# Hawaiian earring

In mathematics, the **Hawaiian earring**
is the topological space defined by the union of circles in the Euclidean plane
with center
and radius
for
endowed with the subspace topology:

The space is homeomorphic to the one-point compactification of the union of a countably infinite family of open intervals.

The Hawaiian earring is a one-dimensional, compact, locally path-connected metrizable space. Although is locally homeomorphic to at all non-origin points, is not semi-locally simply connected at . Therefore, does not have a simply connected covering space and is usually given as the simplest example of a space with this complication.

The Hawaiian earring looks very similar to the wedge sum of countably infinitely many circles; that is, the rose with infinitely many petals, but these 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 (an ε-ball around contains every circle whose radius is less than ε/2); in the rose, a neighborhood of the intersection point might not fully contain any of the circles. Additionally, the rose is not compact: the complement of the distinguished point is an infinite union of open intervals; to those add a small open neighborhood of the distinguished point to get an open cover with no finite subcover.

## Fundamental group

The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all
, the loop
parameterizing the n-th circle is not homotopic to a trivial loop. Thus,
has a nontrivial fundamental group
, sometimes referred to as the *Hawaiian earring group*. The Hawaiian earring group
is uncountable, and it is not a free group. However,
is locally free in the sense that every finitely generated subgroup of
is free.

The homotopy classes of the individual loops
generate the free group
on a countably infinite number of generators, which forms a proper subgroup of
. The uncountably many other elements of
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
circumnavigates the *n*th circle. More generally, one may form infinite products of the loops
indexed over any countable linear order provided that for each
, the loop
and its inverse appear within the product only finitely many times.

It is a result of John Morgan and Ian Morrison that
embeds into the inverse limit
of the free groups with *n* generators,
, where the bonding map from
to
simply kills the last generator of
. However,
is a proper subgroup of the inverse limit since each loop in
may traverse each circle of
only finitely many times. An example of an element of the inverse limit that does not correspond an element of
is an infinite product of commutators
, which appears formally as the sequence
in the inverse limit
.

## First Singular Homology

Katsuya Eda and Kazuhiro Kawamura proved that the abelianisation of , and therefore the first singular homology group is isomorphic to the group

.

The first summand
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
around every circle of
and is precisely the first Cech Singular homology group
. Additionally,
may be considered as the *infinite abelianization* of
, since every element in the kernel of the natural homomorphism
is represented by an infinite product of commutators. The second summand of
consists of homology classes represented by loops whose winding number around every circle of
is zero, i.e. the kernel of the natural homomorphism
. The existence of the isomorphism with
is proven abstractly using infinite abelian group theory and does not have a geometric interpretation.

## Higher dimensions

It is known that is an aspherical space, i.e. all higher homotopy and homology groups of are trivial.

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 -dimensional Hawaiian earring is defined as

Hence, is a countable union of -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 . Alternatively, may be constructed as the Alexandrov compactification of a countable union of s. Recursively, one has that consists of a convergent sequence, is the original Hawaiian earring, and is homeomorphic to the reduced suspension .

For , the -dimensional Hawaiian earring is a compact, -connected and locally -connected. For , it is known that is isomorphic to the Baer-Specker group . For and Barratt and Milnor have proved that the singular homology groups are not zero and even uncountable.[1]

## References

- Barratt, Michael; Milnor, John (1962). "An example of anomalous singular homology".
*Proceedings of the American Mathematical Society*.**13**: 293–297. doi:10.2307/2034486. MR 0137110.

## Further reading

- 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" (PDF),
*Algebraic & Geometric Topology*,**5**: 1585–1587, arXiv:math/0501482, 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.