Barlow surface

In mathematics, a Barlow surface is one of the complex surfaces discovered by Template:Harvs. They are simply connected surfaces of general type with p_g = 0. They are homeomorphic but not diffeomorphic to a projective plane blown up in 8 points. The Hodge diamond for the Barlow surfaces is: Template:Hodge diamond

Detection of exotic smooth structures

According to Freedman's classification, a simply connected oriented closed topological 4-manifold with intersection form $[+1]\oplus 8[-1]$ is homeomorphic to $ℂP^2\mathrm{\#}8\overline{ℂP^2}$. Any such smooth manifold with a non-vanishing Seiberg–Witten invariant has an exotic smooth structure since $ℂP^2\mathrm{\#}8\overline{ℂP^2}$ 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

References

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