[Paper Review] Representation homology of topological spaces

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

Homotopy and Cohomology in Algebraic Topology45 references4 citations

TL;DR

This paper introduces representation homology of topological spaces as a derived extension of representation varieties, using both derived functors on simplicial groups and functor homology, establishing an isomorphism between representation homology of suspensions and higher Hochschild homology. A key contribution is the identification of representation homology for link complements in ℝ³ with ordinary Hochschild homology, yielding a new algebraic invariant of links.

ABSTRACT

In this paper, we introduce and study representation homology of topological spaces, which is a natural homological extension of representation varieties of fundamental groups. We give an elementary construction of representation homology parallel to the Loday-Pirashvili construction of higher Hochschild homology; in fact, we establish a direct geometric relation between the two theories by proving that the representation homology of the 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 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) $. In the case of link complements, we identify the representation homology in terms of ordinary Hochschild homology, which gives a new algebraic invariant of links in $\R^3$.

Motivation & Objective

• To develop a homological extension of representation varieties of fundamental groups, addressing their singularities and limited applicability beyond aspherical spaces.
• To establish a geometric and algebraic framework for representation homology using derived functors and functor categories, inspired by Loday-Pirashvili's higher Hochschild homology.
• To relate representation homology to other homology theories, such as Pontryagin algebras, S¹-equivariant homology of free loop spaces, and stable homology of automorphism groups of free groups.
• To compute representation homology explicitly for key spaces, including spheres, suspensions, Riemann surfaces, and 3-manifolds like lens spaces and link complements.
• To construct a new algebraic invariant of links in ℝ³ by identifying their representation homology with ordinary Hochschild homology.

Proposed method

• Define representation homology via non-abelian derived functors on simplicial groups, generalizing classical representation varieties.
• Provide an equivalent formulation using classical homological algebra in functor categories, parallel to the Loday-Pirashvili construction of higher Hochschild homology.
• Establish a direct isomorphism between the representation homology of the suspension of a pointed connected space and its higher Hochschild homology.
• Construct natural spectral sequences and maps linking representation homology to S¹-equivariant homology of the free loop space and stable homology of Aut(Fn).
• Use model category theory and Bousfield-Kan approximations to derive homotopy colimits and establish derived functors in the context of left model approximations.
• Apply the theory to compute representation homology for specific spaces, including S^n, ΣX, ℂP^n, Riemann surfaces, and 3-manifolds such as link complements and lens spaces L(p,q).

Experimental results

Research questions

• RQ1 How can representation varieties of fundamental groups be extended to a homological theory that captures higher homotopy information?
• RQ2 What is the precise relationship between representation homology and higher Hochschild homology, particularly for suspensions?
• RQ3 Can representation homology be related to other classical homology theories, such as Pontryagin algebras or S¹-equivariant homology of free loop spaces?
• RQ4 What are the explicit computations of representation homology for key topological spaces like spheres, projective spaces, and 3-manifolds?
• RQ5 Does representation homology of link complements in ℝ³ yield a new algebraic invariant of links, and if so, how is it related to Hochschild homology?

Key findings

• Representation homology of the suspension of a pointed connected space is isomorphic to its higher Hochschild homology, establishing a direct geometric link between the two theories.
• For link complements in ℝ³, the representation homology is isomorphic to the ordinary Hochschild homology of a certain algebra, providing a new algebraic invariant of links.
• The representation homology of spheres S^n and suspensions ΣX is computed explicitly in terms of known invariants, including Hochschild homology and derived functors.
• For lens spaces L(p,q), the representation homology is computed explicitly, showing dependence on the parameters p and q.
• The representation homology of Riemann surfaces is computed and shown to relate to the geometry of their fundamental groups and character varieties.
• Spectral sequences and natural maps are constructed that relate representation homology to S¹-equivariant homology of the free loop space and stable homology of Aut(Fn), revealing deeper structural connections.