# Configuration spaces as commutative monoids

Oscar Randal-Williams<EMAIL_ADDRESS>Centre for
Mathematical Sciences
Wilberforce Road
Cambridge CB3 0WB
UK

###### Abstract.

After 1-point compactification, the collection of all unordered configuration
spaces of a manifold admits a commutative multiplication by superposition of
configurations. We explain a simple (derived) presentation for this
commutative monoid object. Using this presentation, one can quickly deduce
Knudsen’s formula for the rational cohomology of configuration spaces, prove
rational homological stability, and understand how automorphisms of the
manifold act on the cohomology of configuration spaces. Similar considerations
reproduce the work of Farb–Wolfson–Wood on homological densities.

##  1\. Introduction

Let $M$ be the interior of a connected compact manifold with boundary. The
1-point compactification of the space $C_{n}(M)$ of unordered configurations
in $M$ may be written as

$C_{n}(M)^{+}=\left[\frac{(M^{+})^{\wedge n}}{\text{locus where two points
coincide}}\right]_{\mathfrak{S}_{n}},$

the quotient formed in pointed spaces. Not-necessarily-disjoint union of
unordered configurations defines a _superposition product_

$C_{n}(M)^{+}\wedge C_{n^{\prime}}(M)^{+}\longrightarrow
C_{n+n^{\prime}}(M)^{+}$

which is associative, commutative, and unital. This gives a unital commutative
monoid object in the symmetric monoidal category
$\mathsf{Top}_{*}^{\mathbb{N}}$ of $\mathbb{N}$-graded pointed spaces:

$\mathbb{C}(M):n\longmapsto C_{n}(M)^{+}.$

The goal of this note is to explain and exploit this algebraic structure.

More generally, let $\pi:L\to M$ be a vector bundle, and let

$C_{n}(M;L)^{+}=\left[\frac{(L^{+})^{\wedge n}}{\text{locus where two points
have the same projection in $M$}}\right]_{\mathfrak{S}_{n}}.$

These assemble in the same way to a unital commutative monoid object
$\mathbb{C}(M;L)$. (Of course more general spaces of labels can be implemented
too, but the above suffices for us.) In the following we write $X[n]$ for the
$\mathbb{N}$-graded pointed space which consists of $X$ in grading $n$ and the
point in all other gradings, and $\mathbf{Com}(X)$ for the free commutative
monoid on $X\in\mathsf{Top}_{*}^{\mathbb{N}}$.

######  Theorem 1.1.

There is a pushout square

${\mathbf{Com}([(L\oplus
L)^{+}]_{\mathfrak{S}_{2}}[2])}$${S^{0}[0]}$${\mathbf{Com}(L^{+}[1])}$${\mathbb{C}(M;L)}$$\scriptstyle{\epsilon}$$\scriptstyle{\Delta}$

of unital commutative monoids in $\mathsf{Top}_{*}^{\mathbb{N}}$, where
$\epsilon$ is the augmentation and $\Delta$ is induced by the inclusion
$[(L\oplus L)^{+}]_{\mathfrak{S}_{2}}\to[L^{+}\wedge
L^{+}]_{\mathfrak{S}_{2}}=\mathbf{Com}(L^{+}[1])(2)$. Furthermore, this square
is a homotopy pushout, i.e. there is an induced equivalence

$\mathbf{Com}(L^{+}[1])\otimes^{\mathbb{L}}_{\mathbf{Com}([(L\oplus
L)^{+}]_{\mathfrak{S}_{2}}[2])}S^{0}[0]\overset{\sim}{\longrightarrow}\mathbb{C}(M;L).$

Recall that the derived relative tensor product may be computed by the two-
sided bar construction, formed in $\mathsf{Top}_{*}^{\mathbb{N}}$, so the
conclusion can equivalently be stated as an equivalence

(1.1) $B(\mathbf{Com}(L^{+}[1]),\mathbf{Com}([(L\oplus
L)^{+}]_{\mathfrak{S}_{2}}[2]),S^{0}[0])\overset{\sim}{\longrightarrow}\mathbb{C}(M;L).$

As one application of this result we will show how to recover Knudsen’s
[Knu17] formula for $H^{*}(C_{n}(M);\mathbb{Q})$ in terms of the compactly-
supported $\mathbb{Q}$-cohomology of $M$ and its cup-product map, which in
particular quickly implies homological stability. As another application we
will show that the action on $H^{*}(C_{n}(M);\mathbb{Q})$ of the group of
proper homotopy self-equivalences of $M$ factors over a surprisingly small
group. Finally, in an appendix written with Quoc P. Ho, we show how similar
considerations reproduces the work of Farb–Wolfson–Wood on homological
densities.

This note was inspired by a speculative remark of Banerjee [Ban21, Remark
1.2], and is my attempt to make sense of it in a specific case. It also has
much to do with the work of Ho [Ho21, Ho20], Petersen [Pet20], Knudsen
[Knu17], Getzler [Get99a, Get99b], Bödigheimer–Cohen–Milgram [BCM93], and
Segal [Seg79].

##  2\. Applications

###  2.1. Homology of configuration spaces

The space $C_{n}(M;L)^{+}$ is the 1-point compactification of the
$n\cdot(\dim(M)+\dim(L))$-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 $(\dim(M)-1)$ trivial line bundles. Thus if $M$ is
$d$-dimensional then 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 (1.1) 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.

