Configuration spaces as commutative monoids

Let $M$ be $d$-dimensional. The space $C_{n}(M;L)^{+}$ is the 1-point compactification of the $n\cdot(d+\dim(L))$-dimensional manifold

$$C_{n}(M;L):=\left[L^{n}\setminus\{(l_{1},\ldots,l_{n})\,|\,\pi(l_{i})=\pi(l_{j})\text{ and }i\neq j\}\right]_{\mathfrak{S}_{n}}.$$

This is a vector bundle over $C_{n}(M)$, but is a manifold itself and is orientable if and only if the manifold $L$ is orientable and even-dimensional. To arrange this, we can take the vector bundle $W$ given by the orientation line of $M$ plus $(d-1)$ trivial line bundles. Thus by Poincaré duality we have

$$H^{*}(C_{n}(M);\mathbbm{k})\cong H^{*}(C_{n}(M;W);\mathbbm{k})\cong\widetilde{H}_{2dn-*}(C_{n}(M;W)^{+};\mathbbm{k}).$$

In view of this, the bar construction description (1.2) can be used, in combination with the homology of free commutative monoids (see [Mil69]), to investigate $H^{*}(C_{n}(M);\mathbbm{k})$. We do not pursue this in general here, but rather focus on the case $\mathbbm{k}=\mathbb{Q}$, where a complete answer is possible, and reproduces a formula of Knudsen.

For an $\mathbb{N}$-graded pointed space we write $H_{n,i}(X):=\widetilde{H}_{i}(X(n))$, and similarly for chains. Write $S^{*}(V)$ for the free graded-commutative algebra on a homologically graded vector space $V$, i.e. $S^{*}(V)=\bigoplus_{n\geq 0}[V^{\otimes n}]_{\mathfrak{S}_{n}}$, where the Koszul sign rule is implemented. If $V$ is equipped with additional $\mathbb{N}$-grading, then this is inherited by $S^{*}(V)$ (but there is no Koszul sign rule associated to the $\mathbb{N}$-grading, only to the homological grading).

We consider $\mathbb{C}(M;W)$. There is a map $\widetilde{C}_{*}(W^{+};\mathbb{Q})[1]\to\widetilde{C}_{*}(\mathbf{Com}(W^{+}[1]);\mathbb{Q})$ and, using the Eilenberg–Zilber maps, it extends to a map of cdga’s

$$S^{*}(\widetilde{C}_{*}(W^{+};\mathbb{Q})[1])\longrightarrow{C}_{*,*}(\mathbf{Com}(W^{+}[1]);\mathbb{Q}),$$

which is an equivalence (since the maps $[(W^{+})^{\wedge n}]_{h\mathfrak{S}_{n}}\to[(W^{+})^{\wedge n}]_{\mathfrak{S}_{n}}$ are rational homology isomorphisms). Similarly, there is an equivalence of cdga’s

$$S^{*}(\widetilde{C}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})[2])\longrightarrow{C}_{*,*}(\mathbf{Com}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}}[2]);\mathbb{Q}).$$

Furthermore, one may choose formality equivalences

	$\displaystyle\widetilde{H}_{*}(W^{+};\mathbb{Q})$	$\displaystyle\longrightarrow\widetilde{C}_{*}(W^{+};\mathbb{Q})$
	$\displaystyle\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})$	$\displaystyle\longrightarrow\widetilde{C}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q}),$

i.e. chain maps inducing the identity on homology, and hence obtain equivalences

	$\displaystyle S^{*}(\widetilde{H}_{*}(W^{+};\mathbb{Q})[1])$	$\displaystyle\longrightarrow S^{*}(\widetilde{C}_{*}(W^{+};\mathbb{Q})[1])$
	$\displaystyle S^{*}(\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})[2])$	$\displaystyle\longrightarrow S^{*}(\widetilde{C}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})[2])$

of cdga’s. In $\mathbb{N}$-grading 2, the map $\Delta$ induces a map

$$\delta_{*}:\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})\overset{\Delta_{*}}{\longrightarrow}\widetilde{H}_{*}([(W^{+})^{\wedge 2}]_{\mathfrak{S}_{2}};\mathbb{Q})\cong[\widetilde{H}_{*}(W^{+};\mathbb{Q})^{\otimes 2}]_{\mathfrak{S}_{2}}.$$

With these choices the square

need not commute, but does commute up to homotopy in the category of cdga’s because the two chain maps $\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})\to\widetilde{C}_{*}([(W^{+})^{\wedge 2}]_{\mathfrak{S}_{2}};\mathbb{Q})$ induce the same map on homology, namely $\delta_{*}$, so are chain homotopic. The bar construction description then gives an identification

$$\mathrm{Tor}_{*}^{S^{*}(\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})[2])}(S^{*}(\widetilde{H}_{*}(W^{+};\mathbb{Q})[1]),\mathbb{Q}[0])\cong H_{*,*}(\mathbb{C}(M;W);\mathbb{Q}).$$

