In addition to the fundamental group and the higher homotopy groups, there are other groups that can be attached to a topological space in a way that is topologically invariant. To motivate them, let us look again at the fundamental group. Using the device of circle representatives as described in Chapter 7, we can think of nontrivial elements of the fundamental group of a space X as equivalence classes of maps from the circle into X that do not extend to the disk. Roughly, the idea of homology theory is to divide out by a somewhat larger equivalence relation, so a map from the circle will represent the zero element if it extends continuously to any surface whose boundary is the circle.
KeywordsExact Sequence Fundamental Group Homology Group Homology Class Free Abelian Group
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