###  2.2. Revisiting Knudsen’s formula

For an $\mathbb{N}$-graded pointed space we write
$H_{n,d}(X)=\widetilde{H}_{d}(X(n))$, and similarly for chains. Write
$S^{*}(V)$ for the free graded-commutative algebra on a graded vector space
$V$. If $V$ is equipped with additional $\mathbb{N}$-grading, 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. With such choices the square

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

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, so are
chain homotopic. The bar construction description then gives an identification

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

The Koszul complex for calculating these Tor groups takes the form

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

with differential given by $\partial(\Sigma x)=\Delta_{*}(x)\in
S^{2}(\widetilde{H}_{*}(W^{+}[1];\mathbb{Q}))$ when
$x\in\widetilde{H}_{*}([(W\oplus W)^{+}]/\mathfrak{S}_{2}[2];\mathbb{Q})$, 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 as 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.

###  2.3. Homological stability

Knudsen has explained [Knu17, Section 5.3] how his formula implies
(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
$[M]\in\widetilde{H}_{d}(M^{+};\mathbb{Q}^{w_{1}})$, a cycle representing
which defines a map

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

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 $(n,2dn-i)$ with $2dn-i>(2d-1)n$, 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. These ranges can be
improved by closer inspection of the structure of (2.1).

###  2.4. The action of automorphisms on unordered configurations

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:

######  Theorem 2.1.

If $M$ is orientable and odd-dimensional, then a pointed homotopy self-
equivalence of $M^{+}$ which acts trivially on
$\widetilde{H}_{*}(M^{+};\mathbb{Q})$ also acts trivially on
$H^{*}(C_{n}(M);\mathbb{Q})$.∎

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. 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 for the two-sided bar construction

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

whose homotopy groups are identified with the
$\widetilde{H}_{*}(C_{n}(M)^{+})$.

The bar construction may be acted upon by the group-like simplicial monoid of
homotopy automorphisms of the data

$\mathcal{Q}:=(\widetilde{\mathbb{Q}}[M^{+}],\delta:\widetilde{\mathbb{Q}}[M^{+}]\to\big{[}\widetilde{\mathbb{Q}}[M^{+}]^{\otimes
2}\big{]}_{\mathfrak{S}_{2}}).$

This simplicial monoid has $p$-simplices given by—using the tensoring $\odot$
of simplicial modules over simplicial sets—pairs of an equivalence
$\phi:\Delta^{p}\odot\widetilde{\mathbb{Q}}[M^{+}]\to\widetilde{\mathbb{Q}}[M^{+}]$
and a homotopy

${\Delta^{p}\odot\widetilde{\mathbb{Q}}[M^{+}]}$${\Delta^{p}\odot{\big{[}\widetilde{\mathbb{Q}}[M^{+}]^{\otimes
2}\big{]}_{\mathfrak{S}_{2}}}}$${{\big{[}(\Delta^{p}\odot\widetilde{\mathbb{Q}}[M^{+}])^{\otimes
2}\big{]}_{\mathfrak{S}_{2}}}}$${\widetilde{\mathbb{Q}}[M^{+}]}$${{\big{[}\widetilde{\mathbb{Q}}[M^{+}]^{\otimes
2}\big{]}_{\mathfrak{S}_{2}}};}$$\scriptstyle{\phi}$$\scriptstyle{\Delta^{p}\odot\delta}$$\scriptstyle{diag_{\Delta^{p}}}$$\scriptstyle{[\phi\otimes\phi]_{\mathfrak{S}_{2}}}$$\scriptstyle{\delta}$$\scriptstyle{h}$

the simplicial structure is given by the evident functoriality in
$\Delta^{p}$. This is neither the simplicial monoid of those automorphisms
which preserve $\delta$ strictly, nor is it the simplicial monoid of those
automorphisms which preserve $\delta$ up to homotopy: rather, a choice of
homotopy is built in. This has the effect that the forgetful map

$\mathrm{Aut}(\mathcal{Q})\longrightarrow\mathrm{Aut}(\widetilde{\mathbb{Q}}[M^{+}])$

is easily seen to be a Kan fibration, with fibre over the identity given by
the group-like simplicial monoid of self-homotopies of the map $\delta$. On
homotopy groups this gives an exact sequence

${\pi_{1}(\mathrm{Hom}(\widetilde{\mathbb{Q}}[M^{+}],\widetilde{\mathbb{Q}}[M^{+}]);id)}$${\pi_{1}(\mathrm{Hom}(\widetilde{\mathbb{Q}}[M^{+}],\big{[}\widetilde{\mathbb{Q}}[M^{+}]^{\otimes
2}\big{]}_{\mathfrak{S}_{2}});\delta)}$${\pi_{0}\mathrm{Aut}(\mathcal{Q})}$${\mathrm{Aut}(\widetilde{H}_{*}(M^{+}),\Delta_{*})}$${1}$$\scriptstyle{\delta\circ-}$

and so an extension

$1\to\mathrm{Hom}(\Sigma\widetilde{H}_{*}(M^{+}),S^{2}(\widetilde{H}_{*}(M^{+}))/\Delta_{*})\to\pi_{0}\mathrm{Aut}(\mathcal{Q})\to\mathrm{Aut}(\widetilde{H}_{*}(M^{+}),\Delta_{*})\to
1.$

The simplicial monoid of proper homotopy self-equivalences of $M$ acts via
$\mathrm{Aut}(\mathcal{Q})$, by letting $\phi$ be the induced map on
($\mathbb{Q}$-linearised) singular simplices, and $h$ be the trivial homotopy,
because such $\phi$ literally commute with the diagonal map $\delta$.

