Hawaiian earring

The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all ${\displaystyle n\geq 1,}$ the loop ${\displaystyle \ell _{n}}$ parameterizing the nth circle is not homotopic to a trivial loop. Thus, ${\displaystyle \mathbb {H} }$ has a nontrivial fundamental group  ${\displaystyle G=\pi _{1}(\mathbb {H} ,(0,0)),}$ sometimes referred to as the Hawaiian earring group. The Hawaiian earring group ${\displaystyle G}$ is uncountable, and it is not a free group. However, ${\displaystyle G}$ is locally free in the sense that every finitely generated subgroup of ${\displaystyle G}$ is free.

The homotopy classes of the individual loops ${\displaystyle \ell _{n}}$ generate the free group ${\displaystyle \langle [\ell _{n}]\mid n\geq 1\rangle }$ on a countably infinite number of generators, which forms a proper subgroup of ${\displaystyle G}$. The uncountably many other elements of ${\displaystyle G}$ 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. More generally, one may form infinite products of the loops ${\displaystyle \ell _{n}}$ indexed over any countable linear order provided that for each ${\displaystyle n\geq 1}$, the loop ${\displaystyle \ell _{n}}$ and its inverse appear within the product only finitely many times.

It is a result of John Morgan and Ian Morrison that ${\displaystyle G}$ embeds into the inverse limit ${\displaystyle \varprojlim F_{n}}$ 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, ${\displaystyle G}$ is a proper subgroup of the inverse limit since each loop in ${\displaystyle \mathbb {H} }$ may traverse each circle of ${\displaystyle \mathbb {H} }$ only finitely many times. An example of an element of the inverse limit that does not correspond an element of ${\displaystyle G}$ is an infinite product of commutators ${\displaystyle \prod _{n=2}^{\infty }[\ell _{1}\ell _{n}\ell _{1}^{-1}\ell _{n}^{-1}]}$, which appears formally as the sequence ${\displaystyle \left(1,[\ell _{1}][\ell _{2}][\ell _{1}]^{-1}[\ell _{2}]^{-1},[\ell _{1}][\ell _{2}][\ell _{1}]^{-1}[\ell _{2}]^{-1}[\ell _{1}][\ell _{3}][\ell _{1}]^{-1}[\ell _{3}]^{-1},\dots \right)}$ in the inverse limit ${\displaystyle \varprojlim F_{n}}$.

Katsuya Eda and Kazuhiro Kawamura proved that the abelianisation of ${\displaystyle G,}$ and therefore the first singular homology group ${\displaystyle H_{1}(\mathbb {H} )}$ is isomorphic to the group

${\displaystyle \left(\prod _{i=1}^{\infty }\mathbb {Z} \right)\oplus \left(\prod _{i=1}^{\infty }\mathbb {Z} {\Big /}\bigoplus _{i=1}^{\infty }\mathbb {Z} \right)}$.

The first summand ${\displaystyle \prod _{i=1}^{\infty }\mathbb {Z} ,}$ is the direct product of infinitely many copies of the infinite cyclic group (the Baer–Specker group). This factor represents the singular homology classes of loops that do not have winding number ${\displaystyle 0}$ around every circle of ${\displaystyle \mathbb {H} }$ and is precisely the first Cech Singular homology group ${\displaystyle {\check {H}}_{1}(\mathbb {H} )}$. Additionally, ${\displaystyle \prod _{i=1}^{\infty }\mathbb {Z} ,}$ may be considered as the infinite abelianization of ${\displaystyle G}$, since every element in the kernel of the natural homomorphism ${\displaystyle G\to \prod _{i=1}^{\infty }\mathbb {Z} }$ is represented by an infinite product of commutators. The second summand of ${\displaystyle H_{1}(\mathbb {H} )}$ consists of homology classes represented by loops whose winding number around every circle of ${\displaystyle \mathbb {H} }$ is zero, i.e. the kernel of the natural homomorphism ${\displaystyle H_{1}(\mathbb {H} )\to \prod _{i=1}^{\infty }\mathbb {Z} }$. The existence of the isomorphism with ${\displaystyle \prod _{i=1}^{\infty }\mathbb {Z} {\Big /}\bigoplus _{i=1}^{\infty }\mathbb {Z} }$ is proven abstractly using infinite abelian group theory and does not have a geometric interpretation.

It is known that ${\displaystyle \mathbb {H} }$ is an aspherical space, i.e. all higher homotopy and homology groups of ${\displaystyle \mathbb {H} }$ are trivial.

The Hawaiian earring can be generalized to higher dimensions. Such a generalization was used by Michael Barratt and John Milnor to provide examples of compact, finite-dimensional spaces with nontrivial singular homology groups in dimensions larger than that of the space. The ${\displaystyle k}$-dimensional Hawaiian earring is defined as

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

Hence, ${\displaystyle \mathbb {H} _{k}}$ is a countable union of 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 ${\displaystyle n\to \infty .}$ Alternatively, ${\displaystyle \mathbb {H} _{k}}$ may be constructed as the Alexandrov compactification of a countable union of disjoint ${\displaystyle \mathbb {R} ^{k}}$s. Recursively, one has that ${\displaystyle \mathbb {H} _{0}}$ consists of a convergent sequence, ${\displaystyle \mathbb {H} _{1}}$ is the original Hawaiian earring, and ${\displaystyle \mathbb {H} _{k+1}}$ is homeomorphic to the reduced suspension ${\displaystyle \Sigma \mathbb {H} _{k}}$.

For ${\displaystyle k\geq 1}$, the ${\displaystyle k}$-dimensional Hawaiian earring is a compact, ${\displaystyle (k-1)}$-connected and locally ${\displaystyle (k-1)}$-connected. For ${\displaystyle k\geq 2}$, it is known that ${\displaystyle \pi _{k}(\mathbb {H} _{k})}$ is isomorphic to the Baer-Specker group ${\displaystyle \prod _{i=1}^{\infty }\mathbb {Z} .}$

For ${\displaystyle q\equiv 1{\bmod {(}}k-1)}$ and ${\displaystyle q>1,}$ Barratt and Milnor showed that the singular homology groups ${\displaystyle H_{q}(\mathbb {H} _{k};\mathbb {Q} )}$ are nontrivial – in fact, uncountable.[1]

• List of topologies

• Cannon, James W.; Conner, Gregory 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, MR 1775710.
• Conner, Gregory; 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, MR 2126731.
• Eda, Katsuya (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, MR 1879978.
• Eda, Katsuya; Kawamura, Kazuhiro (2000), "The singular homology of the Hawaiian earring", Journal of the London Mathematical Society, 62 (1): 305–310, doi:10.1112/S0024610700001071, MR 1772189.
• Fabel, Paul (2005), "The topological Hawaiian earring group does not embed in the inverse limit of free groups", Algebraic & Geometric Topology, 5 (4): 1585–1587, arXiv:math/0501482, Bibcode:2005math......1482F, doi:10.2140/agt.2005.5.1585, MR 2186111.
• Morgan, John W.; Morrison, Ian (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, MR 0868459.