# 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.]

Figure 1
The space mentioned above

Proof.

$\square$