End (topology)

In mathematics , the ends of a topological space are clearly speaking the connected components of the "edge in infinity". Formally, they are defined as equivalence classes of complements of compact sets .

definition

Let be a (locally connected, connected, locally compact, Hausdorffian) topological space. ${\displaystyle X}$

We consider the family of all descending sequences ${\displaystyle {\mathcal {U}}}$

${\displaystyle (U_{1}\supset U_{2}\supset U_{3}\supset \ldots )}$

connected, open sets with compact margins, for which

${\displaystyle \bigcap _{i=1}^{\infty }{\overline {U_{i}}}=\emptyset }$

applies.

On we define an equivalence relation by ${\displaystyle {\mathcal {U}}}$ ${\displaystyle \sim }$

${\displaystyle (U_{1}\supset U_{2}\supset U_{3}\supset \ldots )\sim (V_{1}\supset V_{2}\supset V_{3}\supset \ldots )\Longleftrightarrow \forall n\exists k\colon V_{k}\subset U_{n},U_{k}\subset V_{n}}$.

The equivalence classes of the equivalence relation on hot ends of the topological space . ${\displaystyle \sim }$ ${\displaystyle {\mathcal {U}}}$ ${\displaystyle X}$

The open sets in the respective equivalence class are called neighborhoods of one end.

Characterization via complements of compacts

(Specker, Raymond): A space has at least ends if there is an open set with compact closure whose complement has connected components . ${\displaystyle k}$ ${\displaystyle k}$

Fundamental group of an end

The fundamental group of an end is defined as the projective limit of the fundamental groups of the neighborhoods of the end : ${\displaystyle E}$ ${\displaystyle U_{i}}$ ${\displaystyle E}$

${\displaystyle \pi _{1}(E)=\varprojlim _{i\in I}\pi _{1}(U_{i})}$.

Examples

• The number line has two ends. ${\displaystyle \mathbb {R} ^{1}}$
• It has an end for. ${\displaystyle n\geq 2}$ ${\displaystyle \mathbb {R} ^{n}}$
• Let be the interior of a compact manifold with boundaries , so . Then the ends of the connected components correspond to . ${\displaystyle M}$ ${\displaystyle {\overline {M}}}$ ${\displaystyle \partial {\overline {M}}}$ ${\displaystyle M={\overline {M}}-\partial {\overline {M}}}$ ${\displaystyle M}$ ${\displaystyle \partial {\overline {M}}}$

The Cayley graph of the free group with two generators a and b

• Let be the Cayley graph of a nonabelian free group . Then has an infinite number of ends, there is a bijection of the set of ends onto a Cantor set . ${\displaystyle X}$ ${\displaystyle X}$
• According to a Freudenthal theorem , the Cayley graph of a group has either an infinite number or at most 2 ends.

literature

• Hughes, Bruce; Ranicki, Andrew: Ends of complexes. Cambridge Tracts in Mathematics, 123. Cambridge University Press, Cambridge, 1996. ISBN 0-521-57625-3
• Freudenthal, Hans: About the ends of discrete rooms and groups. Comment. Math. Helv. 17, (1945). 1-38. online (pdf)