Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Coarse cohomology of configuration space and coarse embedding

Arka Banerjee

TL;DR

The paper develops a coarse, equivariant obstruction framework for embeddings by defining the $Z_2$-equivariant coarse cohomology of the configuration space $ ext{Conf}(X)$ as the equivariant coarse cohomology of $X^2- obreak oldsymbol{ obreak oldsymbol{ obreak oldsymbol{ obreak oldsymbol{ obreak obreak } obreak } }$, with obstructions encoded in the coarse van Kampen classes $cvk^n(X)$. It establishes a coarse version of the van Kampen obstruction and a coarse Gysin sequence, and proves a central cobdim theorem: if $X$ coarse embeds into $Y$, then $ ext{cobdim}(X)\le ext{cobdim}(Y)$. The work provides tools to compute these obstructions in concrete settings (e.g., Euclidean spaces, cones, and uniformly acyclic manifolds) via boundedly supported and equivariant Alexander–Spanier theories, and applies coarse duality to bound cobdim from above and below, thereby obstructing coarse embeddings in significant cases. Overall, the results connect coarse geometry, equivariant cohomology, and configuration-space obstructions to yield quantitative embedding obstructions for broad classes of spaces.

Abstract

We introduce a notion of equivariant coarse cohomology of the complement of a subspace in a metric space. We use this cohomology to define a notion of coarse cohomology of the configuration space of a metric space and develop tools to compute this cohomology under certain conditions. As an application of this theory, we show that certain classes in the coarse cohomology of configuration space obstruct coarse embedding between two metric spaces.

Coarse cohomology of configuration space and coarse embedding

TL;DR

The paper develops a coarse, equivariant obstruction framework for embeddings by defining the -equivariant coarse cohomology of the configuration space as the equivariant coarse cohomology of , with obstructions encoded in the coarse van Kampen classes . It establishes a coarse version of the van Kampen obstruction and a coarse Gysin sequence, and proves a central cobdim theorem: if coarse embeds into , then . The work provides tools to compute these obstructions in concrete settings (e.g., Euclidean spaces, cones, and uniformly acyclic manifolds) via boundedly supported and equivariant Alexander–Spanier theories, and applies coarse duality to bound cobdim from above and below, thereby obstructing coarse embeddings in significant cases. Overall, the results connect coarse geometry, equivariant cohomology, and configuration-space obstructions to yield quantitative embedding obstructions for broad classes of spaces.

Abstract

We introduce a notion of equivariant coarse cohomology of the complement of a subspace in a metric space. We use this cohomology to define a notion of coarse cohomology of the configuration space of a metric space and develop tools to compute this cohomology under certain conditions. As an application of this theory, we show that certain classes in the coarse cohomology of configuration space obstruct coarse embedding between two metric spaces.
Paper Structure (19 sections, 40 theorems, 110 equations, 1 figure)

This paper contains 19 sections, 40 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Alexander--Spanier complexes
  4. Coarse inclusion
  5. Coarse intersection
  6. Coarse (co)homology
  7. Equivariant coarse cohomology of the complement
  8. Equivariant coarse cohomology of the complement
  9. Computation of equivariant coarse cohomology
  10. Boundedly supported cohomology of the complement
  11. Equivariant Alexander-Spanier cohomology
  12. Relation between $\mathop{\mathrm{H}}\nolimits {\mathrm B}_{\mathrm G}^{*}$ and $\mathop{\mathrm{H}}\nolimits^{*}_{\mathrm G}$
  13. Relation between $\mathop{\mathrm{H}}\nolimits {\mathrm X}_{\mathrm G}^{*}$ and $\mathop{\mathrm{H}}\nolimits {\mathrm B}_{\mathrm G}^{*}$
  14. Coarse cohomology of the configuration space
  15. Coarse van Kampen obstruction
  16. ...and 4 more sections

Key Result

Theorem 1.1

If $X$ admits a coarse embedding into $Y$, then $\mathop{\mathrm{cobdim}}\nolimits(X)\leq \mathop{\mathrm{cobdim}}\nolimits(Y)$.

Figures (1)

  • Figure 1: A subspace of $\mathbb{R}^2$ that consists of a countable union of circles $\{C_i\}_{i\overset{}{\in}\mathbb{N}}$ and the ray $\mathcal{R}:=[0,\infty)\times \{0\}$ such that the $i^{th}$ circle has radius $i$ and touches $\mathcal{R}$ at $(i^2,0)$ so that the distance between two consecutive circles grows to infinity. This space is uniformly acyclic away from any bounded set.

Theorems & Definitions (96)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Example 2.2
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Definition 3.4
  • Example 3.5
  • Definition 3.6
  • ...and 86 more