Abstract: 
We define a new $E_{\infty}$ operad based on surfaces
with foliations which contains $E_k$ suboperads. We construct CW
models for these operads and provide applications of these models by
giving actions on Hochschild complexes (thus making contact with
string topology), by giving explicit cell representatives
 for the Dyer-Lashof-Cohen operations for the
2-cubes and by constructing new $\Omega$ spectra. The underlying
novel principle is that we can trade genus in the surface
representation vs. the dimension $k$ of the little $k$-cubes.

1. Department of Mathematics
   Purdue University
   150 N. University St.
   West Lafayette, IN 47907-2067, U.S.A.