[Paper Review] Sequential ends of metric spaces

Michael DeLyser, Brendon LaBuz+1 more|arXiv (Cornell University)|Mar 4, 2013

Geometric and Algebraic Topology4 references3 citations

TL;DR

This paper introduces a coarse geometric invariant called 'sequential ends' for metric spaces, defined via equivalence classes of coarse sequences under subsequence relations. It proves that this invariant is equivalent to the previously defined coarse invariant $\sigma(X,x_0)$, establishing a robust, basepoint-independent characterization of large-scale structure in metric spaces under coarse equivalence.

ABSTRACT

We develop an analog to the ends of a metric space for the category of coarse metric spaces and show that it is equivalent to a previously defined coarse invariant.

Motivation & Objective

• To develop a coarse geometric analog to the classical notion of ends in topological spaces.
• To define a new invariant—sequential ends—using coarse sequences and their equivalence under subsequence relations.
• To prove that this new invariant is equivalent to the previously defined coarse invariant $\sigma(X,x_0)$.
• To establish that the invariant is independent of basepoint and preserved under coarse equivalences.

Proposed method

• Define a coarse sequence as a function $f: \mathbb{N} \to X$ that is both bornologous and proper, corresponding to sequences with uniformly bounded jumps and diverging from a basepoint.
• Introduce an equivalence relation $\approx$ on coarse sequences via the symmetric transitive closure of the subsequence relation.
• Define the sequential ends $\sigma(X,x_0)$ as the set of equivalence classes of coarse sequences starting at $x_0$ under $\approx$.
• Prove that any coarse sequence is an $N$-sequence for some $N>0$, and that the equivalence relation can be realized within a uniform $K$-sequence framework.
• Use $K$-chains (finite sequences with adjacent points within distance $K$) to connect points outside large balls, modeling coarse connectivity.
• Establish equivalence between sequential ends and the direct limit $\varinjlim \sigma_N(X,x_0)$, showing independence from basepoint and invariance under coarse equivalence.

Experimental results

Research questions

• RQ1 How can the classical notion of ends in topological spaces be adapted to the coarse category of metric spaces?
• RQ2 What is the precise definition of a coarse analog to a ray, and how does it relate to bornologous and proper functions from $\mathbb{N}$ to $X$?
• RQ3 Is the proposed invariant of sequential ends equivalent to the previously defined coarse invariant $\sigma(X,x_0)$?
• RQ4 Does the sequential ends invariant remain unchanged under coarse equivalences and basepoint changes?

Key findings

• The set of sequential ends $\sigma(X,x_0)$ is equivalent to the coarse invariant $\sigma(X,x_0)$ defined in prior works, confirming consistency across formulations.
• The invariant is independent of the choice of basepoint $x_0$, as shown through equivalence under subsequence relations and $K$-chain constructions.
• For $\mathbb{R}^n$ with $n \geq 2$, the space has exactly one sequential end, reflecting its large-scale connectedness.
• The sequential ends invariant distinguishes non-geodesic spaces: for example, a space with two classical ends but only one sequential end, as in Example 4.3.
• In a space with three distinct large-scale directions (e.g., $X = \{(x,2^n)\} \cup \{(0,y)\}$), the sequential ends count is three, corresponding to positive $x$-axis, negative $x$-axis, and $y$-axis directions.
• The construction of $K$-chains outside large balls allows the proof that sequences in the same large-scale direction are equivalent, even in non-geodesic spaces.