Hawaiian earring

In the mathematical branch of algebraic topology , the Hawaiian earring is the simplest example of a non- semilocal, simply connected space. One also speaks of “wild spaces” or “wild topology”. Its fundamental group and its first homology group are uncountable.

definition

The Hawaiian earring. The picture shows only the ten largest circles.

The Hawaiian earring can be defined as

${\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{1},x_{2})\in \mathbb {R} ^{2}:{\Bigl (}x_{1}-{\frac {1}{n}}{\Bigr )}^{2}+x_{2}^{2}={\frac {1}{n^{2}}}\right\}}$.

It is therefore a countable union of circles that have a single point in common and whose topology comes from a metric in which the diameter of the circles converges towards zero as the diameter increases . ${\displaystyle n}$

Fundamental group and homology

The fundamental group of the Hawaiian earring is uncountable and was described by Eda.

Their abelization is isomorphic to ${\displaystyle H_{1}(X;\mathbb {Z} )}$

${\displaystyle \mathbb {Z} ^{\omega }\oplus \bigoplus _{c}\mathbb {Q} \oplus {\widehat {\bigoplus _{c}\mathbb {Z} }}}$,

where is the direct product of countably many copies of the group of integers , the thickness of the real number line and the -adic completion of the free Abelian group of rank . ${\displaystyle \mathbb {Z} ^{\omega }}$ ${\displaystyle c}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle {\widehat {\bigoplus _{c}\mathbb {Z} }}}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle c}$

The higher homology groups for are zero. ${\displaystyle H_{q}(X;\mathbb {Z} )}$ ${\displaystyle q>1}$

properties

The Hawaiian earring is compact and one-dimensional .

It is neither simply connected nor locally simply connected .

It is not simply connected semilocal .

literature

• K. Eda: The fundamental groups of one-dimensional spaces and the Hawaiian earring , Proc. Amer. Math. Soc. 130, 1515-1522 (2002)
• K. Eda, K. Kawamura: The singular homology of the Hawaiian earring , J. London Math. Soc. 62, 305-310 (2000)