TThe Euler characteristic of configuration spaces
LOUIS HAINAUT ∗ September 25, 2020
Abstract
In this short note we present a generating function computing the com-pactly supported Euler characteristic χ c ( F ( X, n ) , K (cid:2) n ) of the configura-tion spaces on a topologically stratified space X , with K a constructiblecomplex of sheaves on X , and we obtain as a special case a generatingfunction for the Euler characteristic χ ( F ( X, n )) . We also recall how touse existing results to turn our computation of the Euler characteristicinto a computation of the equivariant Euler characteristic. We consider (ordered) configuration spaces of n distinct points on a topo-logical space X , for n ≥ an integer: F ( X, n ) = { ( x , . . . , x n ) ∈ X n | x i (cid:54) = x j for i (cid:54) = j } These spaces admit a free action by the symmetric group S n by permu-tation of the coordinates. One can also consider the quotient space B ( X, n ) = F ( X, n ) / S n . In particular, the map F ( X, n ) → B ( X, n ) is a covering map.We will be interested in computing the Euler characteristic of thespaces F ( X, n ) . In the case of a manifold M , the exponential generatingfunction for the Euler characteristic of configuration spaces on M is ∞ (cid:88) n =0 χ ( F ( M, n )) · t n n ! = (cid:26) (1 + t ) χ ( M ) if dim( M ) is even (1 − t ) − χ ( M ) if dim( M ) is odd . (1)This formula can be found using the fibration M \ { p , . . . , p n − } → F ( M, n ) → F ( M, n − . Using multiplicativity of the Euler characteristic on fibrations (and Mayer-Vietoris to compute the characteristic of the fibers) we can prove induc-tively that χ ( F ( M, n )) = (cid:26) (cid:81) n − i =0 ( χ ( M ) − i ) if dim( M ) is even (cid:81) n − i =0 ( χ ( M ) + i ) if dim( M ) is odd . and these coefficients induce the aforementioned exponential generatingfunctions. ∗ The author gratefully acknowledges support by ERC-2017-STG 759082 a r X i v : . [ m a t h . A T ] S e p e are therefore primarily interested in the case when X is not amanifold. If X is a finite simplicial complex, the following formula wasfound by Gal [Gal01]: ∞ (cid:88) n =0 χ ( F ( X, n ) · t n n ! = (cid:89) σ (1 + ( − d σ (1 − v σ ) t ) ( − dσ (2)where the product runs over all the cells σ of X , d σ denotes the dimensionof σ and v σ the Euler characteristic of its normal link L σ . If we usethe compactly supported Euler characteristic, Getzler [Get95] found thefollowing formula, which applies to any locally compact Hausdorff space: ∞ (cid:88) n =0 χ c ( F ( X, n )) · t n n ! = (1 + t ) χ c ( X ) . This formula will be further discussed in Lemma 2.5. We can alreadynotice that it generalizes formula (1), since for even-dimensional mani-folds M we have χ c ( M ) = χ ( M ) , while for odd-dimensional ones we have χ c ( M ) = − χ ( M ) .The formula we present here generalizes (2) to any topologically strat-ified space of finite type X = ∪ α X α . For each stratum X α , let d α be itsdimension and L α its link. We can express our formula as ∞ (cid:88) n =0 χ ( F ( X, n ) · t n n ! = (cid:89) α (1 + ( − d α (1 − χ ( L α )) · t ) ( − dα χ ( X α ) . (3)We remark that in the special case when X is a finite simplicial com-plex, with the stratification given by its open cells, our formula (3) be-comes the same as Gal’s (2). Remark . Many "reasonable" spaces admit a topological stratification.For example every algebraic variety, or more generally every semialgebraicor subanalytic set admits a topological stratification, since they all admita Whitney stratification.The formula given here is simple and the proof is not too involved,however to the best of our knowledge it has not appeared in the litera-ture. If we compare with (2), our formula seems more convenient as thereis no need to triangulate X and X is not necessarily compact. Another ad-vantage of our formula is that the same argument allows more generally tocompute the compactly supported Euler characteristic χ c ( F ( X, n ) , F (cid:2) n ) for any constructible complex of sheaves of finite type F on X (see The-orem 2.8). The formula (3) is obtained as a special case with F thedualizing complex of X .Our proof uses sheaf cohomology and Verdier duality. Most of thearguments are formal properties, but we need some technical assumptionsto ensure that the dualizing complex of our space has good properties. Remark . During the final stages of writing this paper, we learned thatYuliy Baryshnikov [Bar20] independently discovered results that signifi-cantly overlap with the results presented here.
