Spin structure on mapping torus

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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1 Answer

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
@ Kevin Walker @ Oscar Randal-Williams : I would like to ask if all 4-dimensional compact orientable mapping tori are spin? – Xiao-Gang Wen May 8 '14 at 17:06
@OscarRandal-Williams : do you have a reference for that formula? It seems to be at odds with the transformation rules from projecteuclid.org/euclid.cmp/1104115859, in the sense that your formula allows you to go from any spin structure to any other spin structure, while there it is shown that "even" and "odd" spin structures are distinct under the action of diffeomorphisms (see around 4.25 in the paper), and there is 1 odd spin structure on the torus (the trivial one), and the other 3 are even. – Jan Jitse Venselaar May 8 '14 at 19:51
@Jan: I think that $(u,v)=(1,1)$ is fixed by any operation (the Arf invariant I have in mind is $(1+u)(1+v)$). This uses that if $A,B,C,D$ are the entries of an element of $SL(2,\mathbb{Z})$ then $A+C+AC \equiv 1 \, mod \, 2$ (i.e. $A$ and $C$ cannot both be even). – Oscar Randal-Williams May 8 '14 at 21:08