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Let $X$ be a orientable 4-manifolds.

I know that $X$ can be endowed with $\operatorname{Spin}^\mathbb{C}$ structures by the choice of integral lift of $w_2(X)$ (second Stiefel-Whitney class of tangent bundle of $X$) in $H^2(X;\mathbb{Z})$. (I know that this is always possible.)

Then, my question is what is the set of $\operatorname{Spin}^\mathbb{C}$ structures on $X$?

1st candidate : In Fintushel's Budapest 2004 conference Lecture note, it says that "$\operatorname{Spin}^\mathbb{C}$ structures are classified by the lifts of $w_2(X)$ to $H^2(X;\mathbb{Z})$ up to the action of $H^1(X;\mathbb{Z}/2)$.

2nd candidate: In Taubes' introduction of famous SW->GW JAMS paper, Taubes says that the set of equivalence classes of $\operatorname{Spin}^\mathbb{C}$ structures ($\operatorname{Spin}$, in his notation) has naturally the structure of a principal $H^2(X;\mathbb{Z})$ bundle over a point. (Hence, it should be equal to $H^2(X;\mathbb{Z})$.)

I think that 1st candidate is right answer. Is it right?

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Taubes is in the 4-manifold case, in which it seems the two candidates agree (see Gompf-Stipsitz sections 2.4 and 10.4) – Max Sep 15 '11 at 20:16

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