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We define a new $E_{\infty}$ operad based on surfaces with foliations whichcontains $E_k$ sub-operads. We construct CW models for these operads andprovide applications of these models by giving actions on Hochschild complexes-thus making contact with string topology-, by giving explicit cellrepresentatives for the Dyer-Lashof-Cohen operations for the 2-cubes and byconstructing new $\Omega$ spectra. The underlying novel principle is that wecan trade genus in the surface representation vs. the dimension $k$ of thelittle $k$-cubes.

Dimension vs. Genus: A surface realization of the little k-cubes and an E_{\infty}-operad