Hatcher 2.1.8

Hatcher, Algebraic Topology, Chapter 2, Section 1

8. Construct a 3-dimensional \Delta-complex X from n tetrahedra T_1,\ldots,T_n by the following two steps. First, arrange the tetrahedra in a cyclic pattern as in Figure 1 below, so that each T_i shares a common vertical face with its two neighbors T_{i-1} and T_{i+1}, subscripts being taken mod n. Then, identify the bottom face of T_i with the top face of T_{i+1} for each i. Show the simplicial homology groups of X in dimensions 0, 1, 2, 3 are \mathbb{Z}, \mathbb{Z}_n, 0, \mathbb{Z}, respectively. [The space X is an example of a lens space; see Example 2.43 for the general case.]

Hatcher 2.1.8
Figure 1
The space mentioned above

Proof.

\square

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