Configuration spaces as commutative monoids

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.


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 (1.1) C n (M ) + = (M + ) ∧n locus where two points coincide Sn , the quotient formed in pointed spaces.Not-necessarily-disjoint union of unordered configurations defines a superposition product which is associative, commutative, and unital.This gives a unital commutative monoid object in the symmetric monoidal category Top N * of N-graded pointed spaces: The goal of this note is to explain and exploit this algebraic structure.
In the following, for a pointed space X we write X[n] for the N-graded pointed space which consists of X in grading n and the point in all other gradings, and write Com(Y ) for the free unital commutative monoid on an object Y ∈ Top N * .Theorem 1.1.There is a pushout square of unital commutative monoids in Top N * , where ϵ is the augmentation and ∆ is induced by the diagonal inclusion M + → [M + ∧ M + ] S2 = Com(M + [1])(2).Furthermore, this square is a homotopy pushout, i.e. there is an induced equivalence That the square is a strict pushout of unital commutative monoids is elementary: it means identifying C(M ) as the quotient of the based symmetric power monoid of M + by the ideal given by those tuples which contain a repeated element, which is a reformulation of (1.1).The content of the theorem is that the square is also a homotopy pushout, rendering it amenable to homological calculation.This formula has many applications to the homology of configuration spaces.As one application we will show how to recover Knudsen's [Knu17] formula for H * (C n (M ); Q) in terms of the compactly-supported 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 ); 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 [FWW19] on homological densities.
Context.This note is my attempt to give a topological implementation of some of the sheaf-theoretic ideas of Banerjee [Ban23] in the case of configuration spaces.The applications to the homology of configuration spaces given in Section 2 arise by taking singular chains of the equivalence (1.2) to obtain a derived tensor product description of the chains on C(M ; L): this description will also follow from [Ban] as explained in [Ban23, Remark 1.1].As such, the purpose of this paper is (i) to give a space-level implementation/interpretation of Banerjee's ideas in a specific case, in order to popularise them among topologists, and (ii) to explain how several classical, recent, and new results about the rational homology of configuration spaces can be obtained very efficiently from (1.2) (or its chain-level analogue).Everything I will describe has much to do with the work of Ho [Ho21,Ho20], Petersen [Pet20], Knudsen [Knu17], Getzler [Get99a,Get99b], Kallel [Kal98], Bödigheimer-Cohen-Milgram [BCM93], Segal [Seg79], and Arnol'd [Arn70].
Acknowledgements.I am grateful to Andrea Bianchi, Sadok Kallel, and the anonymous referee for their useful feedback on an earlier version of the paper.ORW was supported by the ERC under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 756444).

Applications
2.1.Homology of configuration spaces.Let M be d-dimensional.The space 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 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 ); k).We do not pursue this in general here, but rather focus on the case k = Q, where a complete answer is possible, and reproduces a formula of Knudsen.
2.2.Revisiting Knudsen's formula.For an N-graded pointed space we write H n,i (X) := H i (X(n)), and similarly for chains.Write S * (V ) for the free gradedcommutative algebra on a homologically graded vector space V , i.e. S * (V ) = n≥0 [V ⊗n ] Sn , where the Koszul sign rule is implemented.If V is equipped with additional N-grading, then this is inherited by S * (V ) (but there is no Koszul sign rule associated to the N-grading, only to the homological grading).

Furthermore, one may choose formality equivalences
i.e. chain maps inducing the identity on homology, and hence obtain equivalences of cdga's.In N-grading 2, the map ∆ induces a map With these choices the square Recall that for a free graded-commutative algebra S * (V ) on a homologically graded vector space V (perhaps equipped with a further N-grading), there is a free resolution of the trivial left S * (V )-module Q given by ϵ : S * (V ⊕ ΣV ) ∼ → Q equipped with the differential given by ∂(Σv) = 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 ΣV id → V , and over Q taking homology commutes with the formation of symmetric powers.Applying this resolution to calculate the Tor groups above gives the complex and extended by the Leibniz rule.This can be simplified as follows.If M is d-dimensional then the Thom isomorphism gives This lets us write the complex as (2.1) where the differential is dual to the map S 2 (H * c (M ; Q w1 )) → H * c (M ; Q) induced by cup product, so following Knudsen we can recognise this complex as the Chevelley-Eilenberg complex for the bigraded Lie algebra After appropriate dualisations and reindexings, this agrees with Knudsen's formula.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

