ //F2x ¯a, ¯cy ` 1 1˘ //F2x ¯ay (b)
Figure 1.6: In picture (a), we see the two-parameter filtration obtained by considering the one-parameter filtrations in Figure1.5together. In the diagram (b), a representation of the corresponding persistence module in degree 0 (connected components) is depicted. For each F2-vector space in the persistence module, the corresponding reference basis is made explicit, e.g., ¯ais the homology class of the vertex a. Each matrix expresses a linear linear map in terms of those bases.
We observe that, for a Lefschetz complex L filtered by L, if z P Z is such that Lz“ L, then ιz,wis an isomorphism for every w ě z.
The Morse reduction result Theorem1.4.9has a counterpart for filtered complexes. The result is implied by Theorem 4.3 in [144].
1.5.8 Theorem (One-parameter compatible Morse reduction). Given a regular cell complex X filtered by aX with filtration grades in Z and a Morse complex pM, κMq compatible with X , the
inclusion-induced maps at each grade u P Z
ϕku: HkpMuq ´Ñ HkpXuq
are isomorphisms for all integers k making all the inclusion-induced maps in each filtration commute.