In the following we will use topologically stratified spaces of finite type .We will recall the definition of topologically stratified spaces [GM83] andsome of their properties. efinition 1.3. An n -dimensional topologically stratified space is a Haus-dorff space X with a filtration ∅ = X − ⊂ X ⊂ X ⊂ . . . ⊂ X n = X of X by closed subspaces such that for each i and each x ∈ X i \ X i − thereexists a neighborhood U x ⊂ X of x , a compact ( n − i − -dimensionaltopologically stratified space L x and a filtration preserving homeomor-phism U x ∼ = R i × CL x with CL x denoting the open cone on L x .The connected components X α of the set differences X i \ X i − arecalled the strata of X . Remark . From the definition of topologically stratified spaces we canderive following useful properties: • It is not required that any of the inclusions X i − ⊂ X i are strict. Inparticular every n -dimensional topologically stratified space admitsan ( n + 1) -dimensional topological stratification with X n +1 = X n . • Since the homeomorphism U x ∼ = R i × CL x is filtration preserving,the strata X α are topological manifolds. • We call the space L x the link of x . We can prove that on any stratum X α we can choose the same link for all points x ∈ X α and we willtherefore write this space L α .We say that a topologically stratified space X is of finite type if thecollection of strata is finite and every stratum is (homeomorphic to) theinterior of a compact manifold with boundary. This latter condition isenough to ensure that the strata have the homotopy type of a finite CW-complex, and in particular that their Betti numbers are finite.All sheaves considered are sheaves of Q -vector spaces, so in particularall sheaves are flat. We will always assume that the complexes of sheavesare cohomologically bounded and that local systems have finite rank. Wewill also need the notion of a constructible complex of sheaves. A sheaf F on a stratified space X = ∪ mi =1 X i is called constructible if its restriction toeach stratum X i is a local system of finite rank. A complex of sheaves F • is called constructible if each cohomology sheaf H p ( F • ) is constructible.We will use χ c ( X, F • ) the compactly supported Euler characteristicwith sheaf coefficients. As always, the Euler characteristic is defined asthe alternate sum of the cohomology groups, so we need to know what is H • c ( X, F • ) , with F • being a complex of sheaves. Definition 1.5.
Let F be a sheaf on a space X . We define the functorof global sections of F with compact support as Γ c ( X, F ) = { s ∈ Γ( X, F ) : supp ( s ) is compact } , the support of a section s ∈ Γ( U, F ) being the set { x ∈ U : s x (cid:54) = 0 } .As for the regular sheaf cohomology, we then define the cohomologygroups with compact supports H • c ( X, F • ) to be the higher direct imageof the functor Γ c ( X, − ) . When F • is the constant sheaf Q concentratedin degree , we may omit the coefficient. Remark . • When X is compact, then Γ c ( X, − ) = Γ( X, − ) and therefore forany complex of sheaves F • we have H • ( X, F • ) = H • c ( X, F • ) . Thismeans in particular that χ ( X, F • ) = χ c ( X, F • ) . When X is a manifold of dimension d , we have χ c ( X ) = ( − d χ ( X ) ,by Poincaré duality. Although the introduction focused on the Euler characteristic χ ( X ) , wewill work with the Euler characteristic with compact support χ c ( X, F ) .We explain later how the former is a special case of the latter. Definition 2.1.
Let X be a topological space and K be a complex ofsheaves on X . We define the generating series e ( X, K ; t ) = (cid:88) n χ c ( F ( X, n ) , K (cid:2) n ) · t n n ! . This expression makes sense if all the compactly supported Euler char-acteristics on the right hand side exist.We will use the result that the Euler characteristic with compact sup-ports χ c ( X, F ) is additive over stratifications. We present here a proof ofthis result for the reader’s convenience. Lemma 2.2.