This implies the following. We continue to assume that $M$ is even-dimensional
and orientable. Let $G$ denote the group of homotopy classes of pointed
homotopy self-equivalences of $M^{+}$ which act as the identity on
$\widetilde{H}_{*}(M^{+};\mathbb{Q})$.

######  Theorem 2.2.

If $M$ is orientable and even-dimensional, then $G$ acts on
${H}^{*}(C_{n}(M);\mathbb{Q})$ via
$\mathrm{Hom}(\Sigma\widetilde{H}_{*}(M^{+};\mathbb{Q}),S^{2}(\widetilde{H}_{*}(M^{+};\mathbb{Q}))/\Delta_{*})$.∎

######  Example 2.3.

When $M$ is a punctured surface one has $\widetilde{H}_{*}(M^{+})=\Sigma
H_{1}(M)\oplus\Sigma^{2}\mathbb{Q}$ so the map
$\Delta_{*}:\widetilde{H}_{*}(M^{+})\to S^{2}(\widetilde{H}_{*}(M^{+}))$ has
the form

$\Sigma
H_{1}(M)\oplus\Sigma^{2}\mathbb{Q}\longrightarrow\Sigma^{2}\Lambda^{2}(H_{1}(M))\oplus\Sigma^{3}H_{1}(M)\oplus\Sigma^{4}\mathbb{Q},$

which in grading 2 is the inclusion of the symplectic form
$\omega\in\Lambda^{2}(H_{1}(M))$ and is zero otherwise. Thus the above is
$\mathrm{Hom}(H_{1}(M),\Lambda^{2}(H_{1}(M))/\langle\omega\rangle)\oplus
H_{1}(M)$. Using Poincaré duality and
$\Lambda^{2}(H_{1}(M))\cong\mathbb{Q}\\{\omega\\}\oplus\Lambda^{2}(H_{1}(M))/\langle\omega\rangle$,
this can be identified with $\mathrm{Hom}(H_{1}(M),\Lambda^{2}(H_{1}(M)))$.
This is the target of the Johnson homomorphism, cf. [Sta23a].

######  Remark 2.4.

The results of this section should also follow from [Sta23a, Theorem 1.2] and
some rational homotopy theory.

##  3\. Proofs

Recall that $X\in\mathsf{Top}_{*}$ is _well-based_ if the basepoint map
$i:*\to X$ is a closed cofibration: under this condition $X\wedge-$ preserves
weak equivalences between well-based spaces, and preserves closed
cofibrations. Let us say that an $\mathbb{N}$-graded based space $Y$ is well-
based if $Y(n)$ is well-based for each $n\in\mathbb{N}$.

Let us write $\mathbb{R}:=\mathbf{Com}(L^{+}[1])$ and
$\mathbb{S}:=\mathbf{Com}([(L\oplus L)^{+}]_{\mathfrak{S}_{2}}[2])$ to ease
notation, so $\Delta:\mathbb{S}\to\mathbb{R}$ makes $\mathbb{R}$ into a
$\mathbb{S}$-module.

######  Lemma 3.1.

$\mathbb{S}$ and $\mathbb{R}$ are well-based. The subspace of
$[(L^{+})^{\wedge p}]_{\mathfrak{S}_{p}}$ of those tuples which do not have
distinct $M$ coordinates is well-based, and this inclusion is a closed
cofibration.

###### Proof.

Recall that $M$ is the interior of a compact manifold with boundary
$\overline{M}$. This admits a collar, showing that $i:M\to\overline{M}$ admits
a homotopy inverse, and so the vector bundle $L\to M$ extends to a vector
bundle over $\overline{M}$, which we also call $L$. Furthermore, choosing an
inner product on this bundle we can form the closed disc bundle
$D(L)\to\overline{M}$, and consider $L$ as lying inside it as the open disc
bundle. Now $D(L)$ is a manifold with boundary $\partial D(L)=S(L)\cup
D(L)|_{\partial\overline{M}}$, and $L^{+}=D(L)/\partial D(L)$.

Observe that $(\overline{M},\partial\overline{M})$ is an compact manifold pair
so (is an ENR pair and hence) can be expressed as a retract of a pair
$(|X_{\bullet}|,|\partial X_{\bullet}|)$ of the geometric realisations of a
simplicial set and a subset. We may pull $L$ back to $|X_{\bullet}|$ using the
retraction; let us call this $L_{X}$. Now $D(L_{X})/S(L_{X})\cup
D(L_{X})|_{|\partial X_{\bullet}|}$ can be given an evident cell-structure (by
induction over the relative cells of $|\partial
X_{\bullet}|\to|X_{\bullet}|$), and $L^{+}=D(L)/\partial D(L)$ is a retract of
it, so is well-based. More generally, for the exterior direct sum
$L_{X}^{\boxplus p}\to|X_{\bullet}^{p}|$ and writing
$\partial|X_{\bullet}^{p}|$ for the subcomplex where some factor lies in
$\partial X_{\bullet}$, there is a cell structure on $D(L_{X}^{\boxplus
p})/S(L_{X}^{\boxplus p})\cup D(L_{X}^{\boxplus
p})|_{\partial|X_{\bullet}^{p}|}$ for which the group $\mathfrak{S}_{p}$ acts
cellularly, and so $[D(L_{X}^{\boxplus p})/S(L_{X}^{\boxplus p})\cup
D(L_{X}^{\boxplus p})|_{\partial|X_{\bullet}^{p}|}]_{\mathfrak{S}_{p}}$ is a
cell complex of which $[(L^{+})^{\wedge p}]_{\mathfrak{S}_{p}}$ is a retract,
and so is well-based. This shows that $\mathbb{R}$ is well-based, and similar
reasoning shows $\mathbb{S}$ is.

