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We compute all loop topological string amplitudes on orientifolds of localCalabi-Yau manifolds, by using geometric transitions involving SO/SpChern-Simons theory, localization on the moduli space of holomorphic maps withinvolution, and the topological vertex. In particular we count Klein bottlesand projective planes with any number of handles in some Calabi-Yauorientifolds.Comment: 40 pages, 18 figures, some corrections in section

Counting Higher Genus Curves with Crosscaps in Calabi-Yau Orientifolds