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I'm confused with the following definition of a spin map.

A spin map is a map $f: N\to M$ between differentiable manifolds such that their second Stiefel-Whitney classes are related $\omega_2(N)=f^*\omega_2(M)$.

My question is why the above condition is equivalent to there existing a Dirac bundle $W\to N$ that is locally isometric to the tensor product of spinor bundles $SN\otimes f^*SM$ of $N$ and $M$?

I'm not sure if the question is too easy on MO. Could you give me some help with the details? Thanks in advance.

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  • $\begingroup$ The bundle $SN\otimes f^*SM$ would be a spinor bundle for $TN\oplus f^*TM$, which has second Stiefel-Whitney class $w_2(TN)+f^*w_2(TM)+w_1(TN)\smile f^*w_1(TM)$. So unless both manifolds are nonorientable, your description is correct. $\endgroup$
    – Sebastian Goette
    Commented Apr 18, 2022 at 15:12

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