Batalin-Vilkovisky algebras and the J-homomorphism

TL;DR

This paper characterizes the algebraic structure of the homology of iterated loop spaces over the rationals, linking the action of SO(n) to the J-homomorphism and analyzing BV-operator behavior in different characteristics.

Contribution

It provides a complete determination of the $H_*f ext{D}_n$-algebra structure on rational homology of iterated loop spaces and relates the SO(n) action to the J-homomorphism.

Findings

The $H_*f ext{D}_n$-algebra structure on $H_*( ext{Ω}^n X; ext{Q})$ is fully determined.

The action of $H_*(SO(n))$ on iterated loop space homology is connected to the J-homomorphism.

The BV-operator vanishes on spherical classes in characteristics other than 2.

Abstract

Let X be a topological space. The homology of the iterated loop space $H_*\Omega^n X$ is an algebra over the homology of the framed n-disks operad $H_*f\mathcal{D}_n$ \cite{Getzler:BVAlg,Salvatore-Wahl:FrameddoBVa}. We determine completely this $H_*f\mathcal{D}_n$-algebra structure on $H_*(\Omega^n X;\mathbb{Q})$. We show that the action of $H_*(SO(n))$ on the iterated loop space $H_*\Omega^n X$ is related to the J-homomorphism and that the BV-operator vanishes on spherical classes only in characteristic other than 2.