In mathematics, a Barlow surface is one of the complex surfaces discovered by . They are simply connected surfaces of general type with pg = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is:
Detection of exotic smooth structures According to Freedman's classification, a simply connected oriented closed topological 4-manifold with intersection form is homeomorphic to . Any such smooth manifold with a non-vanishing Seiberg–Witten invariant has an exotic smooth structure since with the induced smooth structure allows a Riemannian metric of positive scalar curvature (psc metric) and therefore has only vanishing Seiberg–Witten invariants.
See also Hodge theory
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