Spin-c structures on a fixed oriented bundle form an H2 torsor

For any second lift $P_1$, define $L(P_0,P_1{)}_x$ to be the set of Spin${}^c$-equivariant maps from $(P_0{)}_x$ to $(P_1{)}_x$ covering the identity map of the common oriented frame fiber. Since both are lifts of that same frame fiber, two such maps differ by a unique element of the central kernel $U(1)$ from [F1]. In local lift trivializations, the transition functions of $P_0$ and $P_1$ project to the same $\mathrm{S}\mathrm{O}(n)$ functions, so their pointwise quotient lies in central $U(1)$; the cocycle identity for each lift makes these quotients a $U(1)$-valued cocycle. Thus the difference fibers form a principal $U(1)$-bundle $L(P_0,P_1)$ over $X$.

Associate to the principal difference bundle in step 1.1 the standard complex line. By [F2], its unit circle bundle is that principal bundle. The diagonal central action defines the twist $P_0\otimes L=(P_0{\times }_{X}L)\mathrm{/}U(1)$; evaluation of each difference map gives a Spin${}^c$-equivariant isomorphism $P_0\otimes L(P_0,P_1)\cong P_1$. If $P_0\otimes L\cong P_0$, its difference bundle is trivial, so the action is free; the construction for arbitrary $P_1$ proves transitivity. Tensor product of line bundles gives the group law and pullback commutes with the fiberwise construction.

By [F3], the isomorphism class of the associated line $L$ is uniquely identified with $c_1(L)\in H^2(X;\mathbb{Z})$, compatibly with tensor product and pullback. Thus the free transitive line-bundle action is precisely a free transitive $H^2(X;\mathbb{Z})$ action and is natural under $f^{\ast }$. If $X$ is a point, $H^2(X;\mathbb{Z})=0$ and the given lift is the only class in its torsor; no existence of an initial lift is asserted. AC is used only through [A1] and [F3].