Homotopy Groups

The idea of homology groups in the previous chapter was to assign a group structure to cycles that are not boundaries. In homotopy groups, however, we are interested in continuous deformation of maps one to another. Let X and Y be topological spaces and let F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275826/f89ac38d-581c-4b20-9248-de0a1dc00421/content/eq820.tif"/> be the set of continuous maps, from X to Y. We introduce an equivalence relation, called ‘homotopic to’, in T by which two maps f, g ∈ F https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315275826/f89ac38d-581c-4b20-9248-de0a1dc00421/content/eq821.tif"/> are identified if the image f(X) is continuously deformed to g(X) in Y. We choose X to be some standard topological spaces whose structures are well known. For example, we may take the n-sphere Sⁿ as the standard space and study all the maps from Sⁿ to Y to see how these maps are classified according to homotopic equivalence. This is the basic idea of homotopy groups.