In mathematics, the Hawaiian earring H is the topological space defined by the union of circles in the Euclidean plane R² with center (1 / n,0) and radius 1 / n for n = 1,2,3,. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Set theory, the term Union (denoted as ∪ refers to a set operation used in the convergence of set elements to form a resultant set containing the elements of both sets Euclidean geometry is a mathematical system attributed to the Greek Mathematician Euclid of Alexandria. . . . H is homeomorphic to the one-point compactification of a countably infinite family of open intervals. In Mathematics, an interval is a set of Real numbers with the property that any number that lies between two numbers in the set is also included in the set

The Hawaiian earring. Only the ten largest circles are shown.

The Hawaiian earring can be given a complete metric and it is compact. In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has It is path connected but not semilocally simply connected. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of In Mathematics, in particular Topology, a Topological space X is called semi-locally simply connected if every point x

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. In Topology, the wedge sum (sometimes wedge product, though not to be confused with the Exterior product, which also shares this terminology is a "one-point In Mathematics, a rose (also known as a bouquet of circles) is a Topological space obtained by gluing together a collection of circles

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. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

The Hawaiian earring H has the free group of countably infinitely many generators as a proper subgroup of its fundamental group. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. 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 nth circle.

It has been shown that G embeds into the inverse limit of the free groups with n generators, F_n, where the map from F_n to F_{n − 1} that kills the last generator of F_n. In Mathematics, an embedding (or imbedding) is one instance of some Mathematical structure contained within another instance such as a group In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise 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 , the direct product of infinitely many copies of the infinite cyclic group. In Mathematics, more specifically in Abstract algebra, the commutator subgroup or derived subgroup of a group is the Subgroup In Mathematics, one can often define a direct product of objectsalready known giving a new one In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an 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. is a mathematician currently a professor at Waseda University. 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. John Willard Morgan is an American Mathematician, well-known for his contributions to Topology and Geometry. Morrison, A van Kampen theorem for weak joins, Proc. London Math. Soc. (3) 53 (1986), 562–576
• Daniel K. Biss, A Generalized Approach to the Fundamental Group, The American Mathematical Monthly, Vol. 107, No. 8 (Oct. , 2000), pp. 711–720