The word problem for some uncountable groups given by countable words. (English) Zbl 1250.57002
The authors study the fundamental group and the first singular homology group of Griffiths’ space and of the Hawaiian Earrings by using countable reduced tame words. It is shown that two such words represent the same element in the corresponding group if and only if they can be carried to the same tame word by a finite number of word transformations of a given list. Using this the authors construct elements with special properties in these groups. It is shown that the homology groups of Griffiths’ space and of the Hawaiian Earrings contain uncountably many different elements that can be represented by infinite concatenations of countably many commutators of loops. As another application the authors give a short proof that these homology groups contain the direct sum of \(2^{\aleph_0}\) copies of the rational numbers. Finally it is shown that the fundamental group of Griffiths’ space contains the rational numbers.
Reviewer: Stephan Rosebrock (Karlsruhe)
MSC:
| 57M05 | Fundamental group, presentations, free differential calculus |
| 55N10 | Singular homology and cohomology theory |
| 54A05 | Topological spaces and generalizations (closure spaces, etc.) |
Keywords:
word problem for uncountable groups; fundamental group; singular homology group; Hawaiian Earring; Griffiths’ spaceReferences:
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