Map Induced on $H_1(T^2)$ by Rotation by Irrational Angle

Question

Suppose I have a standard flat torus that is the quotient of $\mathbb{R}^2$ by the lattice $\Lambda = (\vec{e}_1, \vec{e}_2)$ and a matrix $A$ that represents rotation by an irrational angle between 0$^o$ and 90$^o$. What would be the induced map on $H_1(T^2)$ of the diffeomorphism $\phi$ of $T^2$ that takes the image of the geodesic at $t$ determined by the vector $\vec{v}_1$ at $(\theta_1, \theta_2) = (0,0)$ to the image of the geodesic at $t$ determined by the vector $A\vec{v}_1$ at $(\theta_1, \theta_2) = (0,0)$? One can't just look at the number of times the geodesics $\gamma_1$ induced by $A\vec{e}_1$ and $\gamma_2$ induced by $A\vec{e}_2$, say, wrap around each circle, because they're not closed geodesics.

Answer 1

Your final sentence, slightly tweaked, gives the reason why this diffeomorphism $A$ does not exist: if yout take $\vec v_1 = \langle 1,0 \rangle$, then the geodesic induced by $\vec v_1$ does wrap around a circle in $T^2$, whereas the geodesic induced by the vector $A \vec v_1$ does not wrap around a circle. But a diffeomorphism must take a circle to a circle, so no such diffeomorphism as you have described can exist.

In general, consider a unit vector $\vec v_1$ making an angle $\theta = 2 \pi s$ from the positive $x$-axis (measured in radians). Consider also the geodesic in $T^2$ that is induced by $\vec v_1$. Then one of two possibilities holds:

• $s$ is a rational number, and the geodesic induced by $\vec v_1$ wraps around a circle in $T^2$, which is not a dense subset of $T^2$.
• $s$ is an irrational number, and the geodesic induced by $\vec v_1$ does not wrap around a circle in $T^2$, in fact that geodesic is a dense subset of $T^2$.

Since a diffeomorphism cannot take a non-dense subset of $T^2$ to a dense subset of $T^2$, and since an "irrational rotation" diffeomorphism as you have described it would take a rational angle to an irrational angle, no such diffeomorphism exists.

I suspect what you are missing is that although a matrix $A \in GL_+(2,\mathbb R)$ does act on the universal covering space $\mathbb R^2$ of the torus $T^2$, a general matrix $A$ does not preserve the orbits of the lattice action of $\mathbb Z^2$ on $T^2$, and hence there is no induced diffeomorphism of $T^2$.

There is, nonetheless, a special subset of $GL_+(2,\mathbb R)$ whose elements do preserve orbits of the lattice action of $\mathbb Z^2$ on $T^2$ and which do induce diffeomorphisms of $T^2$: that subset is precisely the subgroup $GL_+(2,\mathbb Z)$.

(This is just to get the typo edit through.)