Augmentation and Reduced Homology Groups

A useful variation of the definitions above is that of reduced homology groups. To define these, let us take another look at dimension —1. Until now, we have always set C A) = 0. However, as the reader will recall from our discussion in Chapter 2, we do have an extra simplex in dimension —1, unless of course our complex is void. 

Assume that A is not void, and let e denote a chosen generator of C-i A). For n 1, set = d . Set do v) = e for all vertices v, and extend do linearly to the whole group Co A). The process of adding the group C-i A) together with the boundary operator do is called augmentation. 

The difference between the reduced homology groups and the nonreduced ones is not large. In fact, 

In the same vein, we get reduced Betti numbers / j(4). If the complex is not empty and not void, then its reduced Betti numbers are equal to e nonreduced ones in all dimensions except for 0, where we have / o(4) = / o(4) + 1. For the empty complex we see that / i = 1, and all other reduced Betti numbers are equal to 0, whereas for the void complex all of its reduced Betti numbers are equal to 0. 

Finally, the reduced Euler characteristic is defined by setting 

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