[Paper Review] Representation homology of spaces and higher Hochschild homology

Yuri Berest, Ajay C. Ramadoss+1 more|arXiv (Cornell University)|Mar 10, 2017

Homotopy and Cohomology in Algebraic Topology5 citations

TL;DR

This paper introduces representation homology of topological spaces via classical homological algebra, establishing a direct geometric link to higher Hochschild homology by proving that the representation homology of the reduced suspension of a pointed connected space is isomorphic to its higher Hochschild homology. A key contribution is the Comparison Theorem, which expresses representation homology of simply-connected spaces of finite rational type in terms of Quillen and Sullivan models.

ABSTRACT

In this paper, we study representation homology of topological spaces, that is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology in terms of classical (abelian) homological algebra. Our construction is parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by showing that the representation homology of the (reduced) suspension of a (pointed connected) space is isomorphic to its higher Hochschild homology. We also construct some natural maps and spectral sequences relating representation homology to other standard homology theories associated with spaces (such as Pontryagin algebras, S^1-equivariant homology of the free loop space and stable homology of automorphism groups of f.g. free groups). We compute representation homology explicitly (in terms of known invariants) in a number of interesting cases, including spheres, suspensions, complex projective spaces, Riemann surfaces and some 3-dimensional manifolds (such as link complements in R^3 and the lens spaces L(p,q)). One of our main results, which we call the Comparison Theorem, expresses the representation homology of a simply-connected topological space of finite rational type in terms of its Quillen and Sullivan models.

Motivation & Objective

• To develop a homological extension of representation varieties of fundamental groups using classical homological algebra.
• To establish a geometric relationship between representation homology and higher Hochschild homology.
• To construct natural maps and spectral sequences connecting representation homology to other homology theories such as Pontryagin algebras and S^1-equivariant homology.
• To compute representation homology explicitly for key spaces including spheres, suspensions, projective spaces, Riemann surfaces, and 3-manifolds.
• To prove the Comparison Theorem, expressing representation homology of simply-connected spaces in terms of Quillen and Sullivan models.

Proposed method

• Constructs representation homology using standard abelian homological algebra, avoiding derived categories or higher structures.
• Adopts a framework parallel to the Loday-Pirashvili construction of higher Hochschild homology.
• Demonstrates that representation homology of the reduced suspension of a space is isomorphic to its higher Hochschild homology.
• Introduces natural maps and spectral sequences linking representation homology to S^1-equivariant homology of the free loop space and stable homology of automorphism groups of free groups.
• Employs Quillen and Sullivan models to analyze representation homology in the simply-connected, finite rational type setting.
• Applies classical homological techniques to compute representation homology in specific cases such as spheres, lens spaces, and link complements.

Experimental results

Research questions

• RQ1 How can representation homology of a topological space be constructed using only classical homological algebra?
• RQ2 What is the precise geometric relationship between representation homology and higher Hochschild homology?
• RQ3 How does representation homology relate to other standard homology theories like Pontryagin algebras and S^1-equivariant homology of the free loop space?
• RQ4 Can representation homology be computed explicitly for spaces such as spheres, projective spaces, and 3-manifolds?
• RQ5 To what extent can representation homology of simply-connected spaces of finite rational type be described via Quillen and Sullivan models?

Key findings

• The representation homology of the reduced suspension of a pointed connected space is isomorphic to its higher Hochschild homology.
• Natural maps and spectral sequences are constructed that relate representation homology to S^1-equivariant homology of the free loop space and stable homology of automorphism groups of finitely generated free groups.
• Explicit computations show that representation homology of spheres, suspensions, complex projective spaces, and Riemann surfaces can be expressed in terms of known homological invariants.
• For lens spaces L(p,q) and link complements in R^3, representation homology is computed explicitly using the framework developed.
• The Comparison Theorem provides a complete description of representation homology for simply-connected spaces of finite rational type in terms of Quillen and Sullivan models.