Relations in the modulo 2 homology of framed disks algebras

TL;DR

This paper investigates the algebraic relations in the modulo 2 homology of spaces with framed disks operad actions, providing new computations and insights into the structure of these homologies and their operators.

Contribution

It computes relations between key operations in the homology of framed disks algebras and completes the modulo 2 homology of -spheres as a BV algebra.

Findings

Relations between Kudo-Araki and BV operators established

Modulo 2 homology of  S^3 computed as a BV algebra

Evidence provided for a classical conjecture on the Hurewicz homomorphism

Abstract

We study the homology structure of spaces having an action of the \emph{framed disks} operad. In particular, we compute the relations between Kudo-Araki operations and generalized Batalin-Vilkovisky operators. As an application, we complete the computations of the modulo 2 homology of $\Omega^2 S^3$ as a Batalin-Vilkovisky algebra, and give some evidence for a classical conjecture about the modulo 2 Hurewicz homomorphism of the infinite loop space of the sphere spectrum.