Let

C_2 \xrightarrow{\partial_2} C_1 \xrightarrow{\partial_1} C_0

be a chain complexDefinitionChain complex2026-10-04 · Lîm Tsú-thuàn of free finite rank $R$ -modules over a principal ideal domain $R$ . Then

H_1(C_\bullet) \cong R^b \oplus R/(a_1) \oplus \cdots \oplus R/(a_n)

where $a_1, \dots, a_n \in R$ are the invariant factors of $\partial_2$ and where $b = \text{rank} \ker \partial_1 - n$ .

Example Real projective plane [local-0]

Take $R = \mathbb{Z}$ and the real projective plane $\mathbb{R}P^2$ as a $\Delta$ -complexDefinitionDelta complex2026-10-04 · Lîm Tsú-thuàn: two vertices $v, w$ , three edges $a, b, c$ , and two triangles $U, L$ .

Order the vertices of each triangle $[v_0, v_1, v_2]$ so that $v_0 v_1 = c$ in both, $v_1 v_2 = a, v_0 v_2 = b$ in $U$ , and $v_1 v_2 = b, v_0 v_2 = a$ in $L$ . Then

\partial_2 U = a - b + c \qquad \partial_2 L = -a + b + c

In the bases $U, L$ and $a, b, c$ , the Smith normal form of $\partial_2$ is

\begin{pmatrix} 1 & -1 \\ -1 & 1 \\ 1 & 1 \end{pmatrix} \xrightarrow{R_2 + R_1,\ R_3 - R_1} \begin{pmatrix} 1 & -1 \\ 0 & 0 \\ 0 & 2 \end{pmatrix} \xrightarrow{C_2 + C_1,\ R_2 \leftrightarrow R_3} \begin{pmatrix} 1 & 0 \\ 0 & 2 \\ 0 & 0 \end{pmatrix}

so the invariant factors are $a_1 = 1, a_2 = 2$ and $n = 2$ . For $b$ , we have $\partial_1 a = \partial_1 b = w - v$ and $\partial_1 c = 0$ , so $\partial_1$ has rank $1$

\text{rank} \ker \partial_1 = \text{rank}\ C_1 - \text{rank}\ \partial_1 = 3 - 1 = 2

Hence, $b = 2 - 2 = 0$ . Now applied the theorem

H_1(\mathbb{R}P^2) \cong \mathbb{Z}^{2-2} \oplus \mathbb{Z}/(1) \oplus \mathbb{Z}/(2) \cong \mathbb{Z}/2

Lemma. First homology of a short chain complex over a PID [LNMK]

Example Real projective plane [local-0]