On the computation of coarse cohomology

Abstract

The purpose of this article is to relate coarse cohomology of metric spaces with a more computable cohomology. We introduce a notion of boundedly supported cohomology and prove that coarse cohomology of many spaces are isomorphic to the boundedly supported cohomology. Boundedly supported cohomology coincides with compactly supported Alexander--Spanier cohomology if the space is proper and contractible. Our work generalizes an earlier result of Roe which says that the coarse cohomology is isomorphic to the compactly supported Alexander-Spanier cohomology if the space is uniformly contractible. As an application of our main theorem, we obtain that coarse cohomology of the complement can be computed in terms of Alexander-Spanier cohomology for many spaces.

Figures (5)

• Figure 1: A subspace of $\mathbb{R}^2$ that consists of countable union of circles $\{C_i\}_{i\in\mathbb{N}}$ and the ray $r:=[0,\infty)\times \{0\}$ such that the $i^{th}$ circle has radius $i$ and distance between two consecutive circle grows to infinity. This is an example of a space that is uniformly contractible at infinity.
• Figure 2: In the above figure, the space $X\overset{}{\subset}\mathbb{R}^2\overset{}{\subset}\ell^\infty$ is the Warsaw circle and $N(X)$ is a tubular neighborhood of the grey region in $\ell^\infty$ that deforms retract to the grey region. Red line is the convex filling of the simplex $(x,y)$. Any filling of $(x,y)$ in $X$ has to be around the circular part. Hence, there is no homotopy in $N(X)$ between the red convex filling and any filling of $(x,y)$ in $X$. Furthermore, any neighborhood of $x$ contains such a $y$ where this happens.
• Figure 3: $H_t$ and $G_s$ are two contracting homotopy with $p$ and $q$ being their contracting point respectively. The first two pictures are of $\bar{H}(c(x,y))$ and $\bar{H}(c(x,y,z))$. The third picture is of $\bar{G}(\bar{H}(c(x,y)))$.
• Figure 4: In the left picture, the straight line between $x$ and $y$ is $c((x,y))$ and the red singular 1-simplices give the filling $f((x,y))=\bar{H}^{B((x,y))}(\partial(x,y))$ where $H^{B((x,y)}_t$ contracts the set $B((x,y))$ to the point $p$. The striped region is $D((x,y))=\bar{H}^{B(x,y)}(c((x,y)))$. The picture on the right is the support of $\bar{H}^{B((x,y,z))}(D(\partial((x,y,z)))$ which is made of four singular 3-simplices. The one in the center is $\bar{H}^{B(x,y,z)}((x,y,z))$. Other three belong to the support of $\bar{H}^{B((x,y,z))}(D(\partial((x,y,z))))$.
• Figure 5: This is a picture of $g(\sigma )$ where $\sigma =(x,y,z)$. Here $p$ is the contracting point of the homotopy $H^{B((x,y))}$. The grey part is $f(\sigma )$ and striped part is $D(\sigma ')$ where $\sigma '=(x,y)$

Theorems & Definitions (39)

• Theorem 1.1: Roe r93
• Proposition 1.2: BB21
• Theorem 1.3
• Corollary 1.4
• Example 2.1
• Example 2.2
• Proposition 2.3
• proof
• Example 2.4
• Example 2.5