The Hawaiian earring is neither simply connected nor semilocally simply connected since, for all the loop parameterizing the 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 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 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
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. 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. 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 -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 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 groups are nontrivialin fact, uncountable.
