why can $g$ be considered as a linear function?

Question

Here is the question I am trying to understand its solution:

Compute the homology groups $H_n(X, A)$ when $X$ is $S^2$ or $S^1 \times S^1$ and $A$ is a finite set of points in $X.$

Here is the part I am trying to understand $(3)$ in it:

My question is:

In the last $2$ lines of $(3),$ why can $g$ be considered as a linear function and hence using the Rank-Nullity theorem to find the dimension of $H_1(S^1 \times S^1, A)$?

Could anyone help me in answering this question please?

Answer 1

$g$ is the connecting homomorphism produced by the snake lemma. Just as the applications induced on homology by continuous functions between spaces, this homomorphism is linear.