Timeline for What is the "higher version" of chain homotopy in singular homology?
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| Dec 14, 2013 at 18:41 | comment | added | Tim Porter | The combinatorial version of homotopies is then easy to give. Trying to give the higher homotopies simplicially is a case of studying the shuffles .... and they are fun! | |
| Dec 14, 2013 at 18:39 | comment | added | Tim Porter | The singular chain groups that you mention are all free Abelian groups on a simplicial set of singular simplices in $X$. Look at one of the fairly elementary introductions to simplicial set theory (I like the old paper by Curtis:Simplicial Homotopy Theory, Advances in Math., 6,(1971),107 – 209, but there are many other introductions).The structure of product simplicial sets is very simple, so at least to start with, homotopies etc. are quite nice to write down.To understand what higher homotopies might be you can work with the simplicial function spaces and then specialise down to the prisms. | |
| Dec 14, 2013 at 17:52 | comment | added | Zhaoting Wei | Thank you very much for your answer and would you like to explain a little bit about "the additive chain complex groups are not the place to study this as the singular complex given as a simplicial set enables the full structure to be exhibited explicitly."? Are additive chain complex and singular complex the same thing? | |
| Dec 14, 2013 at 16:22 | history | answered | Tim Porter | CC BY-SA 3.0 |