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

Abstract

When $X$ is an associative H-space, the bar spectral sequence computes the homology of the delooping, $H_{\ast }(BX)$. If $X$ is an $n$-fold loop space for $n\ge 2$ 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 $H_{\ast }(BX)$, 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 $H_{\ast }(X)$ and the degree $n-2$ bracket on $H_{\ast }(BX)$. This generalizes a result of Browder and puts it in a computational context.