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1-Dimensional Intrinsic Persistence of Geodesic Spaces

Žiga Virk

Abstract

Given a compact geodesic space $X$ we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of $X$ to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point $c$ corresponds to an isometrically embedded circle of length $3c$, that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base and that Rips and Čech induced persistences are isomorphic up to a factor $3/4$. The theory describes geometric properties of the underlying space encoded and extractable from persistence.

1-Dimensional Intrinsic Persistence of Geodesic Spaces

Abstract

Given a compact geodesic space we apply the fundamental group and alternatively the first homology group functor to the corresponding Rips or Čech filtration of to obtain what we call a persistence. This paper contains the theory describing such persistence: properties of the set of critical points, their precise relationship to the size of holes, the structure of persistence and the relationship between open and close, Rips and Čech induced persistences. Amongst other results we prove that a Rips critical point corresponds to an isometrically embedded circle of length , that a homology persistence of a locally contractible space with coefficients in a field encodes the lengths of the lexicographically smallest base and that Rips and Čech induced persistences are isomorphic up to a factor . The theory describes geometric properties of the underlying space encoded and extractable from persistence.

Paper Structure

This paper contains 24 sections, 42 theorems, 37 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Geometry of Rips complexes
  4. The size of holes
  5. The correspondence theorems
  6. Radius of the enclosing ball as a measurement of a hole
  7. The length of a representative as a measurement of a hole
  8. The diameter of a representative as a measurement of a hole
  9. Fundamental groups of Rips complexes are finitely generated
  10. Loops in the limit
  11. The structure of $\pi_1$-persistence
  12. Lasso decompositions
  13. Localization theorem
  14. The structure of $H_1$-persistence and minimal generators
  15. Minimal loop decompositions
  16. ...and 9 more sections

Key Result

Theorem 1.1

Suppose $X$ is a compact Riemannian manifold, ${\mathbb F}$ is a field and $l_1, l_2, \ldots, l_n$ are the lengths of the shortest generating set of representatives of $H_1(X, {\mathbb F})$. Then the following persistences are isomorphic: where ${\mathcal{S}}(X,a), {\mathcal{L}}(X, a), {\mathcal{D}}(X,a)$ are subspaces of $H_1(X, {\mathbb F})$ generated by loops size less than $a$, with the size

Figures (8)

  • Figure 1: The minimal (with respect to the radius parameter of the complex) configuration of three points on a circle forming a simplex in Rips (left) and Čech (right) complex, whose convex hull contains the center of the circle.
  • Figure 2: An excerpt of the $r$-homotopy from (8) of Proposition \ref{['PropRips']}.
  • Figure 3: A sketch of the decomposition of two $r$-samples of size $4$ from Proposition \ref{['PropDif']}. The solid line (filling $\alpha$) differs from the dashed line (filling $\beta$) by four loops $\alpha_i * \beta_i ^-$ of length less than $2r$.
  • Figure 4: A sketch of a map ${\varphi}$ of Proposition \ref{['PropNull']}. Given a configuration of abstract triangles on the left, construct the appropriate system of geodesics (red lines) of length less than $r$ (as suggested by the red label) on the right. Note that the decomposition into triangles on the left corresponds to the decomposition into loops of length less than $3r$ on the right.
  • Figure 5: An excerpt from the proof of Proposition \ref{['PropNull']}. Given an $r$-null $r$-loop $x_0,x_1,x_2,x_3$ on the left, an $r$-nullhomotopy is given by a triangulation $\Delta$ of the disc (depicted as the rectangle on the left in this figure). In this case $\Delta$ contains an additional vertex $y$ and four triangles in $\operatorname{Rips}\nolimits(X,r)$. Thinking of vertices of $\Delta$ as points in $X$, we may replace the edges of $\Delta$ by geodesics and obtain the same scheme in $X$ with $\alpha$ along the boundary. Loop $\alpha$ is decomposed into four loops of length less than $3r$ and the ordered loops on the right are the four lassos, whose concatenation in the suggested order demonstrates that $\alpha \in {\mathcal{L}}(X, 3r, \pi_1)$.
  • ...and 3 more figures

Theorems & Definitions (108)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Definition 4.1
  • ...and 98 more