(i) Suppose that $(C, d)$ and $\left(C^{\prime}, d^{\prime}\right)$ are chain complexes, and $f, g: C \rightarrow C^{\prime}$ are chain maps. Define what it means for $f$ and $g$ to be chain homotopic.

Show that if $f$ and $g$ are chain homotopic, and $f_{*}, g_{*}: H_{*}(C) \rightarrow H_{*}\left(C^{\prime}\right)$ are the induced maps, then $f_{*}=g_{*}$.

(ii) Define the Euler characteristic of a finite chain complex.

Given that one of the sequences below is exact and the others are not, which is the exact one?

$$\begin{aligned} &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{25} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \\ &0 \rightarrow \mathbb{Z}^{11} \rightarrow \mathbb{Z}^{24} \rightarrow \mathbb{Z}^{19} \rightarrow \mathbb{Z}^{13} \rightarrow \mathbb{Z}^{20} \rightarrow \mathbb{Z}^{23} \rightarrow \mathbb{Z}^{11} \rightarrow 0 \end{aligned}$$

Justify your choice.