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Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories

Hannah Alpert, Fedor Manin

Abstract

We give $\mathbb{Z}$-bases for the homology and cohomology of the configuration space $\operatorname{config}(n,w)$ of $n$ unit disks in an infinite strip of width $w$, first studied by Alpert, Kahle and MacPherson. We also study the way these spaces evolve both as $n$ increases (using the framework of representation stability) and as $w$ increases (using the framework of persistent homology). Finally, we include some results about the cup product in the cohomology and about the configuration space of unordered disks.

Configuration spaces of disks in a strip, twisted algebras, persistence, and other stories

Abstract

We give -bases for the homology and cohomology of the configuration space of unit disks in an infinite strip of width , first studied by Alpert, Kahle and MacPherson. We also study the way these spaces evolve both as increases (using the framework of representation stability) and as increases (using the framework of persistent homology). Finally, we include some results about the cup product in the cohomology and about the configuration space of unordered disks.

Paper Structure

This paper contains 33 sections, 44 theorems, 69 equations, 15 figures, 1 table.

Table of Contents

  1. Introduction
  2. Main results
  3. Proof ideas
  4. Additional results
  5. Combinatorial and algebraic setup
  6. The complex $\mathop{\mathrm{cell}}\nolimits(n)$
  7. Signs of the boundary operator
  8. Twisted algebra structure
  9. The permutohedron and no-$(k+1)$-equal spaces
  10. Generators and relations for twisted algebras
  11. Homology of weighted no-$(k+1)$-equal spaces
  12. Discrete Morse theory
  13. Discrete gradient vector fields on permutohedra
  14. Basis for homology
  15. Decomposing cell$(n,w)$ into layers
  16. ...and 18 more sections

Key Result

Theorem A

For fixed $w$, $H_*(\mathop{\mathrm{config}}\nolimits({-},w); \mathbb{Z})$ forms a finitely generated, noncommutative twisted algebra whose generators live in $H_{\leq \frac{3}{2}w-2}(\mathop{\mathrm{config}}\nolimits(\leq 3w/2,w); \mathbb{Z})$.

Figures (15)

  • Figure 1: The configuration space $\mathop{\mathrm{config}}\nolimits(n, w)$ is the set of ways to arrange $n$ disjoint labeled disks of width $1$ in $\mathbb{R} \times [0, w]$.
  • Figure 2: Some configurations of a wheel with $5$ disks. The first configuration gives a canonical (up to switching the first two) ordering of the disks.
  • Figure 3: The wheels in a filter always cross over and under each other in the same order. This figure shows a filter with three wheels of size 1 and one wheel of size 3. The resulting cycle is an $S^2 \times T^2$.
  • Figure 4: A basic 14-cycle in $\mathop{\mathrm{config}}\nolimits(24,5)$ represented by an $S^1 \times S^0 \times S^2 \times T^{12}$: the three red boxes are filters giving an $S^1$, an $S^0$ and an $S^2$ respectively, and there are 11 additional circular degrees of freedom from spinning the wheels (in black). We also remark: If the disks and switched places, this would no longer represent a basic cycle, by property (iv) in Theorem \ref{['thm:basis']}.The single disk can move freely past all the disks to its left. So in a basic cycle, it has to appear all the way on the right.
  • Figure 5: We can imagine each symbol of $\mathop{\mathrm{cell}}\nolimits(n, w)$ as a configuration in $\mathop{\mathrm{config}}\nolimits(n, w)$ where the numbers in each block are the labels in a column of disks. Pictured are configurations representing the symbol $(\,7\;2 \mid 6 \mid 4\;5\;8\;1\;3\,)$ and its face $(\,7\;2 \mid 6 \mid 4\;5\;1 \mid 8\;3\,)$.
  • ...and 10 more figures

Theorems & Definitions (87)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem
  • Theorem 2.1
  • proof
  • Proposition 2.3
  • proof
  • Theorem 2.4
  • Remark 2.5
  • ...and 77 more