Hawaiian Earring - Fundamental Group

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 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 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, G has a normal subgroup N such that, the direct product of infinitely many copies of the infinite cyclic group (the Baer–Specker 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.