Hawaiian earring

In mathematics, the Hawaiian earring $$H is the topological space defined by the union of circles in the Euclidean plane $$R^2 with center ($$1/n,0) and radius $$1/n for $$n=1,2,3,.... $$H is homeomorphic to the one-point compactification 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 $$H looks very similar (but is not homeomorphic to) to the wedge sum, $$H', of countably infinitely many circles; that is, the rose with infinitely many petals.

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

It has been shown that "G" embeds into the inverse limit of the free groups with "n" generators, $$F_n, where the bonding map from $$F_n to $$F_{n-1} simply kills the last generator of $$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 which 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, it has a normal subgroup "N" such that $$G/N approx prod_{i=0}^infty mathbb{Z}, the direct product of infinitely many copies of the infinite cyclic group. This is called the "infinite abelianization" or "strong abelianization" of the Hawaiian earring, since the subgroup $$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.

References

* J.W. Cannon,G.R. Conner, "The big fundamental group, big Hawaiian earrings, and the big free groups", Topology Appl. 106 (2000), no. 3, 273–291.
* G. Conner, K. Spencer, "Anomalous behavior of the Hawaiian earring group", J. Group Theory 8 (2005), no. 2, 223–227.
* K. Eda, "The fundamental groups of one-dimensional wild spaces and the Hawaiian earring", Proc. Amer. Math. Soc. 130 (2002), no. 5, 1515–1522
* K. Eda, K. Kawamura, "The singular homology of the Hawaiian earring", J. London Math. Soc. (2) 62 (2000), no. 1, 305–310.
* P. Fabel, The topological Hawaiian earring group does not embed in the inverse limit of free groups", Algebr. Geom. Topol. 5 (2005), 1585–1587
* J. W. Morgan, I. Morrison, "A van Kampen theorem for weak joins", Proc. London Math. Soc. (3) 53 (1986), 562–576
* Daniel K. Biss, [http://links.jstor.org/sici?sici=0002-9890(200010)107%3A8%3C711%3AAGATTF%3E2.0.CO%3B2-I "A Generalized Approach to the Fundamental Group"] , The American Mathematical Monthly, Vol. 107, No. 8 (Oct., 2000), pp. 711–720