On the homology of configuration spaces

Bödigheimer, Carl-Friedrich; Cohen, Fred; Taylor, Laurence R.

doi:10.1016/0040-9383(89)90035-9

Cited by 82 publications

(89 citation statements)

References 21 publications

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“…Let us remark that the spaces of finite configurations, which unlike Γ X possess a natural manifold structure, have been actively studied by geometers and topologists, see e.g. [21], [30] and references given therein. The relationship between these works and our L 2 -theory, which is relevant for the spaces of finite configurations too [24], is not clear yet.…”

mentioning

confidence: 99%

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Albeverio¹,

Daletskii²

2006

Publ. Res. Inst. Math. Sci.

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The space Γ X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ X , equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.

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mentioning

confidence: 99%

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Albeverio¹,

Daletskii²

2006

Publ. Res. Inst. Math. Sci.

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“…One can "read off' the homology of the braid groups, certain groups which are related to mapping class groups, or the homology of unordered configuration spaces directly from the homology of the iterated loop space of a sphere. Similar remarks apply to some ordered configuration spaces and explain the connection between their cohomology, a free Lie algebra, and Whitehead products in classical homotopy theory [BCT,BCM,Cl,C2,C3]. Thus a fruitful point of view is that various function spaces provide a coherent and uniform picture of the types of invariants encountered in this book and in earlier work on these subjects.…”

Section: A Partial Synthesismentioning

confidence: 93%

Book Review: Complements of discriminants of smooth maps\/}: {\it Topology and applications

Cohen¹

1994

Bull. Amer. Math. Soc.

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“…Milgram, Löffler, Bödigheimer, Cohen, and Taylor computed the F 2 -homology of the configuration space B(X, a) of subsets of X of order a in terms of the F 2 -homology of X and the dimension of X , for any compact manifold X (possibly with boundary) and any natural number a [1,14]. Since we need explicit generators for the cohomology of B(X, 2) = S 2 X − X , we compute this cohomology directly for X a closed manifold in Theorem 4.2, not relying on their work.…”

Section: Also the Boundary In Hmentioning

confidence: 99%

The Integral Cohomology of the Hilbert Scheme of Two Points

Totaro

2016

Forum of Mathematics, Sigma

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The Hilbert scheme X [a] of points on a complex manifold X is a compactification of the configuration space of a-element subsets of X . The integral cohomology of X [a] is more subtle than the rational cohomology. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology.2010 Mathematics Subject Classification: 14C05 (primary); 55R80 (secondary)For a complex manifold X and a natural number a, the Hilbert scheme X [a] (also called the Douady space) is the space of 0-dimensional subschemes of degree a in X . It is a compactification of the configuration space B(X, a) of a-element subsets of X . The Hilbert scheme is smooth if and only if X has dimension at most 2 or a 3 [3, equation (0.2.1)]. The integral cohomology of the Hilbert scheme is more subtle than the rational cohomology. Markman computed the integral cohomology of the Hilbert schemes X [a] for X of dimension 2 with effective anticanonical divisor [10]. In this paper, we compute the mod 2 cohomology of X [2] for any complex manifold X , and the integral cohomology of X [2] when X has torsion-free cohomology. In one way, things are unexpectedly good: the Hilbert scheme X [2] has torsionfree cohomology if X does (Theorem 2.2). On the other hand, the details are intricate, and it was not clear that complete answers would be possible. The behavior of the inclusion of the exceptional divisor E X into X [2] is related to the Steenrod operations on the mod 2 cohomology of X (Theorem 2.1). To explain one difficulty: some cohomology classes on X [2] can be defined as the classes

show abstract

On the homology of configuration spaces

Cited by 82 publications

References 21 publications

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

$L^2$-Betti Numbers of Inﬁnite Conﬁguration Spaces

Book Review: Complements of discriminants of smooth maps\/}: {\it Topology and applications

The Integral Cohomology of the Hilbert Scheme of Two Points

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