TITLE: Spin structure on mapping torus
QUESTION [8 upvotes]: I would like to know if, given a spin manifold $X$ and an orientation-preserving diffeomorphism $f : X \longrightarrow X,$ we can naturally endow the mapping torus $M_f = X \times [0, 1] / (x, 0) \sim (f(x), 1)$ with a spin structure.
In the case that interests me particularly, $X$ is simply the two-dimensional torus and $f$ is a classifying map for an automorphism of ${\mathbb Z}^2.$
Thank you for any answer !

REPLY [3 votes]: You can do this iff the spin structures $\mathfrak{s}$ and $f^*(\mathfrak{s})$ are isomorphic.
When $X$ is the 2-torus the set of Spin structures is naturally in bijection with $\mathbb{Z}/2 \oplus \mathbb{Z}/2$, but $SL_2(\mathbb{Z})$ does not act in the usual way. In fact it doesn't act linearly at all, but affinely:
$$
\begin{bmatrix}
A &C \\
B& D
\end{bmatrix} : \begin{bmatrix}
u \\
v
\end{bmatrix} \mapsto \begin{bmatrix}
A &C \\
B& D
\end{bmatrix} \cdot \begin{bmatrix}
u \\
v
\end{bmatrix} + \begin{bmatrix}
AC \\
BD
\end{bmatrix}.
$$
Using this formula you can check if your $f$ preserves a given Spin structure.