Let X = ∪ mi =1 X i be a space with a finite stratification and K be a complex of sheaves on X . Then we have χ c ( X, K ) = m (cid:88) i =1 χ c ( X i , K ) Proof.
The key result is that whenever X is partitioned into two subsets U and Z with U open and Z closed, then χ c ( X, K ) = χ c ( U, K ) + χ c ( Z, K ) . (4)More information about this identity can be found in [Dim04, Remark2.4.5]We prove Lemma 2.2 by induction on the number of strata. This resultis true whenever the (finite) stratification satisfies the two conditions thatevery stratum is locally closed and that the closure of each stratum isa union of strata. These conditions are clearly satisfied in the case of atopological stratification. • If the stratification has one stratum there is nothing to prove. • If the stratification has two or more strata, by the conditions men-tioned above we can assume without loss of generality that the stra-tum X is closed, and let X (cid:48) be the union of all remaining strata.The identity (4) gives that χ c ( X, K ) = χ c ( X , K ) + χ c ( X (cid:48) , K ) , andwe conclude by employing the induction hypothesis on X (cid:48) .If L is a complex of sheaves whose cohomology sheaves are local sys-tems, we define its Euler characteristic χ ( L ) as the alternating sum χ ( L ) = (cid:88) s ∈ Z ( − s rk ( H s ( L )) . The ranks are well-defined since the cohomology sheaves are local systems. emark . If L, L (cid:48) both have cohomology sheaves which are local sys-tems, then χ ( L ⊗ L (cid:48) ) = χ ( L ) χ ( L (cid:48) ) by the Künneth theorem for complexesof sheaves. Lemma 2.4.
Let X be the interior of a compact manifold with boundaryand let L be a complex of sheaves on X whose cohomology sheaves arelocal systems. Then χ c ( X, L ) = χ c ( X ) χ ( L ) .Proof. We first consider the special case when L is a single local systemin degree . In that case χ ( L ) = rk ( L ) and we can prove the result asfollows: Since χ c ( X, − ) = ( − dim( X ) χ ( X, − ) , we can work with the usualcohomology. Moreover X is homotopy equivalent to a compact manifoldwith boundary (its closure), and these spaces have the homotopy type of afinite CW-complex, so X is homotopy equivalent to a finite CW-complex.We can therefore reduce the statement to the case when X is a finiteCW-complex, and this case can be proven by using cellular cochains (seealso [Dim04, Proposition 2.5.4]).For the general case we use the spectral sequence relating sheaf coho-mology and hypercohomology to obtain χ c ( X, L ) = (cid:88) i ( − i rk ( H ic ( X, L ))= (cid:88) p,q ( − p + q rk ( H pc ( X, H q ( L )))= (cid:88) q ( − q χ c ( X, H q ( L ))= χ c ( X ) χ ( L ) . Lemma 2.5.
Let X be a Hausdorff topological space. Then e ( X, Q ; t ) = (1 + t ) χ c ( X ) Proof.
This formula is already known. We give an argument for thereader’s convenience.The space X n admits a stratification whose strata are the configu-ration spaces F ( X, T ) for T any partition of the set { , , . . . n } . Theconfiguration space F ( X, T ) is defined as the space F ( X, T ) = { ( x , . . . , x n ) ∈ X n | x i = x j ⇔ i ∼ T j } or in English F ( X, T ) is the subspace of X n where two coordinates x i and x j are equal if and only if i and j belong to the same block of T . ByLemma 2.2 we obtain χ c ( X n ) = (cid:88) T ∈ Π n χ c ( F ( X, T )) . Now on the one hand by multiplicativity we have χ c ( X n ) = χ c ( X ) n ,and on the other hand, if | T | denotes the number of blocks of the partition T , we have an identity χ c ( F ( X, T )) = χ c ( F ( X, | T | )) . Therefore the aboveequality becomes χ c ( X ) n = n (cid:88) k =1 S ( n, k ) χ c ( F ( X, k )) ith S ( n, k ) a Stirling number of the second kind, defined as the numberof partitions of n in k parts. Using (signed) Stirling numbers of the firstkind s ( k, n ) , we can invert the relation between χ c ( X ) n and χ c ( F ( X, k )) to obtain χ c ( F ( X, k )) = k (cid:88) n =1 s ( k, n ) χ c ( X ) n . Finally we use the generating function for the Stirling numbers of thefirst kind to obtain ∞ (cid:88) k =0 χ c ( F ( X, k )) t k k ! = 1 + ∞ (cid:88) k =1 k (cid:88) n =1 s ( k, n ) χ c ( X ) n t k k ! = (1 + t ) χ c ( X ) . Lemma 2.6.
