ABSTRACT
Among the topological invariants the Euler characteristic is a quantity readily computable by the ‘polyhedronization’ of space. The homology groups are refinements, so to speak, of the Euler characteristic. Moreover, we can easily read off the Euler characteristic from the homology groups. Let us look at figure 3.1. In figure 3.1(a), the interior is included but not in figure 3.1(b). How do we characterize this difference? An obvious observation is that the three edges of figure 3.1(a) form a boundary of the interior while the edges of figure 3.1(b) do not (the interior is not a part of figure 3.1(b)). Clearly the edges in both cases form a closed path (loop), having no boundary. In other words, the existence of a loop that is not a boundary of some area implies the existence of a hole within the loop. This is our guiding principle in classifying spaces here: find a region without boundaries, which is not itself a boundary of some region. This principle is mathematically elaborated into the theory of homology groups.
