Quantifying Transversality by Measuring the Robustness of Intersections

Abstract

By definition, transverse intersections are stable under infinitesimal perturbations. Using persistent homology, we extend this notion to a measure. Given a space of perturbations, we assign to each homology class of the intersection its robustness, the magnitude of a perturbation in this space necessary to kill it, and then we prove that the robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for contours of smooth mappings.

On Computability and Triviality of Well Groups

“…There are maps f with φ f trivial but nontrivial U 0 ( f, r ). 11 But this can be detected by the above mentioned extendability criterion. We do not present an example where…”

“…If the parameter r varies, well groups U ( f, r ) naturally fit into a zig-zag sequence of homomorphisms called well module that can be converted into a well diagram, a multiset of real numbers indicating the death of homology classes of X r supported by all Z ∈ Z r ( f ) as r increases [11]. In this section, we show that the subgroups V := φ H * (X, A ∪ B) are not only subgroups of well groups for fixed r but they naturally form a persistence module which is a sub-module of the well module in some sense.…”

“…Recently, the concept of well groups was designed to quantify transversality of a map K → Y with respect to a subspace Y of Y . Following [11], we assume 1 a subspace P of C(K , Y ) with a metric d. Any h ∈ P for which d(h, f ) ≤ r is called an r -perturbation of f and we define X r ⊆ K to be the union of g −1 (Y ) over all r -perturbations h of f . The inclusion h −1 (Y ) → X r induces a map in homology j h :…”

“…Let B ⊆ K be a pair of compact Hausdorff spaces and f : K → R n continuous. Let X := | f | −1 [0, r ] where | f | denotes the function x → | f (x)|; this is the smallest space containing zero sets of all r -perturbations h of f and coincides with X r from the notation of [11].…”

“…This direction is rather orthogonal to the matters of this paper. 3 In [4,11], well groups were defined by means of singular homology. But then, once we allow arbitrary continuous perturbations, to the best of our knowledge, no f : K → R n with nontrivial U j ( f, r ) for j > 0 would be known.…”