For the second statement,

$inc:F:=\text{fat diagonal of }|X_{\bullet}|^{p}=|\text{fat diagonal of
}X_{\bullet}^{p}|\longrightarrow|X_{\bullet}^{p}|=|X_{\bullet}|^{p}$

is the inclusion of a $\mathfrak{S}_{p}$-CW-subcomplex, and so has a
$\mathfrak{S}_{p}$-equivariant open neighbourhood $U$ which equivariantly
deformation retracts to it. This may be chosen to preserve the subcomplexes
where some factor lies in $|\partial X_{\bullet}|$. Thus it lifts to a
$\mathfrak{S}_{p}$-equivariant deformation retraction of an open neighbourhood
of $L_{X}^{\boxplus p}|_{F}\to L_{X}^{\boxplus p}$, and descends to the
quotient by the subcomplexes where some factor lies in $|\partial
X_{\bullet}|$. As it is equivariant, it descends further to the
$\mathfrak{S}_{p}$-quotient. That is, it proves the claim for
$(\overline{M},\partial\overline{M},L)$ replaced by $(|X_{\bullet}|,|\partial
X_{\bullet}|,L_{X})$; as the former data is a retract of the latter, the claim
follows. ∎

######  Lemma 3.2.

$\mathbb{R}$ is a flat $\mathbb{S}$-module, in the sense that
$\mathbb{R}\otimes_{\mathbb{S}}-$ preserves weak equivalences between left
$\mathbb{S}$-modules whose underlying objects are well-based.

###### Proof.

Recall that $\mathbb{R}(n)=[(L^{+})^{\wedge n}]_{\mathfrak{S}_{n}}$. Define a
filtration of $\mathbb{R}$ by $F_{0}\mathbb{R}=\mathbb{S}$ and

$F_{p}\mathbb{R}(n):=F_{p-1}\mathbb{R}(n)\cup\mathrm{Im}\big{(}(L^{+})^{\wedge
p}\wedge((L\oplus L)^{+})^{\wedge(n-p)/2}\to\mathbb{R}(n)\big{)},$

where the latter term is only taken when it makes sense: for $n-p$ even. This
is a filtration by right $\mathbb{S}$-modules. One checks that the diagram

${F_{p-2}\mathbb{R}(p)[p]\otimes\mathbb{S}}$${F_{p-1}\mathbb{R}}$${\mathbb{R}(p)[p]\otimes\mathbb{S}}$${F_{p}\mathbb{R}}$

is a pushout (in $\mathsf{Top}_{*}^{\mathbb{N}}$ and so in right
$\mathbb{S}$-modules), where the horizontal maps are induced by the
$\mathbb{S}$-module structure and the adjoints of the map
$inc:F_{p-2}\mathbb{R}(p)\to F_{p-1}\mathbb{R}(p)$, and the map
$id:\mathbb{R}(p)\to F_{p}\mathbb{R}(p)$.

We prove by induction on $p$ that $F_{p}\mathbb{R}$ is a flat
$\mathbb{S}$-module in the indicated sense. As $F_{0}\mathbb{R}=\mathbb{S}$
these properties hold for $p=0$. For $\mathbb{M}$ a left $\mathbb{S}$-module
whose underlying object is well-based, applying
$-\otimes_{\mathbb{S}}\mathbb{M}$ to the square above gives a pushout square

(3.1)
${F_{p-2}\mathbb{R}(p)[p]\otimes\mathbb{M}}$${F_{p-1}\mathbb{R}\otimes_{\mathbb{S}}\mathbb{M}}$${\mathbb{R}(p)[p]\otimes\mathbb{M}}$${F_{p}\mathbb{R}\otimes_{\mathbb{S}}\mathbb{M}.}$

The map $F_{p-2}\mathbb{R}(p)\to\mathbb{R}(p)$ is the inclusion of the
subspace of those $p$-tuples of points in $M$ labelled by $L$ which do not
have distinct $M$ coordinates, so is a closed cofibration from a well-based
space by the second part of Lemma 3.1. As $\mathbb{M}$ is assumed well-based,
the left-hand vertical map in (3.1) is a closed cofibration in each grading,
and so this square is also a homotopy pushout. A weak equivalence
$f:\mathbb{M}\overset{\sim}{\to}\mathbb{M}^{\prime}$ then induces a map of
homotopy pushout squares which is a weak equivalence on all but the bottom
right corner, by inductive assumption, so also induces a weak equivalence on
this corner.

Thus each $F_{p}\mathbb{R}$ is flat in the indicated sense, so $\mathbb{R}$ is
too because $F_{p}\mathbb{R}\to\mathbb{R}$ is an isomorphism when evaluated on
$n<p$, so
$F_{p}\mathbb{R}\otimes_{\mathbb{S}}\mathbb{M}\to\mathbb{R}\otimes_{\mathbb{S}}\mathbb{M}$
is too. ∎

######  Lemma 3.3.

The induced map $\mathbb{R}\otimes_{\mathbb{S}}S^{0}[0]\to\mathbb{C}(M;L)$ is
an isomorphism.

