Hatcher, *Algebraic Topology*, Chapter 2, Section 1

**8. Construct a 3-dimensional -complex from tetrahedra by the following two steps. First, arrange the tetrahedra in a cyclic pattern as in Figure 1 below, so that each shares a common vertical face with its two neighbors and , subscripts being taken mod . Then, identify the bottom face of with the top face of for each . Show the simplicial homology groups of in dimensions 0, 1, 2, 3 are , , 0, , respectively. [The space is an example of a ***lens space*; see Example 2.43 for the general case.]

**Figure 1**

The space mentioned above

*Proof.*

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