Let X be the interior of a compact manifold with boundaryand let L be a complex of sheaves whose cohomology sheaves are localsystems. Then e ( X, L ; t ) = (1 + χ ( L ) · t ) χ c ( X ) . Proof.
We first notice that χ c ( F ( X, n ) , L (cid:2) n ) = χ c ( F ( X, n )) · χ ( L ) n dueto Lemma 2.4 and Remark 2.3. Then using Lemma 2.5 we obtain e ( X, L ; t ) = e ( X, Q ; χ ( L ) · t ) = (1 + χ ( L ) · t ) χ c ( X ) . Proposition 2.7.
Let X = (cid:83) mj =1 X j be a stratified space and K be acomplex of sheaves on X . Then e ( X, K ; t ) = m (cid:89) j =1 e ( X j , K | X j ; t ) . Proof.
The stratification ( X j ) induces a stratification of F ( X, n ) withstrata of the form m (cid:89) j =1 F ( X j , n j ) such that n + . . . + n m = n and n j ≥ for each j . Moreover, therestriction of K (cid:2) n to each such stratum is (cid:2) mj =1 ( K | X j ) (cid:2) n j . Once these two observations have been made, we can first compute fora fixed nχ c ( F ( X, n ) , K (cid:2) n ) · t n n != (cid:88) n + ... + n m = n (cid:32) nn , . . . , n m (cid:33) χ c (cid:32) m (cid:89) j =1 F ( X j , n j ) , (cid:2) mj =1 ( K | X j ) (cid:2) n j (cid:33) · t n n != (cid:88) n + ... + n m = n m (cid:89) j =1 χ c ( F ( X j , n j ) , ( K | X j ) (cid:2) n j ) · t n j n j ! nd thus we obtain for the exponential generating function e ( X, K ; t ) = (cid:88) n χ c ( F ( X, n ) , K (cid:2) n ) · t n n != m (cid:89) j =1 (cid:88) n χ c ( F ( X j , n ) , ( K | X j ) (cid:2) n ) · t n n != m (cid:89) j =1 e ( X j , K | X j ; t ) . We are now ready to state the main result:
Theorem 2.8.
Let X = ∪ mi =1 X i be a topologically stratified space of finitetype and K be a constructible complex of sheaves on X . Then e ( X, K ; t ) = m (cid:89) i =1 (1 + χ ( K | X i ) · t ) χ c ( X i ) Proof.