###### Proof.

By definition of the relative tensor product there is a coequaliser diagram

${\mathbb{R}\otimes\mathbb{S}}$${\mathbb{R}}$${\mathbb{R}\otimes_{\mathbb{S}}S^{0}[0]}$$\scriptstyle{\alpha}$$\scriptstyle{\beta}$

in $\mathsf{Top}_{*}^{\mathbb{N}}$, where $\alpha$ is given by the
$\mathbb{S}$-module structure on $\mathbb{R}$, and $\beta$ is induced by the
augmentation $\epsilon:\mathbb{S}\to S^{0}[0]$. The image of
$\mathbb{R}\otimes\mathrm{ker}(\epsilon)(n)\to\mathbb{R}(n)=[(L^{+})^{\wedge
n}]_{\mathfrak{S}_{n}}$ is precisely the image of $(L^{+})^{\wedge
n-2}\wedge(L\oplus L)^{+}\to[(L^{+})^{\wedge n}]_{\mathfrak{S}_{n}}$, whose
cofibre is by definition $\mathbb{C}(M;L)$. ∎

###### Proof of Theorem 1.1.

Apply Lemma 3.2 to the weak equivalence
$B(\mathbb{S},\mathbb{S},S^{0}[0])\overset{\sim}{\to}S^{0}[0]$, giving an
equivalence
$B(\mathbb{R},\mathbb{S},S^{0}[0])\overset{\sim}{\to}\mathbb{R}\otimes_{\mathbb{S}}S^{0}[0]$,
and the latter is isomorphic to $\mathbb{C}(M)$ by Lemma 3.3. ∎

######  Remark 3.4.

It is possible to fool oneself into thinking that the above argument can be
adapted to the case of ordered configuration spaces. It cannot, because the
statement (in the guise of the equivalence (1.1), for example) is false in
this case. One can verify this directly in the case $M=*$ with trivial
0-dimensional Euclidean bundle, in grading 3.

##  Appendix A Homological densities
by Quoc P. Ho and Oscar Randal-Williams

###  A.1. Spaces of 0-cycles

It is easy to generalise Theorem 1.1 to the following variant of configuration
spaces, called “spaces of 0-cycles” by Farb–Wolfson–Wood [FWW19]. Let $m,k\geq
1$, and for $n_{1},n_{2},\ldots,n_{m}\in\mathbb{N}$ let

$Z^{k}_{n_{1},\ldots,n_{m}}(M)\subset\mathrm{Sym}_{n_{1},\ldots,n_{m}}(M):=[M^{n_{1}}]_{\mathfrak{S}_{n_{1}}}\times[M^{n_{2}}]_{\mathfrak{S}_{n_{2}}}\times\cdots\times[M^{n_{m}}]_{\mathfrak{S}_{n_{m}}}$

be the open subspace of those
$(\\{x_{1}^{1},\ldots,x_{n_{1}}^{1}\\},\\{x_{1}^{2},\ldots,x_{n_{2}}^{2}\\},\ldots,\\{x_{1}^{m},\ldots,x_{n_{m}}^{m}\\})$
such that no $x^{i}_{j}$ has multiplicity $\geq k$ in all of these $m$
multisets. That is, $Z^{k}_{n_{1},\ldots,n_{m}}(M)$ is the configuration space
of particles of $m$ different colours, $n_{i}$ having colour $i$, which may
collide except that no point of $M$ may carry $\geq k$ points of every colour.
The 1-point compactifications $Z_{n_{1},\ldots,n_{m}}(M)^{+}$ again have a
composition product

$Z^{k}_{n_{1},\ldots,n_{m}}(M)^{+}\wedge
Z^{k}_{n^{\prime}_{1},\ldots,n^{\prime}_{m}}(M)^{+}\longrightarrow
Z^{k}_{n_{1}+n^{\prime}_{1},\ldots,n_{m}+n^{\prime}_{m}}(M)^{+},$

giving a commutative monoid $\mathbb{Z}^{m,k}(M)$ in $\mathbb{N}^{m}$-graded
pointed spaces. Just as before, we can introduce labels in a vector bundle
$L\to M$, giving $Z^{k}_{n_{1},\ldots,n_{m}}(M;L)$ and
$\mathbb{Z}^{m,k}(M;L)$. Writing
$1_{i}=(0,\ldots,0,1,0\ldots,0)\in\mathbb{N}^{m}$ with the 1 in the $i$th
position, there is a pushout square

(A.1) ${\mathbf{Com}([(L^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}}[k,k,\ldots,k])}$${S^{0}[0,\ldots,0]}$${{\displaystyle\mathbf{Com}(\bigvee_{i=1}^{m}L^{+}[1_{i}])}}$${\mathbb{Z}^{m,k}(M;L)}$$\scriptstyle{\epsilon}$$\scriptstyle{\Delta}$

of unital commutative monoids in $\mathsf{Top}_{*}^{\mathbb{N}^{m}}$, where
$\Delta$ is now induced by the inclusion $[(L^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}}\to[(L^{+})^{\wedge
k}]_{\mathfrak{S}_{k}}\wedge\cdots\wedge[(L^{+})^{\wedge
k}]_{\mathfrak{S}_{k}}=\mathbf{Com}(\bigvee_{i=1}^{m}L^{+}[1_{i}])(k,\ldots,k)$.
The same argument as Theorem 1.1 shows that there is an equivalence

(A.2)
$\mathbf{Com}(\vee_{i=1}^{m}L^{+}[1_{i}])\otimes^{\mathbb{L}}_{\mathbf{Com}([(L^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}}[k,\ldots,k])}S^{0}[0,\ldots,0]\overset{\sim}{\longrightarrow}\mathbb{Z}^{m,k}(M;L).$

###  A.2. Revisiting homological densities

This can be used to revisit the work of Farb–Wolfson–Wood [FWW19] and Ho
[Ho21] on homological densities, and in particular to explain coincidences of
homological densities at the level of topology rather than algebra, as
proposed in [Ho21, 1.5.1].

