Filtrations induced by continuous functions

In Persistent Homology and Topology, filtrations are usually given by introducing an ordered collection of sets or a continuous function from a topological space to $\R^n$. A natural question arises, whether these approaches are equivalent or not. In this paper we study this problem and prove that, while the answer to the previous question is negative in the general case, the approach by continuous functions is not restrictive with respect to the other, provided that some natural stability and completeness assumptions are made. In particular, we show that every compact and stable $1$-dimensional filtration of a compact metric space is induced by a continuous function. Moreover, we extend the previous result to the case of multidimensional filtrations, requiring that our filtration is also complete. Three examples show that we cannot drop the assumptions about stability and completeness. Consequences of our results on the definition of a distance between filtrations are finally discussed

On Harder-Narasimhan filtrations and their compatibility with tensor products

We attach buildings to modular lattices and use them to develop a metric approach to Harder-Narasimhan filtrations. Switching back to a categorical framework, we establish an abstract numerical criterion for the compatibility of these filtrations with tensor products. We finally verify our criterion in three cases, one of which is new

C^0-topology in Morse theory

Let $f$ be a Morse function on a closed manifold $M$, and $v$ be a Riemannian gradient of $f$ satisfying the transversality condition. The classical construction (due to Morse, Smale, Thom, Witten), based on the counting of flow lines joining critical points of the function $f$ associates to these data the Morse complex $M_*(f,v)$. In the present paper we introduce a new class of vector fields ($f$-gradients) associated to a Morse function $f$. This class is wider than the class of Riemannian gradients and provides a natural framework for the study of the Morse complex. Our construction of the Morse complex does not use the counting of the flow lines, but rather the fundamental classes of the stable manifolds, and this allows to replace the transversality condition required in the classical setting by a weaker condition on the $f$-gradient (almost transversality condition) which is $C^0$-stable. We prove then that the Morse complex is stable with respect to $C^0$-small perturbations of the $f$-gradient, and study the functorial properties of the Morse complex. The last two sections of the paper are devoted to the properties of functoriality and $C^0$-stability for the Novikov complex $N_*(f,v)$ where $f$ is a circle-valued Morse map and $v$ is an almost transverse $f$-gradient.Comment: 22 pages, Latex file, one typo correcte

Towards Persistence-Based Reconstruction in Euclidean Spaces

Manifold reconstruction has been extensively studied for the last decade or so, especially in two and three dimensions. Recently, significant improvements were made in higher dimensions, leading to new methods to reconstruct large classes of compact subsets of Euclidean space $\R^d$. However, the complexities of these methods scale up exponentially with d, which makes them impractical in medium or high dimensions, even for handling low-dimensional submanifolds. In this paper, we introduce a novel approach that stands in-between classical reconstruction and topological estimation, and whose complexity scales up with the intrinsic dimension of the data. Specifically, when the data points are sufficiently densely sampled from a smooth $m$-submanifold of $\R^d$, our method retrieves the homology of the submanifold in time at most $c(m)n^5$, where $n$ is the size of the input and $c(m)$ is a constant depending solely on $m$. It can also provably well handle a wide range of compact subsets of $\R^d$, though with worse complexities. Along the way to proving the correctness of our algorithm, we obtain new results on \v{C}ech, Rips, and witness complex filtrations in Euclidean spaces