位相幾何学の続き

以前にTorusの話を行いましたからその説明を済ませます。
Given that we have determined H0, H1, and H2, the homology groups on the torus, we now turn our attention to the Betti numbers h0, h1, and h2. The two groups isomorphic to C2, H0 and H2, should have a single element in their basis and Betti number of one. H1, on the other hand, is isomorphic to the Klein four group. It would seem that, given elements a and b, all members of the group can be generated (a, b, a+b, null). Thus, h1 is two. This agrees with the solution of problem 8 on page 163, which asks the reader to prove that the first Betti number is 2k for the connected sum of k tori.