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 Hawaiian earring. Only the ten largest circles are shown.

The space is homeomorphic to the one-point compactification of the union of a countable family of disjoint 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 (0, 0) 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

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The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all   the loop   parameterizing the nth 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 nth 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

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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

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It is known that   is an aspherical space, i.e. all higher homotopy and homology groups of   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  -dimensional Hawaiian earring is defined as

 

Hence,   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 to zero as   Alternatively,   may be constructed as the Alexandrov compactification of a countable union of disjoint  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 showed that the singular homology group   is a nontrivial uncountable group for each such  .[1]

See also

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References

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  1. ^ Barratt, Michael; Milnor, John (1962). "An example of anomalous singular homology". Proceedings of the American Mathematical Society. 13 (2): 293–297. doi:10.1090/s0002-9939-1962-0137110-9. MR 0137110.

Further reading

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