two maps are not homotopic equivalent

Question

Let $S^1\times S^1=\{(e^{2\pi ix},e^{2\pi iy})\in\mathbb{C^2}\mid x,y\in\mathbb{R}\}$,$g,h:S^1\times S^1\rightarrow S^1\times S^1$ are defined as follows: $g(e^{2\pi ix},e^{2\pi iy})=(e^{2\pi i(2x+3y)},e^{2\pi i(x+2y)}),h(e^{2\pi ix},e^{2\pi iy})=(e^{2\pi i(2x+3y)},e^{2\pi i(x+y)})$,I want to prove they are not homotopic equivalent.I think the induced homology group by $g,h$ are not isomorphic.Can anyone show me the

Answer 1

Spaces may or may not be homotopy equivalent. Maps may or may not be homotopic.

Homotopic maps induce equal maps on homology groups. The first homology group $H_1(S^1\times S^2)\equiv \Bbb Z\times \Bbb Z$. With respect to the standard basis, $g_*$ and $h_*$ have matrices $\pmatrix{2&3\\1&2}$ and $\pmatrix{2&3\\1&1}$ respectively. The matrices are different, so the maps $g$ and $h$ are not homotopic.