Suppose that $M$ is an oriented riemannian manifold and choose transition functions $\varphi_{ij}:U_i \cap U_j \to SO(n)$ for the tangent bundle. They satisfy the cocycle condition $\varphi_{ij} \varphi_{jk} \varphi_{ki}=1$. One can lift them to $\psi_{ij}:U_i \cap U_j \to spin(n)$ and these lifted functions need not satisfy cocycle condition. However the element $\alpha_{ijk}=\psi_{ij} \psi_{jk} \psi_{ki}:U_i \cap U_j \cap U_k \to spin(n)$ is central and one can show that $\alpha$ defines Cech $2$-cocycle. One can also show that its cohomology class $[\alpha] \in H^2(M,\mathbb{Z}_2)$ is independent from the choice of transition functions and from the choice of the liftings. I would like to understand why if the class $[\alpha]$ is trivial then $M$ is spin manifold. Here is the attempt:
suppose that $[\alpha]$ is trivial therefore $\alpha=\partial \beta$ where $\partial$ is the boundary in Cech theory. Suppose that $\alpha$ was defined in terms of $\varphi_{ij}$'s and liftings $\psi_{ij}$'s. Consider another liftings $\psi_{ij}':=\beta_{ij}\psi_{ij}$. One can check that now the cocycle condition is felfilled for $\psi_{ij}'$ therefore $\psi_{ij}'$'s can serve for a transition functions for some principial $spin(n)$-bundle $P$ over $M$. However to obtain spin structure we need also a surjective mapping $\eta:P \to SO(M)$ which satisfies $\eta(p \cdot g)=\eta(p) \cdot \rho(g)$ where $\rho:spin(n) \to SO(n)$ is the $2:1$ covering map.

How this map $\eta$ is constructed?

EDIT: I think that the correct way to define this map is the following: from transition functions one construct principial bundle as $P=\bigsqcup_{i}(U_i \times spin(n))/ \sim$ where we glue $(x,g) \sim (x,\psi_{ij}(x)h)$. Therefore it is natural to put $\eta:P \to SO(M)$ as $\eta([(x,g)]):=[(x,\rho(g))]$ where $\rho$ is as above (one checks that this is well defined).

  • 2
    $\begingroup$ I can't answer your question, but I want to applaud you for the clarity with which you asked it. :) $\endgroup$ – John Hughes Jan 8 '17 at 15:20
  • $\begingroup$ I made an edit to this post $\endgroup$ – truebaran Jan 10 '17 at 0:52

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