The spaces $Z^{k}_{n_{1},\ldots,n_{m}}(M;L)$ are $\mathbb{Q}$-homology
manifolds, being open subspaces of a product of coarse moduli spaces
$[L^{n}]_{\mathfrak{S}_{n}}$ of the orbifolds $L^{n}/\\!\\!/\mathfrak{S}_{n}$.
As before, we suppose $M$ is $d$-dimensional and take $L=W$ to be given by the
sum of the orientation line of $M$ plus $(d-1)$ trivial lines: then the
$Z^{k}_{n_{1},\ldots,n_{m}}(M;W)$ are orientable $\mathbb{Q}$-homology
manifolds, of dimension $2d\cdot\sum n_{i}$. Again they are vector bundles
over $Z^{k}_{n_{1},\ldots,n_{m}}(M)$, so Poincaré duality gives

$H^{*}(Z^{k}_{n_{1},\ldots,n_{m}}(M))\cong
H^{*}(Z^{k}_{n_{1},\ldots,n_{m}}(M;W))\cong\widetilde{H}_{2d\sum
n_{i}-*}(Z^{k}_{n_{1},\ldots,n_{m}}(M;W)^{+}).$

On the other hand, the bar construction formula above together with the
argument of Section 2.2 identifies the multigraded vector space
$H_{*,*}(\mathbb{Z}^{m,k}(M;W))$ with

$\mathrm{Tor}_{*}^{S^{*}(\widetilde{H}_{*}([(W^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}})[k,\ldots,k])}(S^{*}(\bigoplus_{i=1}^{m}\widetilde{H}_{*}(W^{+})[1_{i}]),\mathbb{Q}[0,\ldots,0]).$

####  A.2.1. Odd-dimensional manifolds

As in Section 2.2 we have $\widetilde{H}_{*}([(W^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}})\cong[\Sigma^{dmk}\widetilde{H}_{*}(M^{+})]_{\mathfrak{S}_{k}^{m}}$
by the Thom isomorphism. If $d$ is odd then the permutation group
$\mathfrak{S}_{k}^{m}$ acts on the Thom class via
$\mathfrak{S}_{k}^{m}\leq\mathfrak{S}_{mk}\overset{sign}{\to}\mathbb{Z}^{\times}$,
so acts nontrivially if $k\geq 2$ and trivially if $k=1$. If $k\geq 2$ this
means that $\widetilde{H}_{*}([(W^{\oplus
mk})^{+}]_{\mathfrak{S}_{k}^{m}})=0$, showing that

$H_{*,*}(\mathbf{Com}(\vee_{i=1}^{m}W^{+}[1_{i}]))\overset{\sim}{\longrightarrow}H_{*,*}(\mathbb{Z}^{m,k}(M;W))$

in this case. Using Poincaré duality on both sides gives [FWW19, Theorem 1.4],
except that that theorem is erroneously claimed for all $k\geq 1$. We will
return to the case $k=1$ below.

####  A.2.2. Even-dimensional manifolds

If $d$ is even then $\mathfrak{S}_{k}^{m}$ acts trivially on
$\Sigma^{dmk}\widetilde{H}_{*}(M^{+})$, and using the Thom isomorphism to
identify $\widetilde{H}_{*}(W^{+})\cong\Sigma^{d}\widetilde{H}_{*}(M^{+})$
too, the Koszul complex for computing the $\mathrm{Tor}$-groups above is

$(S^{*}\big{(}\bigoplus_{i=1}^{m}\Sigma^{d}\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}})[1_{i}]\oplus\Sigma^{dmk+1}\widetilde{H}_{*}(M^{+};(\mathbb{Q}^{w_{1}})^{\otimes
mk})[k,\ldots,k]\big{)},\partial).$

The differential $\partial$ is induced by the map

$\Sigma^{dmk}\widetilde{H}_{*}(M^{+};(\mathbb{Q}^{w_{1}})^{\otimes mk})\to
S^{k}(\Sigma^{d}\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}}))\otimes\cdots\otimes
S^{k}(\Sigma^{d}\widetilde{H}_{*}(M^{+};\mathbb{Q}^{w_{1}}))$

obtained by linearly dualising the cup product map

(A.3) $H^{*}_{c}(M;\mathbb{Q}^{w_{1}})^{\otimes mk}\longrightarrow
H^{*}_{c}(M;(\mathbb{Q}^{w_{1}})^{\otimes mk}),$

and so is trivial if (and only if) all $mk$-fold cup products of
($w_{1}$-twisted) compactly-supported cohomology classes on $M$ vanish.