and choosing a cycle representing this element provides a map
Multiplication by this element defines a map 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 * , * (C(M ; W ); Q)/σ for the mapping cone of left multiplication by σ, the discussion above shows that its homology is calculated by a complex As M is connected, if we assume that d ≥ 3 then the bigraded vector spaces 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 ); Q) → H i (C n (M ); Q) being surjective for i < n and an isomorphism for i < n − 1.For d = 2 the same considerations give surjectivity for i < 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]).
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 ); Q) via their action on H * (M ; 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 odddimensional.Then ) 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 H * (M + ; Q) also acts trivially on 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 Q-modules rather than chain complexes.We write − ⊙ − for the tensoring of this category over simplicial sets.For a space X let us abbreviate Q The discussion in the previous section, ignoring the formality step and translated to simplicial Q-modules, shows that given the simplicial module Q[M + ] and the map δ : whose bigraded homotopy groups are identified with H * (C * (M ) + ; Q).
A homeomorphism of M , or a pointed homotopy self-equivalence of M + , induces an equivalence ϕ : is commutative, which induces a self-equivalence of the two-sided bar construction (2.2).This corresponds to the induced action on H * (C * (M ) + ; Q).However a weaker kind of data suffices to get an induced equivalence on two-sided bar constructions.An equivalence ϕ : of simplicial commutative rings as above where the right-hand square commutes and the left-hand squares commutes up to the homotopy S * (h) : ).This data suffices to obtain a self-equivalence χ(ϕ, h) of the two-sided bar construction (2.2), as the zig-zag Let (ϕ ′ , h ′ ) be another such datum, and suppose that there is a homotopy Φ : ϕ ⇒ ϕ ′ such that the 2-cell Then one may check that χ(ϕ ′ , h ′ ) is homotopic to χ(ϕ, h).If we let Γ denote the set of (ϕ, h)'s modulo the equivalence relation (ϕ, h) ∼ (ϕ ′ , h ′ ) when there exists a homotopy Φ having the above property, then composition of maps and pasting of homotopies makes Γ into a group, which acts on the two-sided bar construction (2.2) in the homotopy category of simplicial Q-modules (and so also acts on its homotopy groups).A pointed homotopy self-equivalence of M + acts on the two-sided bar construction through Γ, via elements of the special form [(ϕ, Id)].
We may analyse the group Γ as follows.There is a homomorphism to the group of homotopy classes of homotopy self-equivalences of Q[M + ].Using the Dold-Kan theorem the latter can be identified with the group of homotopy classes of homotopy self-equivalences of C * (M + ; Q), and using a formality equivalence this is identified with the group Aut( H * (M + ; Q)) of automorphisms of the graded vector space H * (M + ; Q).Such an automorphism is in the image of ρ precisely when it preserves the map δ * : The kernel of ρ consists of those [(ϕ, h)] such that ϕ is homotopic to the identity: by definition of the equivalence relation ∼ such an element may be written as [(Id, h ′ )] where h ′ is obtained from h and a homotopy Φ : ϕ ⇒ id by the 2-cell diagram (2.3).Such an h ′ is a self-homotopy of the map δ, so an element of π 1 (map( )); δ).The ambiguity in h ′ when representing [(ϕ, h)] ∈ Ker(ρ) as [(Id, h ′ )] comes from the choice of the homotopy Φ, so h ′ is well-defined modulo the ambiguity coming from the self-homotopies π 1 (map( Q[M + ], Q[M + ]); id) of the identity map.In conclusion, this discussion establishes an exact sequence Using the Dold-Kan theorem and a formality equivalence again we can identify the first map in this sequence with and so describe Γ by an extension 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 H * (M + ; Q).Theorem 2.2.If M is orientable and even-dimensional, then G acts on which in grading 2 is the inclusion of the symplectic form ω ∈ Λ 2 (H 1 (M ; Q)) and is zero otherwise.Thus the above is Hom(H . 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.

