[Paper Review] Symmetric Homology of Algebras

Shaun V. Ault, Fiedorowicz, Zbigniew|ArXiv.org|Aug 11, 2007

Homotopy and Cohomology in Algebraic Topology10 references3 citations

TL;DR

This paper introduces symmetric homology of algebras as an analog of cyclic homology, replacing cyclic groups with symmetric groups via the framework of crossed simplicial groups. It defines symmetric homology using the symmetric bar construction and Tor functors over the category ΔS, establishing foundational spectral sequences and relating it to cyclic homology through a natural map, with key results showing HS₀(A) = A/[A,A] and a chain-level description in low degrees.

ABSTRACT

In this note, we outline the general development of a theory of symmetric homology of algebras, an analog of cyclic homology where the cyclic groups are replaced by symmetric groups. This theory is developed using the framework of crossed simplicial groups and the homological algebra of module-valued functors. The symmetric homology of group algebras is related to stable homotopy theory. Two spectral sequences for computing symmetric homology are constructed. The relation to cyclic homology is discussed and some conjectures and questions towards further work are discussed.

Motivation & Objective

• To develop a homology theory for algebras analogous to cyclic homology, but based on symmetric groups instead of cyclic groups.
• To establish a homological framework using crossed simplicial groups, particularly ΔS, the category of symmetric sets.
• To define symmetric homology via the symmetric bar construction and Tor over k[ΔS], generalizing the cyclic homology construction.
• To relate symmetric homology to stable homotopy theory and cyclic homology, and to construct spectral sequences for computation.
• To provide explicit chain-level descriptions of symmetric homology in low degrees and conjecture further structural properties.

Proposed method

• Uses the category ΔS of sets with ordered preimages under morphisms, constructed via Pirashvili’s operadic framework from the associative operad.
• Defines the symmetric bar construction B^sym A: ΔS → k-modules, sending [n] to A^⊗(n+1), with morphisms acting by ordered products on fibers.
• Applies homological algebra of module-valued functors, computing symmetric homology as Tor^ΔS_*(k̲, B^sym A), where k̲ is the constant functor to k.
• Constructs two spectral sequences for computing symmetric homology using the filtration by symmetric group degrees and the skeletal filtration of ΔS.
• Relies on the fact that ΔS is a crossed simplicial group with underlying symmetric groups Σ_{n+1}, enabling the use of standard homological algebra over k[ΔS].
• Uses the Vrećica–Živaljević connectivity theorem to construct finite projective resolutions of the constant functor k̲ over k[ΔS], with control on degrees.

Experimental results

Research questions

• RQ1 How can cyclic homology be generalized by replacing cyclic groups with symmetric groups in the homological framework?
• RQ2 What is the structure of symmetric homology for associative algebras, and how does it relate to the algebraic structure of the algebra?
• RQ3 Can spectral sequences be constructed to compute symmetric homology, and what are their convergence properties?
• RQ4 How does symmetric homology relate to cyclic homology, and what is the nature of the natural map between them?
• RQ5 What is the explicit chain-level description of symmetric homology in low degrees, and how does it reflect the algebraic symmetrization of the algebra?

Key findings

• HS₀(A) is isomorphic to A/[A,A], the symmetrization of A, confirming that symmetric homology captures the abelianization of the algebra.
• In degree 1, symmetric homology is the homology of a partial chain complex with terms A⊗A⊗A and (A⊗A⊗A⊗A)⊕A, with explicit differentials involving products and cyclic permutations.
• The map HC*(A) → HS*(A) is induced by a natural transformation from the functor b̲ to the constant functor k̲, reflecting the fact that cyclic homology is a quotient of symmetric homology.
• The symmetric bar construction extends to a covariant functor on ΔS, enabling the definition of symmetric homology via Tor, in contrast to the contravariant cyclic bar construction.
• The category ΔS is realized as the category of operators for the associative operad, providing a conceptual foundation for the symmetric bar construction.
• A finite projective resolution of the constant functor k̲ over k[ΔS] exists, with projectives P_m for m ≤ (3/2)(i+1) in degree i, due to the Vrećica–Živaljević connectivity theorem.