When this cup product map is trivial, so $\partial$ is trivial, the above just
gives a formula for $H_{*,*}(\mathbb{Z}^{m,k}(M;W))$. Using Poincaré duality,
and reindexing, to express this in terms of
$H^{*}(Z^{k}_{n_{1},\ldots,n_{m}}(M))$ and
$H^{*}(\mathrm{Sym}_{n_{1},\ldots,n_{m}}(M))$ we obtain an identity of
multigraded vector spaces

$H^{*}(Z^{k}_{\bullet}(M))\cong H^{*}(\mathrm{Sym}_{\bullet}(M))\otimes
S^{*}(\Sigma^{d(mk-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes
mk-1})[k,\ldots,k]).$

There are stabilisation maps
$\sigma_{i}:H^{*}(Z^{k}_{n_{1},\ldots,n_{m}}(M))\to
H^{*}(Z^{k}_{n_{1},\ldots,n_{i}+1,\ldots,n_{m}}(M))$ analogous to those
constructed in Section 2.3, similarly for
$H^{*}(\mathrm{Sym}_{n_{1},\ldots,n_{m}}(M))$, and both stabilise as
$n_{j}\to\infty$, just as in Section 2.3: this recovers [FWW19, Theorem 1.7].
We may take the colimit of all these stabilisations to obtain

$H^{*}(Z^{k}_{\infty,\ldots,\infty}(M))\cong
H^{*}(\mathrm{Sym}_{\infty,\ldots,\infty}(M))\otimes
S^{*}(\Sigma^{d(mk-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes mk-1})).$

Writing $P_{Z^{m,k}}(t)$ and $P_{Sym^{m}}(t)$ for the Poincaré series of
$H^{*}(Z^{k}_{\infty,\ldots,\infty}(M))$ and
$H^{*}(\mathrm{Sym}_{\infty,\ldots,\infty}(M))$ respectively, this discussion
identifies the _homological density_ ${P_{Z^{m,k}}(t)}/{P_{Sym^{m}}(t)}$ with
the Poincaré series of
$S^{*}(\Sigma^{d(mk-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes mk-1}))$.
This visibly only depends on the product $mk$, giving “coincidences between
homological densities”: this recovers [FWW19, Theorem 1.2]; in fact it also
recovers the stronger Theorem 3.6 of that paper.

####  A.2.3. Odd-dimensional manifolds, $k=1$

Just as in the even-dimensional case, if the cup product map (A.3) is zero
then one gets an explicit description of $H^{*}(Z_{\bullet}(M))$, and the
homological density is given by the Poincaré series of the graded vector space
$S^{*}(\Sigma^{d(m-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes m-1}))$. It
follows from Section A.2.1 that the homological density is $1$ for $k>1$, so
for odd-dimensional manifolds it is _not true_ that the homological density
depends only on $mk$.

####  A.2.4. Euler characteristic

If the cup product map (A.3) is not zero, and either $d$ is even or $d$ is odd
and $k=1$, then there is instead a nontrivial differential on the multigraded
vector space

$H^{*}(\mathrm{Sym}_{\bullet}(M))\otimes
S^{*}(\Sigma^{d(mk-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes
mk-1})[k,\ldots,k]),$

of degree $(+1,0)$, whose homology is $H^{*}(Z_{*,\ldots,*}(M))$. Then one
would not expect $\frac{P_{Z}(t)}{P_{Sym}(t)}$ to agree with the Poincaré
series of $S^{*}(\Sigma^{d(mk-1)-1}{H}^{*}(M;(\mathbb{Q}^{w_{1}})^{\otimes
mk-1}))$, and indeed it does not [FWW19, Remark 1.6]. However, as Euler
characteristic commutes with taking homology we have the identity

$\sum_{n_{1},\ldots,n_{m}\geq
0}\chi(Z_{n_{1},\ldots,n_{m}}(M))s_{1}^{n_{1}}\cdots
s_{m}^{n_{m}}=\big{(}\prod_{i=1}^{m}(1-s_{i})\big{)}^{-\chi(M)}\cdot(1-(s_{1}\cdots
s_{m})^{k})^{\chi(M,(\mathbb{Q}^{w_{1}})^{\otimes mk-1})}$

in $\mathbb{Z}[[s_{1},\ldots,s_{m}]]$. The left-hand factor is
$\sum_{n_{1},\ldots,n_{m}\geq
0}\chi(\mathrm{Sym}_{n_{1},\ldots,n_{m}}(M))s_{1}^{n_{1}}\cdots
s_{m}^{n_{m}}$. This recovers [FWW19, Theorem 1.9 1.].

###  A.3. Spectral densities

The construction of homological densities can be promoted to the level of
spectra, addressing [Ho21, 1.5.1], as follows. Let us _assume that $M$ is
even-dimensional and orientable_: then we can dispense with twisting by the
vector bundle $W\to M$. We consider $\mathbb{Z}^{m,k}(M)$ with its
$\mathbb{N}^{m}$-grading reduced to an $\mathbb{N}$-grading via
$\text{sum}:\mathbb{N}^{m}\to\mathbb{N}$. Collapsing the complement of a small
neighbourhood of a point in $M$ gives a map $M^{+}\to S^{d}$, inducing a map
of commutative monoids

$\mathbf{Com}(\bigvee_{i=1}^{m}M^{+}[1])\longrightarrow\mathbf{Com}(S^{d}[1]).$

If $X$ is a left $\mathbf{Com}(S^{d}[1])$-module, it is equipped with maps
$S^{d}\wedge X(n)\to X(n+1)$ and so we can define the spectrum
$\overline{X}:=\operatorname*{hocolim}_{n\to\infty}S^{-nd}\wedge\Sigma^{\infty}X(n)$.
Using these two constructions we may therefore form the spectrum