Proof of Theorem 1.2
Recall that X ∈ Top * is well-based if the basepoint map i : * → X is a closed cofibration: under this condition X ∧ − preserves weak equivalences between wellbased spaces, and preserves closed cofibrations.Let us say that an N-graded based space Y is well-based if Y (n) is well-based for each n ∈ N.
Let us write R := Com(L + [1]) and ) to ease notation, so ∆ : S → R makes R into a S-module.Lemma 3.1.S and R are well-based.The subspace of [(L + ) ∧p ] Sp 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 M .This admits a collar, showing that i : M → M admits a homotopy inverse, and so the vector bundle L → M extends to a vector bundle over M , which we also call L. Furthermore, choosing an inner product on this bundle we can form the closed disc bundle D(L) → M , and consider L as lying inside it as the open disc bundle.Now D(L) is a manifold with boundary ∂D(L) = S(L) ∪ D(L)| ∂M , and Observe that (M , ∂M ) is an compact manifold pair so (is an ENR pair and hence) can be expressed as a retract of a pair (|X Sp is a retract, and so is well-based.This shows that R is well-based, and similar reasoning shows S is.
For the second statement, Proof.
where the latter term is only taken when it makes sense: for n − p even.This is a filtration by right S-modules.One checks that the diagram is a pushout (in Top N * and so in right S-modules), where the horizontal maps are induced by the S-module structure and the adjoints of the map inc : F p−2 R(p) → F p−1 R(p), and the map id : R(p) → F p R(p).
We prove by induction on p that F p R is a flat S-module in the indicated sense.As F 0 R = S these properties hold for p = 0.For M a left S-module whose underlying object is well-based, applying − ⊗ S M to the square above gives a pushout square (3.1) 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 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 : M ∼ → M ′ 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 In the case that L is the 0-dimensional vector bundle, the filtration stage F p R(n) consists of those elements in the nth based symmetric power [(M + ) ∧n ] Sn containing at most p unrepeated elements.Up to reindexing, this is the same as the filtration used by Arnol'd [Arn70] and by Segal [Seg79].
Proof.By definition of the relative tensor product there is a coequaliser diagram Remark 3.5.It is possible to fool oneself into thinking that the above argument can be adapted to the case of ordered configuration spaces, considered in the category of symmetric sequences of pointed spaces, in order to prove a statement analogous to the equivalence (1.2) in this category.Unfortunately, that statement is false.One can verify this directly in the case M = * with trivial 0-dimensional Euclidean bundle, in grading 3.If there is an analogue for ordered configuration spaces, its statement must be more complicated.
Appendix A. Homological densities by Quoc P. Ho and Oscar Randal-Williams A.1.Spaces of 0-cycles.It is easy to generalise Theorem 1.2 to the following variant of configuration spaces, called "spaces of 0-cycles" by Farb-Wolfson-Wood [FWW19].Let m, k ≥ 1, and for n 1 , n 2 , . . ., n m ∈ N let Z k n1,...,nm (M ) ⊂ Sym n1,...,nm (M ) : be the open subspace of those ({x 1 1 , . . ., x 1 n1 }, {x 2 1 , . . ., x 2 n2 }, . . ., {x m 1 , . . ., x m nm }) such that no x i j has multiplicity ≥ k in all of these m multisets.That is, Z k n1,...,nm (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 ≥ k points of every colour.The 1-point compactifications Z n1,...,nm (M ) + again have a composition product ..,nm+n ′ m (M ) + , giving a commutative monoid Z m,k (M ) in N m -graded pointed spaces.Just as before, we can introduce labels in a vector bundle L → M , giving Z k n1,...,nm (M ; L) and Z m,k (M ; L).Writing 1 i = (0, . . ., 0, 1, 0 . . ., 0) ∈ N m with the 1 in the ith position, there is a pushout square (A.1) of unital commutative monoids in Top N m * , where ∆ is now induced by the inclusion The same argument as Theorem 1.2 shows that there is an equivalence 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 n1,...,nm (M ; L) are Q-homology manifolds, being open subspaces of a product of coarse moduli spaces [L n ] Sn of the orbifolds L n / /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 n1,...,nm (M ; W ) are orientable Q-homology manifolds, of dimension 2d • n i .Again they are vector bundles over Z k n1,...,nm (M ), so Poincaré duality gives On the other hand, the bar construction formula above together with the argument of Section 2.2 identifies the multigraded vector space H * , * (Z m,k (M ; W )) with Tor in this case.Using Poincaré duality on both sides gives [FWW19, Theorem 1.4], except that that theorem is erroneously claimed for all k ≥ 1.We will return to the case k = 1 below.
A.2.2.Even-dimensional manifolds.If d is even then S m k acts trivially on Σ dmk H * (M + ), and using the Thom isomorphism to identify H * (W + ) ∼ = Σ d H * (M + ) too, the Koszul complex for computing the Tor-groups above is The differential ∂ is induced by the map obtained by linearly dualising the cup product map ), and so is trivial if (and only if) all mk-fold cup products of (w 1 -twisted) compactlysupported cohomology classes on M vanish.
When this cup product map is trivial, so ∂ is trivial, the above just gives a formula for H * , * (Z m,k (M ; W )). Using Poincaré duality, and reindexing, to express this in terms of H * (Z k n1,...,nm (M )) and H * (Sym n1,...,nm (M )) we obtain an identity of multigraded vector spaces
• |, |∂X • |) of the geometric realisations of a simplicial set and a subset.We may pull L back to |X • | using the retraction; let us call this L X .Now D(L X )/S(L X ) ∪ D(L X )| |∂X•| can be given an evident cell-structure (by induction over the relative cells of |∂X • | → |X • |), and L + = D(L)/∂D(L) is a retract of it, so is well-based.More generally, for the exterior direct sum L ⊞p X → |X p • | and writing ∂|X p • | for the subcomplex where some factor lies in ∂X • , there is a cell structure on D(L ⊞p X )/S(L ⊞p X ) ∪ D(L ⊞p X )| ∂|X p • | for which the group S p acts cellularly, and so [ inclusion of a S p -CW-subcomplex, and so has a 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 |∂X • |.Thus it lifts to a S p -equivariant deformation retraction of an open neighbourhood of L ⊞p X | F → L ⊞p X , and descends to the quotient by the subcomplexes where some factor lies in |∂X • |.As it is equivariant, it descends further to the S p -quotient.That is, it proves the claim for (M , ∂M , L) replaced by (|X • |, |∂X • |, L X ); as the former data is a retract of the latter, the claim follows.□ Lemma 3.2.R is a flat S-module, in the sense that R ⊗ S − preserves weak equivalences between left S-modules whose underlying objects are well-based.