Abstract Motivated by applications in Topological Data Analysis, we consider decompositions of a simplicial complex induced cover its vertices. We study how the homotopy type such approximates itself. The difference between and an approximation is quantitatively measured means so called obstruction complexes. Our general machinery then specialized to clique complexes, Vietoris-Rips complexes me...

We present an algorithm for the computation of Vietoris-Rips persistence barcodes and describe its implementation in software Ripser. The method relies on implicit representations coboundary operator filtration order simplices, avoiding explicit construction storage matrix. Moreover, it makes use apparent pairs, a simple but powerful constructing discrete gradient field from total simplices sim...

Before starting, we’ll say a few words of motivation. We wish to understand the infinite Galois groupG = Gal(Q/Q), which is very important in number theory. In particular, we wish to understand G-modules, namely modules over some ring that are endowed with an action of G. Such objects tend to be complicated. Recall that in algebraic topology, we define the homology and cohomology of simplicial ...

The persistence of homological features in simplicial complex representations of big datasets in R n resulting from Vietoris-Rips or Čech filtrations is commonly used to probe the topological structure of such datasets. In this paper, the notion of homological persistence in simplicial complexes obtained from power filtrations of graphs is introduced. Specifically, the rth complex, r ≥ 1, in su...

In this paper, we present image segmentation algorithms based on tools from computational algebraic topology and Morse theory. We build our implementations on a very general clustering algorithm [4], developed by Chazal et al., which has been adapted for both grayscale and color image segmentation. By building up a simplicial complex incrementally filtering its simplices by values of a scalar f...

Considering a commutative unital quantale L as the truth value table and using the tool of L-generalized convergence structures of stratified L-filters, this paper introduces a kind of fuzzy upper topology, called fuzzy S-upper topology, on L-preordered sets. It is shown that every fuzzy join-preserving L-subset is open in this topology. When L is a complete Heyting algebra, for every completel...

The set C(G) of closed subgroups of a locally compact group G has a natural topology which makes it a compact space. This topology has been defined in various contexts by Vietoris, Chabauty, Fell, Thurston, Gromov, Grigorchuk, and many others. The purpose of the talk was to describe the space C(G) first for a few elementary examples, then for G the complex plane, in which case C(G) is a 4–spher...

in this paper‎, ‎we introduce a new model of point free topology‎ ‎for point-set topology‎. ‎we study the basic concepts in this‎ ‎structure and will find that it is naturally close to point-set‎ ‎topology‎.‎

this paper introduces a new approach to topology, based on category theory and universal algebra, and called categorically-algebraic (catalg) topology. it incorporates the most important settings of lattice-valued topology, including poslat topology of s.~e.~rodabaugh, $(l,m)$-fuzzy topology of t.~kubiak and a.~v{s}ostak, and $m$-fuzzy topology on $l$-fuzzy sets of c.~guido. moreover, its respe...