$\Delta^{m,k}:=\overline{\mathbf{Com}(S^{d}[1])\otimes^{\mathbb{L}}_{\mathbf{Com}(\bigvee_{i=1}^{m}M^{+}[1])}\mathbb{Z}^{m,k}(M)}.$

By analogy with [Ho21, Section 7.5] we propose $\Delta^{m,k}$ as a spectral
form of the stable density of $Z^{k}_{n_{1},\ldots,n_{m}}(M)$ in
$\mathrm{Sym}_{n_{1},\ldots,n_{m}}(M)$. At the level of $\mathbb{Q}$-chains it
recovers the construction from the proof of Theorem 7.5.1 of [Ho21]. We can
prove the spectral form of that theorem analogously: as $\mathbb{N}$-graded
objects, there is an evident map from (A.1) to the analogous square for
$\mathbb{Z}^{1,mk}(M)$ which induces a map of spectra
$\Delta^{m,k}\to\Delta^{1,mk}$, and this is an equivalence by (A.2) as both
are identified with
$\overline{\mathbf{Com}(S^{d}[1])\otimes^{\mathbb{L}}_{\mathbf{Com}(M^{+}[mk])}S^{0}[0]}$.

This may be simplified for $mk\geq 2$ as follows. The map
$M^{+}\to[(S^{d})^{\wedge mk}]_{\mathfrak{S}_{mk}}$ with which the derived
tensor product is formed factors over $(S^{d})^{\wedge mk}$ so is
nullhomotopic when $mk\geq 2$, and so
$\mathbf{Com}(S^{d}[1])\otimes^{\mathbb{L}}_{\mathbf{Com}(M^{+}[mk])}S^{0}[0]$
is equivalent to
$\mathbf{Com}(S^{d}[1])\otimes(S^{0}[0]\otimes^{\mathbb{L}}_{\mathbf{Com}(M^{+}[mk])}S^{0}[0])$
as a left $\mathbf{Com}(S^{d}[1])$-module. In this situation the
$\overline{(-)}$ construction gives

$\displaystyle\Delta^{m,k}$ $\displaystyle\simeq\bigvee_{n\geq
0}S^{-nd}\wedge\Sigma^{\infty}(S^{0}[0]\otimes^{\mathbb{L}}_{\mathbf{Com}(M^{+}[mk])}S^{0}[0])(n)$
$\displaystyle\simeq\bigvee_{n\geq
0}S^{-nd}\wedge\Sigma^{\infty}\mathbf{Com}(S^{1}\wedge M^{+}[mk])(n).$

Acknowledgements. ORW supported by the ERC under the European Union’s Horizon
2020 research and innovation programme (grant agreement No. 756444).

## References

  * [Ban21] O. Banerjee, _Filtration of cohomology via symmetric semisimplicial spaces_ , https://arxiv.org/abs/1909.00458, 2021.
  * [BCM93] C.-F. Bödigheimer, F. R. Cohen, and R. J. Milgram, _Truncated symmetric products and configuration spaces_ , Math. Z. 214 (1993), no. 2, 179–216.
  * [Bia20] A. Bianchi, _Splitting of the homology of the punctured mapping class group_ , J. Topol. 13 (2020), no. 3, 1230–1260.
  * [FWW19] B. Farb, J. Wolfson, and M. M. Wood, _Coincidences between homological densities, predicted by arithmetic_ , Adv. Math. 352 (2019), 670–716.
  * [Get99a] E. Getzler, _The homology groups of some two-step nilpotent Lie algebras associated to symplectic vector spaces_ , https://arxiv.org/abs/math/9903147, 1999.
  * [Get99b] by same author, _Resolving mixed Hodge modules on configuration spaces_ , Duke Math. J. 96 (1999), no. 1, 175–203.
  * [Ho20] Q. P. Ho, _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ , https://arxiv.org/abs/2004.00252, 2020.
  * [Ho21] by same author, _Homological stability and densities of generalized configuration spaces_ , Geom. Topol. 25 (2021), no. 2, 813–912.
  * [Knu17] B. Knudsen, _Betti numbers and stability for configuration spaces via factorization homology_ , Algebr. Geom. Topol. 17 (2017), no. 5, 3137–3187.
  * [Loo23] E. Looijenga, _Torelli group action on the configuration space of a surface_ , J. Topol. Anal. 15 (2023), no. 1, 215–222.
  * [Mil69] R. J. Milgram, _The homology of symmetric products_ , Trans. Amer. Math. Soc. 138 (1969), 251–265.
  * [Pet20] D. Petersen, _Cohomology of generalized configuration spaces_ , Compos. Math. 156 (2020), no. 2, 251–298.
  * [Seg79] G. Segal, _The topology of spaces of rational functions_ , Acta Math. 143 (1979), no. 1-2, 39–72.
  * [Sta23a] A. Stavrou, _Cohomology of configuration spaces of surfaces as mapping class group representations_ , Trans. Amer. Math. Soc. 376 (2023), no. 4, 2821–2852.
  * [Sta23b] by same author, _Homology of configuration spaces of surfaces as mapping class group representations_ , Ph.D. thesis, University of Cambridge, 2023.