If we wish to distinguish between the sphere and the torus, we have already seen one way of doing so using the fundamental group. Any loop in the sphere can be continuously shrunk to a point, in other words the sphere is simply connected, whereas this is not the case for the torus. The fundamental group is a very valuable tool, but it has a significant defect. Remember that the fundamental group of a polyhedron depends only on the 2-skeleton of the underlying complex, making it ideal for studying questions which are essentially two-dimensional (say distinguishing between two surfaces), but leaving it impotent in the face of a problem such as showing that S3 and S4 are not homeomorphic.
KeywordsFundamental Group Homology Group Betti Number Homology Class Homotopy Type
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