Question

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 !

Answer 1

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.