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The Singular_homology article of Wikipedia mentions singular n-simplex doesn't have to be injective. But when used for integration on chains, it is always injective or it doesn't make sense. Then what is the non-injective use case when the non-injectiveness is important for the application?

jw_
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  • Why does it have to be injective for integration (either meaning applying a singular cochain to a chain, or integrating the pullback of the differential form (smooth singular homology) if you want to prove de Rham's theorem) to make sense? – user10354138 Sep 23 '21 at 13:38
  • @user10354138 The Differential_form article of Wikipedia says "A k-chain is a formal sum of smooth embeddings D → M. " which make me think it is also injective when integrating. Also I thought integration on chain is cut the manifold into pieces each of which is diffeomorphism to standard k-simplex then integrate them on $R^n$. Although you can make this map non injective too but it doesn't bring any essential effect. – jw_ Sep 24 '21 at 00:27
  • That is a different type of chain. Any homology theory will have its own chains. – user10354138 Sep 24 '21 at 00:48
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    While wikipedia can be great at giving you a quick and dirty introduction to a mathematical subject, it can be dangerous to overintepret the content of a wikipedia page. Integration over a non-injective singular simplex makes perfect sense as long as the defining map is smooth: pull back the form and integrate on the domain. The theory of de Rham cohomology uses such integrals all the time, to prove the major theorems, for example. – Lee Mosher Sep 24 '21 at 20:33

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