Recall that for a free graded-commutative algebra $S^{*}(V)$ on a homologically graded vector space $V$ (perhaps equipped with a further $\mathbb{N}$-grading), there is a free resolution of the trivial left $S^{*}(V)$-module $\mathbb{Q}$ given by $\epsilon:S^{*}(V\oplus\Sigma V)\overset{\sim}{\to}\mathbb{Q}$ equipped with the differential given by $\partial(\Sigma v)=v$ and extended by the Leibniz rule. It is usually called the Koszul resolution. It is indeed a resolution because it is the free graded-commutative algebra on the acyclic chain complex $\Sigma V\overset{id}{\to}V$, and over $\mathbb{Q}$ taking homology commutes with the formation of symmetric powers. Applying this resolution to calculate the Tor groups above gives the complex

$$\left(S^{*}\big{(}\widetilde{H}_{*}(W^{+};\mathbb{Q})[1]\oplus\Sigma\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})[2]\big{)},\partial\right)$$

with differential given by $\partial(\Sigma x)=\Delta_{*}(x)\in S^{2}(\widetilde{H}_{*}(W^{+}[1];\mathbb{Q}))$ for $x\in\widetilde{H}_{*}([(W\oplus W)^{+}]/\mathfrak{S}_{2}[2];\mathbb{Q})$, and extended by the Leibniz rule. This can be simplified as follows. If $M$ is $d$-dimensional then the Thom isomorphism gives $\widetilde{H}_{*}(W^{+};\mathbb{Q})=\Sigma^{d}\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}})$, where $\mathbb{Q}^{w_{1}}$ is the orientation local system of $M$. It also gives $\widetilde{H}_{*}((W\oplus W)^{+};\mathbb{Q})=\Sigma^{2d}\widetilde{H}_{*}(M^{+};\mathbb{Q})$. The involution swapping the two $W$ factors acts as $(-1)^{d}$ on the Thom class, so because the map $[(W\oplus W)^{+}]_{h\mathfrak{S}_{2}}\to[(W\oplus W)^{+}]_{\mathfrak{S}_{2}}$ is a rational equivalence we find

$$\widetilde{H}_{*}([(W\oplus W)^{+}]_{\mathfrak{S}_{2}};\mathbb{Q})=\begin{cases}\Sigma^{2d}\widetilde{H}_{*}(M^{+};\mathbb{Q})&d\text{ even}\\ 0&d\text{ odd}.\end{cases}$$

This lets us write the complex as

(2.1)

$$\left(S^{*}\big{(}\Sigma^{d}\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}})[1]\oplus\begin{cases}\Sigma^{2d+1}\widetilde{H}_{*}(M^{+};\mathbb{Q})&d\text{ even}\\ 0&d\text{ odd}\end{cases}[2]\big{)},\partial\right),$$

where the differential is dual to the map $S^{2}(H^{*}_{c}(M;\mathbb{Q}^{w_{1}}))\to H^{*}_{c}(M;\mathbb{Q})$ induced by cup product, so following Knudsen we can recognise this complex as the Chevelley–Eilenberg complex for the bigraded Lie algebra $H^{*}_{c}(M;\mathrm{Lie}(\Sigma^{d-1}\mathbb{Q}^{w_{1}}[1]))$. Thus

$$H^{2nd-*}(C_{n}(M);\mathbb{Q})\cong\widetilde{H}_{*}(C_{n}(M;W)^{+};\mathbb{Q})\cong H^{*}_{\mathrm{Lie}}(H^{*}_{c}(M;\mathrm{Lie}(\Sigma^{d-1}\mathbb{Q}^{w_{1}}[1])))(n).$$

After appropriate dualisations and reindexings, this agrees with Knudsen’s formula.

Stability for the homology of configuration spaces is by now a classical subject, with a large number of contributions by many authors: notable examples are [Arn70, Seg79, Chu12, RW13, BM14, CP15, KM15, Knu17]. In particular Knudsen has explained [Knu17, Section 5.3] how his formula implies rational (co)homological stability for the spaces $C_{n}(M)$. Let us briefly review this from the point of view taken here.

There is a canonical element $[M]\in\widetilde{H}_{d}(M^{+};\mathbb{Q}^{w_{1}})\cong\widetilde{H}_{2d}(W^{+};\mathbb{Q})$, and choosing a cycle representing this element provides a map

$$\sigma:\Sigma^{2d}\mathbb{Q}[1]\longrightarrow{C}_{*,*}(\mathbf{Com}(W^{+}[1]);\mathbb{Q})\longrightarrow{C}_{*,*}(\mathbb{C}(M;W);\mathbb{Q}).$$

Multiplication by this element defines a map

$$(\sigma\cdot-)_{*}:\widetilde{H}_{n-1,2d(n-1)-i}(\mathbb{C}(M;W);\mathbb{Q})\longrightarrow\widetilde{H}_{n,2dn-i}(\mathbb{C}(M;W);\mathbb{Q})$$

which under Poincaré duality gives a map $H^{i}(C_{n-1}(M);\mathbb{Q})\to H^{i}(C_{n}(M);\mathbb{Q})$; this can be checked to be the transfer map which sums over all ways of forgetting one of the $n$ points, see [Knu17, Section 5.2] [Sta23b, Section 2.6].

