During the 1930s, Marston Morse developed a vast generalization of what is commonly known as Morse theory relating the critical points of a semi-continuous functional with the topology of its sublevel sets. Morse and Tompkins applied this body of work, referred to as functional topology, to prove the Unstable Minimal Surface Theorem in the setting defined by Douglas' solution to Plateau's Problem. Several concepts introduced by Morse in this context can be seen as early precursors to the theory of persistent homology, which by now has established itself as a popular tool in applied and theoretical mathematics. In this article, we provide a modern redevelopment of the homological aspects of Morse's functional topology from the perspective of persistence theory. We adjust several key definitions and prove stronger statements, including a generalized version of the Morse inequalities, in order to allow for novel uses of persistence techniques in functional analysis and symplectic geometry. As an application, we identify and correct a mistake in the proof of the Unstable Minimal Surface Theorem by Morse and Tompkins.

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We present an algorithm for computing the barcode of the image of a morphism in persistent homology induced by an inclusion of filtered finite-dimensional chain complexes. The algorithm makes use of the clearing optimization and can be applied to inclusion-induced maps in persistent absolute homology and persistent relative cohomology for filtrations of pairs of simplicial complexes. The clearing optimization works particularly well in the context of relative cohomology, and using previous duality results we can translate the barcodes of images in relative cohomology to those in absolute homology. This forms the basis for our implementation of image persistence computations for inclusions of filtrations of Vietoris-Rips complexes in the framework of the software Ripser. matching construction yields a partial bijection between the barcodes of H * (L • ) and H * (K • ), which can be used to bound the distance between these two barcodes from above. The induced matching is defined in terms of the image persistence barcode, i.e., the barcode of im H * (f • ), motivating the problem of computing this barcode. A first algorithm for this problem has been given by for the special case where f • is of the formThe authors also propose a method for getting rid of the intersection assumption using a mapping cylinder construction. This construction, however, might not be computationally feasible and we will show that this additional step is in fact not necessary.Cohen-Steiner et al. [12] propose applications of image persistence for recovering the persistent homology of a noisy function on a noisy domain, see also the related work by Chazal et al. [7]. Recently, Reani and Bobrowski [15] proposed a method that includes the computation of induced matchings in order to pair up common topological features in different data sets, with applications to statistical bootstrapping. Furthermore, the computation of image barcodes is used in a distributed algorithm for persistent homology based on the Mayer-Vietoris spectral sequence by Álvaro Torras Casas [6]. Image persistence of endomorphisms such as Steenrod squares on the persistent (co)homology of a single filtration has also been proposed by Lupo et al. [14] as a tool to get more comprehensive invariants than the standard persistent (co)homology barcodes.Despite the usefulness of image persistence, there are a few aspects that have prevented these techniques from being widely used in applications so far. Specifically, to the best of our knowledge, there is no publicly available implementation at this moment. Furthermore, computation using the known algorithms is slow in comparison to modern algorithms for a single filtration. Indeed, computing usual persistent homology for larger data sets arising in real-world applications only became feasible in recent years due to optimizations that exploit various structural properties and algebraic identities of the problem [9, 16, 1]. Our goal is to adapt these speed-ups to the computation of images and induced matchings.The basic algorithm ...

We show that the interleaving distance between the persistent singular homology and the persistent Čech homology of a homologically locally connected filtration consisting of paracompact Hausdorff spaces is 0.