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

Abstract

We define a new $E_{\infty }$ operad based on surfaces with foliations which contains $E_k$ sub-operads. 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.

36 pages, 15 figures, new version, some more explanations and clarifications added

References in corpus (2)

• String Topology: Background and Present State
• Strings and the stable cohomology of mapping class groups