This work is a mixture of two different worlds: Persistent Homology and A∞-(co)algebras. The former is a new tool in Topological Data Analysis, whilst the latter is a classical concept in Homological Perturbation Theory.
Persistent Homology (in the sense of [1,2,3]) is a topological technique which has proved to be useful in areas such as digital imaging [4], sensor networks [5], molecular modelling [6], dynamical systems [7,8] and speech pattern analysis [9]. It allows one to extract global structural information about data sets (specially high dimensional ones) which may contain noise.
In this dissertation, we enhance the power of Persistent Homology through the use of A∞-(co)algebras [10]. These are structures with which we can endow the (co)homology of a topological space in order to obtain information of its homotopy type beyond its homology groups and its cohomology algebra.
Throughout this abstract, the word “homology” will refer to "homology with coefficients in some field".
If we want to study a data set, we can build a point cloud (i.e., a finite metric space) to represent it so that the details we find about the structure of the cloud can be translated into patterns in the data. Specifically, if we view the point cloud “P” as a reasonable sample of points of an unknown metric space “M”, the shape of M will tell us about the structure of P. In a situation like this, persistent homology (that we recall in Chapter 1) allows us to approximate the Betti numbers of M.
We work with A∞-(co)algebras (that we recall in Chapter 2) with the aim of providing more information about the space M. To do so, we start by showing how powerful A∞-(co)algebras are (Section 2.4) and choose a part of every A∞-(co)algebra to focus on (Section 3.1), e.g., given an A∞-coalgebra "{Δ_n}_n" on the homology of a topological space "X", we can look at the kernel of each of the linear maps Δ_n restricted to some homology degree. Any such kernel is a vector subspace of the corresponding homology group but its dimension is not a homotopy invariant of X (Section 3.2.1). Despite this, we show situations in which this dimension can give information on the homotopy type of X (Sections 3.2.2 and 3.2.3).
Next step is adding persistence to the scene. The Fundamental Theorem of Persistent Homology (of which we give a new proof in Section 1.2) claims that given a family of nested topological spaces with finite dimensional homology groups, the sequence formed by the homology of each of the spaces, with all maps induced by the corresponding inclusions, can be decomposed as a “barcode” in a unique way (up to reordering of bars). The point is to find analogues of this classical result to capture the information of the vector subspaces we are focusing on, like the kernels of the maps Δ_n mentioned above. We first prove a decomposition result of this kind (Section 3.3) for nested topological spaces and A∞-coalgebra structures on their homology satisfying the following condition, that we call C: the maps induced in homology by the inclusions send elements in the kernel of Δ_n to elements in the kernel of the corresponding Δ_n. We then provide a construction that guarantees that for certain families of nested topological spaces, we can choose an A∞-coalgebra on the homology of each term so that condition C holds (Section 3.4). Notice that when the chosen A∞-coalgebra structures do not satisfy C, an interesting intermittent behavior takes place (Section 3.5), but in spite of this, we can still prove a decomposition result by using zigzag persistence techniques (Section 3.6).
On the computational side, we modify an algorithm by P. Real et al. [11] to compute A∞-coalgebra structures on the homology of finite CW complex by building a discrete vector field (Section 2.3), and we provide an abstract algorithm to compute the barcode decomposition that encapsulates the information of the kernels of the maps Δ_n in an A∞-coalgebra (Section 3.7).
To make this work as self-contained as possible, we have added an Appendix with a collection of tools of Rational Homotopy Theory that we use.
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