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Hawaiian Earring


HawaiianEarring

The plane figure formed by a sequence of circles C_1, C_2, C_3, ... that are all tangent to each other at the same point and such that the sequence of radii converges to zero. In the figure above, C_n is chosen to be the circle with center (1/n^2,0) and radius 1/n^2.

The topology of this set and of its generalizations to higher dimensions has been intensively studied in recent years (Eda 2000, Eda and Kawamura 2002ab). This research was motivated by the following striking discovery: although the fundamental group of the circle is Z, the fundamental group of the figure eight is Z*Z, where * denotes the free product, and in general the fundamental group of the n-petalled rose curve is Z*Z*..._()_(n) (Massey 1989, pp. 123-125), the fundamental group of the Hawaiian ring is not a free group (Higman 1952, de Smit 1992, Black 1996).


See also

Fundamental Group, Tangent Circles

This entry contributed by Margherita Barile

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References

Black, S. R. The Hawaiian Earring. Masters thesis. Oregon State University. November, 1996. http://www.ivygreen.ctc.edu/sblack/Research%20&%20Development/Thesis/Thesis.htm.Cannon, J. W. and Conner, G. R. "The Combinatorial Structure of the Hawaiian Earring Group." Topol. Appl. 106, 225-271, 2000.de Smit, B. "The Fundamental Group of the Hawaiian Earring Is Not Free." Int. J. Algebra Comput. 2, 33-38, 1992.Eda, K. "The Fundamental Groups of One-Dimensional Wild Spaces and the Hawaiian Earring." Proc. Amer. Math. Soc. 130, 1515-1522, 2002.Eda, K. and Kawamura, K. "Homotopy and Homology Groups of the n-Dimensional Hawaiian Earring." Fund. Math. 165, 17-28, 2000a.Eda, K. and Kawamura, K. "The Singular Homology of the Hawaiian Earring." J. London Math. Soc. 62, 305-310, 2000b.Higman, G. "Unrestricted Free Products and Varieties of Topological Groups." J. London Math. Soc. 27, 73-81, 1952.Massey, W. S. Algebraic Topology: An Introduction, 8th ed. New York: Springer-Verlag, 1989.

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Hawaiian Earring

Cite this as:

Barile, Margherita. "Hawaiian Earring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/HawaiianEarring.html

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