So far, I think algebraic topology is turning out to be the best thing since sliced bread. However, I'm having a bit of difficulty with homology, for one particular reason.(adsbygoogle = window.adsbygoogle || []).push({});

Consider, as an example, the first homology group of ##S^1##. The definition of the free abelian group (or, in general, the ##R##-module) of ##n##-chains is the free abelian group generated by ALL ##n##-simplices. Why do we not include ALL paths in ##S^1## in this definition? Are these not also 1-simplices?

Thank you!

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# Why Are Homology Groups Not MUCH Larger?

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