On the Cohomology of Configuration Spaces on Surfaces

Napolitano, Fabien

doi:10.1112/s0024610703004617

Cited by 14 publications

(24 citation statements)

References 15 publications

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“…For an arbitrary S with ∂S = ∅, Conf(S, n) is a K(π, 1) and the stability for the surface braid group B S n = π 1 Conf(S, n) recovers the dimension 2 case of [40], Proposition A.1 (see also [37]). The proof also extends easily to wreath products G ≀ B S n , formed using the natural map B S n → Σ n .…”

Section: ×Smentioning

confidence: 99%

Stabilization for mapping class groups of 3-manifolds

Hatcher

Wahl²

2010

Duke Math. J.

122

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Abstract. We prove that the homology of the mapping class group of any 3-manifold stabilizes under connected sum and boundary connected sum with an arbitrary 3-manifold when both manifolds are compact and orientable. The stabilization also holds for the quotient group by twists along spheres and disks, and includes as particular cases homological stability for symmetric automorphisms of free groups, automorphisms of certain free products, and handlebody mapping class groups. Our methods also apply to manifolds of other dimensions in the case of stabilization by punctures.The main result of this paper is a homological stability theorem for mapping class groups of 3-manifolds, where the stabilization is by connected sum with an arbitrary 3-manifold. More precisely, we show that given any two compact, connected, oriented 3-manifolds N and P with ∂N = ∅, the homology groupis independent of the number n of copies of P in the connected sum, as long as n ≥ 2i + 2, i.e. each homology group stabilizes with P . We also prove an analogous result for boundary connected sum, and a version for the quotient group of the mapping class group by twists along spheres and disks, a group closely related to the automorphism group of the fundamental group of the manifold.Homological stability theorems were first found in the sixties for symmetric groups by Nakaoka [36] and linear groups by Quillen, and now form the foundation of modern algebraic K-theory (see for example [28, Part IV] and [42]). Stability theorems for mapping class groups of surfaces were obtained in the eighties by Harer and Ivanov [14,25] and recently turned out to be a key ingredient to a solution of the Mumford conjecture about the homology of the Riemann moduli space [30]. The other main examples of families of groups for which stability has been known are braid groups [1] and automorphism groups of free groups [17,18].The present paper extends significantly the class of groups for which homological stability is known to hold. It suggests that it is a widespread phenomenon among families of groups containing enough 'symmetries'. In addition to the already mentioned stability theorems for mapping class groups of 3-manifolds, corollaries of our main result include stability for handlebody subgroups of surface mapping class groups, symmetric automorphism groups of free groups, and automorphism groups of free products * n G for many groups G. Using similar techniques we obtain stability results also for mapping class groups π 0 Diff(M −{n points} rel ∂M ) for M any m-dimensional manifold with boundary, m ≥ 2 (even the case m = 2 is new here), as well as for the fundamental group π 1 Conf(M, n) of the configuration space of n unordered points in M . Our paper thus also unifies previous known results as we recover stability for braid groups (as π 1 Conf(D 2 , n)), symmetric groups (as π 1 Conf(D 3 , n)) and automorphism groups of free groups (as Aut( * n Z)).

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Section: ×Smentioning

confidence: 99%

Stabilization for mapping class groups of 3-manifolds

Hatcher

Wahl²

2010

Duke Math. J.

122

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“…The new boundary maps ∆ were computed by Napolitano [Nap03]: We define a new operator D : A r n → A r−1 n−1 by…”

Section: Configuration Spaces Of the Spherementioning

confidence: 99%

“…The tables 3 and 4 were computed with the help of the computer algebra systems Sage ["Th15] and Magma [BCP97]. The cohomology groups H r (C n (S 2 ), Z) have already been determined for n ≤ 9 by Sevryuk [Sev84] and Napolitano [Nap03]. Table 3.…”

Section: Some Tablesmentioning

confidence: 99%

“…Cohomology of C n (S 2 ). Using a cellular decomposition of C n (S 2 ) by Napolitano [Nap03], we compute the cohomology groups of C n (S 2 ) with Z/pZ-coeffcients in this paper.…”

Section: Introductionmentioning

confidence: 99%

“…We will first explain the computations of the cohomology of C n (C) with Z/pZcoefficients by Fuks [Fuk70] and Vainshtein [Vai78] and discuss their cell complex. Afterwards, we present the extension of this cell complex that Napolitano [Nap03] used to calculate H * (C n (S 2 ), Z) for n ≤ 9. The main idea of this paper is the construction of a chain homotopy that simplifies Napolitano's complex.…”

Section: Introductionmentioning

confidence: 99%

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Integral cohomology of configuration spaces of the sphere

Schiessl

2019

Homology, Homotopy and Applications

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Homological stability for configuration spaces of manifolds

2011

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Let C_n(M) be the configuration space of n distinct ordered points in M. We prove that if M is any connected orientable manifold (closed or open), the homology groups H_i(C_n(M); Q) are representation stable in the sense of [Church-Farb]. Applying this to the trivial representation, we obtain as a corollary that the unordered configuration space B_n(M) satisfies classical homological stability: for each i, H_i(B_n(M); Q) is isomorphic to H_i(B_{n+1}(M); Q) for n > i. This improves on results of McDuff, Segal, and others for open manifolds. Applied to closed manifolds, this provides natural examples where rational homological stability holds even though integral homological stability fails. To prove the main theorem, we introduce the notion of monotonicity for a sequence of S_n--representations, which is of independent interest. Monotonicity provides a new mechanism for proving representation stability using spectral sequences. The key technical point in the main theorem is that certain sequences of induced representations are monotone.Comment: 33 pages. v3: improved stable range. Final version, to appear in Inventiones Mathematica

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On the Cohomology of Configuration Spaces on Surfaces

Cited by 14 publications

References 15 publications

Stabilization for mapping class groups of 3-manifolds

Stabilization for mapping class groups of 3-manifolds

Integral cohomology of configuration spaces of the sphere

Homological stability for configuration spaces of manifolds

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