Representation Homology of Topological Spaces

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, ${{\mathbb{S}}}^1$-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 ${\mathbb{R}}^3$ 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 ${\mathbb{R}}^3$.

Representation homology of simply connected spaces

“…\end{array} \right.} \end{equation*}The above isomorphisms were found by a different method in [5] (see [5, Proposition 4.3]).…”

“…The main aim of this paper is to compute the representation homology for an arbitrary simply connected space $$X$$ over a field $$k$$ of characteristic 0. From [5], we know that the representation homology of such a space is a rational homotopy invariant (that is, it depends only on the homotopy type of the rationalization $$ X_{\mathbb {Q}}$$ of $$X$$); on the other hand, by a fundamental theorem of Sullivan [53], the homotopy type of $$X_{\mathbb {Q}}$$ is completely determined by its algebraic model: a commutative cochain DG algebra $$ {\mathcal {A}}_X$$, called the Sullivan model of $$X$$. This leads us to the natural question.…”

“…In fact, there are several natural models that can be used in practical computations (see [6]). In this paper, we will use most exclusively the so‐called Kan loop group model $$\, \Gamma = \mathbb {G}{X}$$, which is a semi‐free simplicial group functorially attached to the space $$X$$ (see [28, chapter V] or [5, section 2] for a brief summary of this construction). Since semi‐free simplicial groups are cofibrant objects in $$\mathtt {sGr}$$, formula (1.6) simplifies in this case to $$\begin{equation} \mathrm{HR}_\ast (X,G) \, = \,\pi _\ast (\mathbb {G}{X})_{G}.$$…”

“…Now, to define representation homology with coefficients in a commutative Hopf algebra $$ {\mathcal {H}}$$ we simply precompose the corresponding functor $$ \underline{\mathcal {H}}: \mathtt {Gr}\rightarrow \mathtt {Comm}_k$$ with the Kan loop group model of a given space $$X$$: the result is the simplicial commutative algebra $$\begin{equation*} \mathcal {H}(\mathbb {G}{X}):\ \Delta ^{\rm op} \xrightarrow {\mathbb {G}{X}} \mathtt {Gr}\xrightarrow {\underline{{\mathcal {H}}}} \mathtt {Comm}_k \end{equation*}$$whose homology we denote by $$\begin{equation*} \mathrm{HR}_\ast (X,\mathcal {H}) := \pi _*[\mathcal {H}(\mathbb {G}{X})] = \mathrm{H}_\ast [N \mathcal {H}(\mathbb {G}{X})]. \end{equation*}$$For $${\mathcal {H}}= \mathcal {O}(G)$$, where $$G$$ is an affine algebraic group scheme over $$k$$, it is easy to show that there is a natural isomorphism (see [5, Proposition 4.1]): …”

“…\end{equation*}Regarding $$ \underline{\mathcal {O}(G)}$$ and $$ \underline{k[\Gamma ]}$$ as linear functors on $$\mathfrak {G}$$ (with values in $$\mathtt {Vect}_k$$), we can form their tensor product $$\,k[\Gamma ] \otimes _{\mathfrak {G}} \mathcal {O}(G) $$. It turns out that there is a natural isomorphism: $$\, k[\Gamma ] \otimes _{\mathfrak {G}} \mathcal {O}(G)\,\cong \,\mathcal {O}[{\rm {Rep}}_G(\Gamma )] $$; more generally, it is shown in [5] that $$\begin{equation} \mathrm{HR}_\ast (\mathrm{B}\Gamma ,\,G) \,\cong \,{\rm {Tor}}_\ast ^{\mathfrak {G}}(k[\Gamma ],\,\mathcal {O}(G))\ , \end{equation}$$where $$ {\rm {Tor}}^{\mathfrak {G}}_\ast$$ is the (homology of the) classical derived tensor product …”