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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 !

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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.

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But even if $f$ preserves the spin structure, there are two spin structures on the mapping torus which restrict to the original spin structure on the fiber. So there's no way to "naturally endow" the mapping torus with a spin structure (as in the original question). Unless of course you choose a lifting of $f$ to a map of spin bundles. –  Kevin Walker Aug 12 '11 at 16:42
Ah, I had missed "naturally". In that case: what Kevin said. –  Oscar Randal-Williams Aug 12 '11 at 22:46
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