From Proposition 2.7, we obtain that e ( X, K ; t ) = m (cid:89) i =1 e ( X i , K | X i ; t ) . Then, for each stratum X i we apply Lemma 2.6 to obtain m (cid:89) i =1 e ( X i , K | X i ; t ) = m (cid:89) i =1 (1 + χ ( K | X i ) · t ) χ c ( X i ) . We now explain how to recover the Euler characteristics χ ( F ( X, n )) from previous theorem.We define the dualizing complex D Q of X as the sheafification of thepresheaf of cochain complexes U (cid:55)→ C • c ( U, Q ) ∨ sending U to the dual of the complex of cochains with compact supporton U . Using [Ive86, Proposition V.2.4], we see that this definition corre-sponds to the more usual one with the exceptional inverse image functor.A standard result of Verdier duality is that H • c ( X, D Q ) = H −• ( X, Q ) ∨ (see[Dim04, Theorem 3.3.10], using the fact that D ( D Q ) ∼ = Q ). This means inparticular that χ c ( X, D Q ) = χ ( X ) . We also notice that for a topologicallystratified space X , the dualizing complex has constructible cohomologysince for any stratum X α of dimension i and any point x ∈ X α , x admitsa basis of neighborhoods all homeomorphic to R i × CL α . Finally we easilysee from the definition that D Q is cohomologically bounded.The remarks made in the previous paragraph actually allow us to provethat for any stratum X α and any point x ∈ X α we have H − p ( D Q | X α ) | x ∼ = H pc ( R i × CL α ) ∨ and therefore χ ( D Q | X α ) = χ c ( R i × CL α )= ( − i χ c ( CL α )= ( − i (1 − χ ( L α )) . orollary 2.9. Specializing the Theorem 2.8 with K = D Q gives theformula for the Euler characteristic (cid:88) n χ ( F ( X, n )) · t n n ! = (cid:89) α (1 + χ ( D Q | X α ) · t ) χ c ( X α ) = (cid:89) α (1 + ( − d α (1 − χ ( L α )) · t ) ( − dα χ ( X α ) . Let us remark that knowing the Euler characteristic χ ( F ( X, n )) also de-termines χ ( B ( X, n )) as well as χ S n ( F ( X, n )) ∈ R ( S n ) . The main obser-vation that we will use here is that for any positive integer n , the space F ( X, n ) admits a free action of the symmetric group S n by permutationof the coordinates. Proposition 2.10.
Let X be a topologically stratified space. Then χ ( B ( X, n )) = χ ( F ( X, n )) /n ! and χ S n ( F ( X, n )) = χ ( B ( X, n )) · Q [ S n ] . In particular, the equivariant Euler characteristic of F ( X, n ) is a mul-tiple of the regular representation. This result is a special case of a more general result about the Eulercharacteristic of spaces with a free action of a finite group, which seemsto have been first written down by Zarelua [Zar68, Theorem 1]. The mainingredient of the proof is the Lefschetz trace formula, which implies that atrace Tr ( g ) = (cid:80) i ( − i Tr ( g | H i ( F ( X, n ))) is for all g ∈ S n , g (cid:54) = e . Therest of the proof uses standard results of character theory on the doublesum (cid:88) i (cid:88) g ∈ S n ( − i Tr ( g | H i ( F ( X, n ))) . We finish by showing how to use our formula for a space to which thepreviously known formulas do not apply. Note that in the proof we as-sumed that the strata where connected, but in computations we can allownon-connected strata as long as the link L x is the same for each point ofthe stratum. This doesn’t change the result because of Lemma 2.2.Consider the subspace X ⊂ R formed by two planes Π , Π intersect-ing on a line l . The space X is obviously not a manifold, and it also cannotbe represented by a finite simplicial complex since it is non-compact, soGal’s formula does not apply.We can take the stratification X = X ∪ X , with X being the disjointunion of the four half-planes, and X being the intersection line. In thissituation, the link L is the empty space, while the link L is a discretespace with four points. We therefore obtain e ( X, D Q ; t ) = (1 + t ) · (1 + ( − − · t ) − = (1 + t ) · (1 + 3 t ) − . eferences [Bar20] Y. Baryshnikov. “Euler Characteristics of Exotic Config-uration Spaces”. http://publish.illinois.edu/ymb/files/2020/05/euler.pdf . 2020.[Dim04] A. Dimca. Sheaves in Topology . Springer-Verlag BerlinHeidelberg, 2004. doi : .[Gal01] Ś. R. Gal. “Euler characteristic of the configuration spaceof a complex”. In: Colloquium Mathematicum doi : .[Get95] E. Getzler. “Mixed Hodge structures of configuration spaces”.arXiv:alg-geom/9510018. 1995.[GM83] M. Goresky and R. MacPherson. “Intersection homologyII”. In: Inventiones mathematicae
Cohomology of Sheaves . Springer-Verlag BerlinHeidelberg New York Tokyo, 1986. doi : .[Zar68] A. Zarelua. “On finite groups of transformations”. In:1969Proc. Internat. Sympos. on Topology and its Applications(Herceg-Novi, 1968)