Hochschild-Pirashvili homology on suspensions and representations of $Out(F_n)$

Abstract

We show that the Hochschild-Pirashvili homology on any suspension admits the so called Hodge splitting. For a map between suspensions $f:\Sigma Y\to \Sigma Z$, the induced map in the Hochschild-Pirashvili homology preserves this splitting if $f$ is a suspension. If $f$ is not a suspension, we show that the splitting is preserved only as a filtration. As a special case, we obtain that the Hochschild-Pirashvili homology on wedges of circles produces new representations of $Out(F_n)$ that do not factor in general through $GL(n,Z)$. The obtained representations are naturally filtered in such a way that the action on the graded quotients does factor through $GL(n,Z)$.

22 pages. Compared to the first version the presentation and especially the introduction have been improved

References in corpus (1)

• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma

Cited by in corpus (3)

• Higher Hochschild homology and exponential functors
• Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups
• On analytic contravariant functors on free groups