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

TL;DR

This paper introduces a new $E_{ty}$ operad based on surface foliations, connecting surface genus with the little $k$-cubes dimension, and explores its applications in string topology and spectra.

Contribution

It constructs a novel $E_{ty}$ operad using surface models that include $E_k$ sub-operads and develops CW models with multiple applications.

Findings

Provides explicit cell representatives for Dyer-Lashof-Cohen operations

Establishes actions on Hochschild complexes linked to string topology

Constructs new $$ spectra

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.