The Bracket in the Bar Spectral Sequence for an Iterated Loop Space

TL;DR

This paper demonstrates that the bar spectral sequence for an iterated loop space preserves the Browder bracket structure, revealing a Poisson algebra structure and connecting brackets across different homology degrees.

Contribution

It shows that the spectral sequence filtration respects the Poisson algebra structure, generalizing Browder's result and linking brackets on homology of X and its delooping.

Findings

Spectral sequence is a Poisson algebra.

Bracket structures are preserved across the spectral sequence.

Connects brackets in homology of X and BX.

Abstract

When $X$ is an associative H-space, the bar spectral sequence computes the homology of the delooping, $H_{*}(BX)$. If $X$ is an $n$-fold loop space for $n\geq2$ this is a spectral sequence of Hopf algebras. Using machinery by Sugawara and Clark, we show that the spectral sequence filtration respects the Browder bracket structure on $H_{*}(BX)$, and so it is moreover a spectral sequence of Poisson algebras. Through the bracket on the spectral sequence, we establish a connection between the degree $n-1$ bracket on $H_{*}(X)$ and the degree $n-2$ bracket on $H_{*}(BX)$. This generalizes a result of Browder and puts it in a computational context.