The homotopy groups of a manifold or more general topological space M, which we shall shortly define, represent (as will become evident) the most important of the invariants (under homeomorphisms) of the space M. The one-dimensional homotopy group of M is, by definition, just the fundamental group π1(M,x0). The zero-dimensional homotopy group π0(M,x0) does not, generally speaking, exist: its elements are, by somewhat loose analogy with the general definition of the homotopy groups given below, the pathwise connected components of the space M, from amongst which there is distinguished a “trivial” element, namely the component containing the base point x0; however only in certain cases does this set come endowed with a natural group structure.
KeywordsFundamental Group Normal Bundle Homotopy Class Homotopy Group Loop Space
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