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The fundamental group of the circle is $\mathbb{Z}$ because: It is a topological group, so it's its fundamental group is Abelian by the Eckmann-Hilton argument. Thus it's its fundamental group and first singular homology group coincide by the Hurewicz theorem. Since singular homology is the same as simplicial homology, I can just do the one line of computation to obtain the result. |
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1 | [made Community Wiki] | ||
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The fundamental group of the circle is $\mathbb{Z}$ because: It is a topological group, so it's fundamental group is Abelian by the Eckmann-Hilton argument. Thus it's fundamental group and first singular homology group coincide by the Hurewicz theorem. Since singular homology is the same as simplicial homology, I can just do the one line of computation to obtain the result. |
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