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Hawaiian Earring

• The Hawaiian earring is the plane compactum consisting of infinitely many circles (jpg-file). In 1950's there were a few researches about the fundamental group of the Hawaiian earring by H.B. Griffiths and G. Higman. In 1998 K. Kawamura and I succeeded to get an explicit presentation of the singular homology group of the Hawaiian earring, i.e. the abelianization of the fundamental group. It turns out to be the direct sum of the following groups: (1) The direct product of countably many copies of the integer group; (2) The direct sum of the continuum many copies of the rational group; (3) The Z-adic completion of the free abelian group of the continuum rank. (These results are in the paper `The singular homology of the Hawaiian earring' in the list.)
• A well-known result due to M. Barratt and J. Milnor is that the 3-dimensional singular homology group of the 2-dimensional Hawaiian earring is non-trivial. We proved that the n-dimensional homotopy group of the n-dimensional Hawaiian earring is isomorphic to the direct product of countably many copies of the integer group and the k-dimensional homotopy group of the n-dimensional Hawaiian earring is trivial in case n is strictly greater than 1 and k is strictly less than n. Consequently, the n-dimensional singular homology group of the n-dimensional Hawaiian earring is isomorphic to the direct product of countably many copies of the integer group.
• In 1998 I proved: If the fundamental groups of everywhere non-semi-locally simply connected one dimensional Peano continua (*) are isomorpohic, then they are homeomorphic. The proof uses the fundamental group of the Hawaiian earring. Based on this theorem, in 1999 Greg Conner and I introduced a new construction of a space from an abstract group using the fundamental group of the Hawaiian earring. This detects a wildness of a space from its fundamental group and for a space (*) the construction from its fundamental group produces the space itself. Consequently, one dimensional fractals with non-trivial fundamental groups and also finite or countable direct products of such fractals can be recovered from their fundamental groups. This new construction of a space also makes possible to treat with certain properties of spaces in group-theoretic terms.
• In December of 2008 I proved that the fundamental groups of one dimensional Peano continua determine their homotopy types. (pdf-file is available. Also another pdf-file is available.) But, there are still unknown things even about the fundamental group of the Hawaiian earring.
• In June of 2012, contradicting to my conjecture, I've proved that the first singular homology groups of one dimensional Peano continua are isomorphic to free abelian groups of finite rank or the first singular homology group of the Hawaiian earring. (pdf-file is available.)

Specker Phenomenon

• The name "Specker phenomenon" was given by A. Blass in 1994, but the phenomenon itself was discovered by E. Specker in 1950. Specker proved that each homomorphism from the countable direct product of copies of the integer group to the integer group factors through a finite generated free abelian group which is a canonical summand. This appears a surprising fact, when we replace the range of a homomorphism to the rational group. In that case there exist 2 to the continuum many homomorphisms, while there exist only countably many and natural homomorphisms when the range is the integer group. In 1952 G. Higman proved the non-commutative variant of Specker's theorem. In topological terms Higman's theorem says: Any homomorphism from the fundamental group of the Hawaiian earring to that of a Bouquet is induced from a continuous map from the Hawaiian earring to the Bouquet. Using infinitary word theoretic argument as well as the Higman theorem, I proved that every endomorphism on the fundamental group of the Hawaiian earring is conjugate to an endomorphism induced by a continuous map (in a published paper in the list). This is a prototype of above statements which are some-what unexpected concerning fundamental groups of wild one dimensional Peano continua. This non-commutative variation of the Specker phenomenon is a key concept for the investigation of fundamental groups of wild spaces.
• Specker's theorem was investigated and generalized in abelian group theory and there are some applications of slender abelian groups to Topology. But there was no application of its non-commutative variation, i.e. Higman's theorem. The main reason seems to be owing to the following. The one is an obvious fact that uncountable non-commutative groups are out of interests of many mathematicians. Another may be that the presentation of the fundamental group due to H. B. Griffiths was done after the publication of the Higman paper, but there was no serious mentioning to the Higman paper in the Griffiths paper. It's likely that H. B. Griffiths did not notice it.
• Now the most interesting problem in this area is to characterize the n-slenderness, i.e. the non-commutative version of the slenderness in abelian group theory. For a long time I've thought that finitely generated torsionfree groups should be n-slender, but Greg Conner informed me that the rational is a subgroup of a finitely presented group, which can be proved using Higman's old result. Hence, the n-slenderness is in question even for finitely presented groups? The n-slenderness is closed under free products, but I don't know whether the n-slenderness is closed under the amalgamated free product.