Filtered Hirsch Algebras

TL;DR

This paper develops a filtered Hirsch model for higher order homotopy commutative differential graded algebras, exploring Massey products, their finite order properties, and applications to loop homology and Hochschild cohomology.

Contribution

It introduces a filtered Hirsch model for special differential graded algebras and investigates properties of Massey products, including their finite order and vanishing behavior over various fields.

Findings

Finite order of symmetric Massey products when x^2=0 over integers.

Vanishing of Massey products over fields of characteristic zero.

Lifting of Kraines formula to mod p cohomology.

Abstract

Motivated by the cohomology theory of loop spaces, we consider a special class of higher order homotopy commutative differential graded algebras and construct the filtered Hirsch model for such an algebra $A$. When $x\in H(A)$ with $\mathbb{Z}$ coefficients and $x^{2}=0,$ the symmetric Massey products $% \langle x\rangle ^{n}$ with $n\geq 3$ have a finite order (whenever defined). However, if $\Bbbk $ is a field of characteristic zero, $\langle x\rangle ^{n}$ is defined and vanishes in $H(A\otimes \Bbbk )$ for all $n$. If $p$ is an odd prime, the Kraines formula $\langle x\rangle ^{p}=-\beta \mathcal{P}_{1}(x)$ lifts to $H^{\ast }(A\otimes {\mathbb{Z}}_{p}).$ Applications of the existence of polynomial generators in the loop homology and the Hochschild cohomology with a $G$-algebra structure are given.