Dimension vs. genus: A surface realization of the little $k$-cubes and an $E_{\infty}$ operad
Volume 85 / 2009
Banach Center Publications 85 (2009), 241-274
MSC: Primary 55P48; Secondary 55P35, 16E40, 18D50, 32G15.
DOI: 10.4064/bc85-0-17
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.
