Note that the $2$-torus $T^2$ can be seen as a quotient space $\Bbb R^2/\Bbb Z^2$ of $\Bbb R^2$. Then any $2\times 2$ integer matrix $A=(\begin{smallmatrix} a & b\\ c & d \end{smallmatrix})$ gives a well-defined map $A:T^2\to T^2$. On the other hand, we have $H_1(T^2)=\Bbb Z^2$ and $H_2(T^2)=\Bbb Z$. What I want to show is, the map $A_*:H_2(T^2)\to H_2(T^2)$ induced by $A$, is given by $\Bbb Z\xrightarrow{\times \det(A)} \Bbb Z$, multiplication by $\det(A)$, and the map $A_*:H_1(T^2)\to H_1(T^2)$ is given by $\Bbb Z^2 \xrightarrow{A} \Bbb Z$.
Actually I want to use this result in Exercise 30 of section 2.2, in Hatcher's Algebraic Topology. (http://pi.math.cornell.edu/~hatcher/AT/AT.pdf) Parts (c) to (e) would become easy then.
The map on $H_1$ seems to be computed if we use the identification $\pi_1(T^2)=H_1(T^2)$ (which is after section 2.2, though), but I have no idea for the map on $H_2$. (Maybe a local degree argument?, but I'm not sure) Thanks in advance.
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