Writing ${C}_{*,*}(\mathbb{C}(M;W);\mathbb{Q})/\sigma$ for the mapping cone of left multiplication by $\sigma$, the discussion above shows that its homology is calculated by a complex

$$\left(S^{*}\big{(}\Sigma^{d}\frac{\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}})}{\langle[M]\rangle}[1]\oplus\begin{cases}\Sigma^{2d+1}\widetilde{H}_{*}(M^{+};\mathbb{Q})&d\text{ even}\\ 0&d\text{ odd}\end{cases}[2]\big{)},\partial\right).$$

As $M$ is connected, if we assume that $d\geq 3$ then the bigraded vector spaces $\Sigma^{d}\frac{\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}})}{\langle[M]\rangle}[1]$ and $\Sigma^{2d+1}\widetilde{H}_{*}(M^{+};\mathbb{Q})[2]$ both vanish in bidegrees $(n,j)$ satisfying $j>(2d-1)n$, and hence so does the free graded-commutative algebra on them. This translates to $H^{i}(C_{n-1}(M);\mathbb{Q})\to H^{i}(C_{n}(M);\mathbb{Q})$ being surjective for $i<n$ and an isomorphism for $i<n-1$. For $d=2$ the same considerations give surjectivity for $i<\tfrac{1}{2}n$ and so on (a more careful analysis gives a slope 1 range in this case too, see [Knu17, Proof of Theorem 1.3]).

Analysing the complex (2.1) can also establish other kinds of stability results, e.g. [BY21, KMT23, Yam23].

Using Knudsen’s formula it is possible to mislead yourself into thinking that homeomorphisms of $M$ (or indeed pointed homotopy self-equivalences of $M^{+}$) act on $H_{*}(C_{n}(M);\mathbb{Q})$ via their action on $H_{*}(M;\mathbb{Q})$: in other words, that such maps which act trivially on the homology of $M$ also act trivially on the homology of $C_{n}(M)$. This is not true: in the case of surfaces see Bianchi [Bia20, Section 7], Looijenga [Loo23], and the complete analysis given by Stavrou [Sta23a].

From the point of view taken here this phenomenon can be explained as follows. For simplicity suppose that $M$ is orientable, and first suppose that it is odd-dimensional. Then $H^{*}(C_{n}(M);\mathbb{Q})\cong\widetilde{H}_{2dn-*}(C_{n}(M;M\times\mathbb{R}^{d})^{+};\mathbb{Q})$ and the analysis of Section 2.2 applied to $\mathbb{C}(M;M\times\mathbb{R}^{d})$ shows that $\mathbf{Com}(S^{d}\wedge M^{+}[1])\to\mathbb{C}(M;M\times\mathbb{R}^{d})$ is a rational homology isomorphism. So we find:

The even-dimensional case is more interesting. As $M$ is assumed orientable, in this case the twisting by $W$ can be dispensed with. It is technically convenient here—for reasons of symmetric monoidality—to work in the category of simplicial $\mathbb{Q}$-modules rather than chain complexes. We write $-\odot-$ for the tensoring of this category over simplicial sets. For a space $X$ let us abbreviate $\mathbb{Q}[X]:=\mathbb{Q}[\mathrm{Sing}_{\bullet}(X)]$, and if it is based then let $\widetilde{\mathbb{Q}}[X]=\mathbb{Q}[X]/\mathbb{Q}[*]$. The discussion in the previous section, ignoring the formality step and translated to simplicial $\mathbb{Q}$-modules, shows that given the simplicial module $\widetilde{\mathbb{Q}}[M^{+}]$ and the map $\delta:\widetilde{\mathbb{Q}}[M^{+}]\to\big{[}\widetilde{\mathbb{Q}}[M^{+}]^{\otimes 2}\big{]}_{\mathfrak{S}_{2}}$ induced by the diagonal $M^{+}\to M^{+}\wedge M^{+}$, we may form the two-sided bar construction

(2.2)

$$B(S^{*}(\widetilde{\mathbb{Q}}[M^{+}][1]),S^{*}(\widetilde{\mathbb{Q}}[M^{+}][2]),\mathbb{Q}[0])$$

whose bigraded homotopy groups are identified with $\widetilde{H}_{*}(C_{*}(M)^{+};\mathbb{Q})$.

A homeomorphism of $M$, or a pointed homotopy self-equivalence of $M^{+}$, induces an equivalence $\phi:\widetilde{\mathbb{Q}}[M^{+}]\to\widetilde{\mathbb{Q}}[M^{+}]$ such that $\delta\circ\phi=([\phi\otimes\phi]_{\mathfrak{S}_{2}})\circ\delta$, meaning that the diagram of simplicial commutative rings

is commutative, which induces a self-equivalence of the two-sided bar construction (2.2). This corresponds to the induced action on $\widetilde{H}_{*}(C_{*}(M)^{+};\mathbb{Q})$.