Structure of semi-continuous $q$-tame persistence modules

Vietoris–Rips persistent homology, injective metric spaces, and the filling radius

“…Nevertheless, in Theorem 2.9, we are able to establish that, if X is totally bounded, then its Vietoris-Rips persistent homology has a (unique) persistence barcode. This is achieved without invoking Crawley-Boevey's theorem and instead through combining our main (isomorphism) theorem (see Theorem 4.1) with a recent result by Schmahl [74,Theorem 1.2]. The proof of Theorem 2.9 can be found in the extended (arXiv) version of this paper [62,Section 5].…”

Coarse nodal count and topological persistence

“…which is a genuine isomorphism (not only an isomorphism in the observable category), see [61] for details. Moreover, all bars in the above barcode are of the form [a, b) or [a, +∞), a, b ∈ .…”

“…Moreover, all bars in the above barcode are of the form [a, b) or [a, +∞), a, b ∈ . We also note that for a continuous function f : X → on a compact Hausdorff space X , V * ( f ) is q-tame, bounded from the left, upper semi-continuous, and has bounded spectrum, see [61]. Therefore, this generality would suffice for our considerations in Sections 4 and 5.…”

“…As before, denote by V * ( f ) t = H * ({ f < t}). Due to lower semicontinuity 1 of V * ( f ), the bars in ( V * ( f )) are of the form (a, b] or (a, +∞) for a, b ∈ and moreover V * ( f ) ∼ = ⊕ I∈ ( V * ( f )) I , see [61] for details. Note that this is a genuine isomorphism of persistence modules, while without the lower semicontinuity assumption we would only have an isomorphism in the observable category, as explained in Remark 2.10.…”

“…Firstly, we notice that δ (p) ≤ C n k n for all δ > 0 implies the finiteness of (p) with the desired bound. Indeed, due to upper semi-continuity of V (p), there are no bars of length zero in (p), see [61] for details. Since the bound does not depend on δ the claim follows.…”