Higher Hochschild homology is not a stable invariant

TL;DR

Higher Hochschild homology, unlike classical homology, is not invariant under suspension, demonstrating it does not behave as a stable invariant in homotopy theory.

Contribution

The paper provides the first counterexample showing higher Hochschild homology is not a stable invariant, challenging previous assumptions.

Findings

Higher Hochschild homology is not preserved under suspension.

Counterexample disproves the stability of higher Hochschild homology.

Challenges the analogy between higher Hochschild homology and classical homology.

Abstract

Higher Hochschild homology is the analog of the homology of spaces, where the context for the coefficients -- which usually is that of abelian groups -- is that of commutative algebras. Two spaces that are equivalent after a suspension have the same homology. We show that this is not the case for higher Hochschild homology, providing a counterexample to a behavior so far observed in stable homotopy theory.