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The fact that every compact oriented 4-manifold admits spinc structures was proved long ago by Hirzebruch and Hopf. However, the usual proof is neither direct nor transparent. This article gives a new proof using twistor spaces that is simpler and more geometric. After using these ideas to clarify various aspects of four-dimensional geometry, we then explain how related ideas can be used to understand both spin and spinc structures in any dimension.
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Dearricott, O.; Tuschmann, W.; Nikolayevsky, Y.; Leistner, T.; Crowley, D. (Ed.)The author has elsewhere given a complete classification of the compact oriented Einstein 4-manifolds that satisfy W⁺ (⍵, ⍵) > 0 for some self-dual harmonic 2-form ⍵, where W⁺ denotes the self-dual Weyl curvature. In this article, similar results are obtained when W⁺ (⍵ , ⍵) ≥ 0, provided the self-dual harmonic 2-form ⍵ is transverse to the zero section of Λ⁺→ M. However, this transversality condition plays an essential role in the story; dropping it leads one into wildly different territory where entirely different phenomena predominate.more » « less
