Hawaiian earring

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In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ with center ${\displaystyle ({\tfrac {1}{n}},0)}$ and radius ${\displaystyle {\tfrac {1}{n}}}$ for ${\displaystyle n=1,2,3,\ldots }$. The space H is homeomorphic to the one-point compactification of the union of a countably infinite family of open intervals.

The Hawaiian earring can be given a complete metric and it is compact. It is path connected but not semilocally simply connected.

The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but those two spaces are not homeomorphic. The difference between their topologies is seen in the fact that, in the Hawaiian earring, every open neighborhood of the point of intersection of the circles contains all but finitely many of the circles. It is also seen in the fact that the wedge sum is not compact: the complement of the distinguished point is a union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.

Video Hawaiian earring

Fundamental group

The Hawaiian earring is not simply connected, since the loop parametrising any circle is not homotopic to a trivial loop. Thus, it has a nontrivial fundamental group G.

The Hawaiian earring H has the free group of countably infinitely many generators as a proper subgroup of its fundamental group. G contains additional elements, which arise from loops whose image is not contained in finitely many of the Hawaiian earring's circles; in fact, some of them are surjective. For example, the path that on the interval ${\displaystyle [2^{-n},2^{-n+1}]}$ circumnavigates the nth circle.

It has been shown that G embeds into the inverse limit of the free groups with n generators, ${\displaystyle F_{n}}$, where the bonding map from ${\displaystyle F_{n}}$ to ${\displaystyle F_{n-1}}$ simply kills the last generator of ${\displaystyle F_{n}}$. However G is not the complete inverse limit but rather the subgroup in which each generator appears only finitely many times. An example of an element of the inverse limit that is not an element of G is an infinite commutator.

G is uncountable, and it is not a free group. While its abelianisation has no known simple description, G has a normal subgroup N such that

${\displaystyle {\frac {G}{N}}\approx \prod _{i=0}^{\infty }\mathbb {Z} ,}$

the direct product of infinitely many copies of the infinite cyclic group (the Baer-Specker group). This is called the infinite abelianization or strong abelianization of the Hawaiian earring, since the subgroup N is generated by elements where each coordinate (thinking of the Hawaiian earring as a subgroup of the inverse limit) is a product of commutators. In a sense, N can be thought of as the closure of the commutator subgroup.

Maps Hawaiian earring

Higher dimensions

Michael Barratt and John Milnor generalized the Hawaiian earring to higher dimensions, thereby constructing compact finite-dimensional spaces whose singular homology groups do not vanish in arbitrarily high degree and even have uncountable dimension. The ${\displaystyle k}$-dimensional Hawaiian earring is defined as

${\displaystyle X=\bigcup _{n\in \mathbb {N} }\left\{(x_{0},x_{1},\ldots ,x_{k})\in \mathbb {R} ^{k+1}:\left(x_{0}-{\frac {1}{n}}\right)^{2}+x_{1}^{2}+\cdots +x_{k}^{2}={\frac {1}{n^{2}}}\right\}.}$

So it is a countable union of ${\displaystyle k}$-spheres, which have one single point in common, and the topology is given by a metric in which the sphere's diameters converge towards zero for ${\displaystyle n\to \infty }$. For ${\displaystyle k=1}$, this reduces to the original Hawaiian earring. The ${\displaystyle k}$-dimensional Hawaiian earrings are compact, ${\displaystyle (k-1)}$-connected and locally ${\displaystyle (k-1)}$-connected. In addition, they are the Alexandrov compactification of a countable union of ${\displaystyle \mathbb {R} ^{k}}$s.

For ${\displaystyle q\equiv 1{\bmod {(}}k-1)}$ and ${\displaystyle q>1,}$ Barratt and Milnor have proved that the singular homology groups ${\displaystyle H_{q}(X;\mathbb {Q} )}$ are not zero and even uncountable.

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References

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Further reading

• Cannon, J. W.; Conner, G. R. (2000), "The big fundamental group, big Hawaiian earrings, and the big free groups", Topology and its Applications, 106 (3): 273-291, doi:10.1016/S0166-8641(99)00104-2.
• Conner, G.; Spencer, K. (2005), "Anomalous behavior of the Hawaiian earring group", Journal of Group Theory, 8 (2): 223-227, doi:10.1515/jgth.2005.8.2.223.
• Eda, K. (2002), "The fundamental groups of one-dimensional wild spaces and the Hawaiian earring" (PDF), Proceedings of the American Mathematical Society, 130 (5): 1515-1522, doi:10.1090/S0002-9939-01-06431-0.
• Eda, K.; Kawamura, K. (2000), "The singular homology of the Hawaiian earring", Journal of the London Mathematical Society, 62 (1): 305-310, doi:10.1112/S0024610700001071.
• Fabel, P. (2005), "The topological Hawaiian earring group does not embed in the inverse limit of free groups" (PDF), Algebraic & Geometric Topology, 5: 1585-1587, arXiv:math/0501482, doi:10.2140/agt.2005.5.1585.
• Morgan, J. W.; Morrison, I. (1986), "A van Kampen theorem for weak joins", Proceedings of the London Mathematical Society, 53 (3): 562-576, doi:10.1112/plms/s3-53.3.562.

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