The Cohomology Algebra of Unordered Configuration Spaces

Félix, Yves; Tanré, Daniel

doi:10.1112/s0024610705006794

Cited by 20 publications

(36 citation statements)

References 16 publications

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“…In particular, as Félix and Thomas show in that paper, this spectral sequence does not always collapse when k 4. Also the fixed point CDGA, F.A; k/ † k , is a DGmodule model of the unordered configuration space, and Félix and Tanré proved in [6] that the associated spectral sequence does collapse.…”

Section: Introductionmentioning

confidence: 99%

A remarkable DGmodule model for configuration spaces

Lambrechts

Stanley

2008

Algebr. Geom. Topol.

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Let M be a simply connected closed manifold and consider the (ordered) configuration space F.M; k/ of k points in M . In this paper we construct a commutative differential graded algebra which is a potential candidate for a model of the rational homotopy type of F.M; k/. We prove that our model it is at least a † k -equivariant differential graded model.We also study Lefschetz duality at the level of cochains and describe equivariant models of the complement of a union of polyhedra in a closed manifold. 55P62, 55R80

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Section: Introductionmentioning

confidence: 99%

A remarkable DGmodule model for configuration spaces

Lambrechts

Stanley

2008

Algebr. Geom. Topol.

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“…It is known from [15] that Haefliger's metastable range condition (7) holds for m ≥ 16, as well as for m = 4, 8,9,10,13 (for the case m = 10, see Table 1 in Section 7, and the comments therein). But the equality TC S (P m ) = Emb(P m ) is also known for m ≤ 2 and, from Theorem 1.4, for m = 3.…”

Section: Tc Tc S and Calculus Of Functors Of Embeddingsmentioning

confidence: 99%

“…The case of mod p coefficients in part 1 of Project 1 was treated by Haefliger [20] (p=2) and Félix-Tanré [13] (odd prime p). Haefliger's method is reviewed in Section 8.…”

Section: (Profit:)mentioning

confidence: 99%

“…T 2 E P 13 22 (P 13 )) would also have a nontrivial Munson secondary obstruction to lift to T 3 E P 14 22 (P 14 ) (resp. T 3 E P 13 22 (P 13 )). The point then is that, modulo usual primary and secondary indeterminacy considerations, this could lead to the (again new, but now optimal) potential nonembedding result P 14 ⊂ R 22 (respectively P 13 ⊂ R 22 and P 14 ⊂ R 22 ).…”

Section: Putting the Projects To Workmentioning

confidence: 99%

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Symmetric topological complexity as the first obstruction in Goodwillie’s Euclidean embedding tower for real projective spaces

González

2011

Trans. Amer. Math. Soc.

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Abstract. As a first goal, it is explained why the Goodwillie-Weiss calculus of embeddings offers new information about the Euclidean embedding dimension of P m only for m ≤ 15. Concrete scenarios are described in these low-dimensional cases, pinpointing where to look for potential-but criticalhigh-order obstructions in the corresponding Taylor towers. For m ≥ 16, the relation TC S (P m ) ≥ n is translated into the triviality of a certain cohomotopy Euler class which, in turn, becomes the only Taylor obstruction to producing an embedding P m ⊂ R n . A speculative bordism-type form of this primary obstruction is proposed as an analogue of Davis' BP -approach to the immersion problem of P m . A form of the Euler class viewpoint is applied to show that TC S (P 3 ) = 5, as well as to suggest a few higher-dimensional projective spaces for which the method could produce new information. As a second goal, the paper extends Farber's work on the motion planning problem in order to develop the notion of a symmetric motion planner for a mechanical system S. Following Farber's lead, this concept is connected to TC S (C(S)), the symmetric topological complexity of the state space of S. The paper ends by sketching the construction of a concrete 5-local-rules symmetric motion planner for P 3 . Main results Recall the Schwarz genus genus(p) of a fibration p : E → B ([26]); it is one less 1 than the smallest number of open sets U covering B in such a way that p admits a (continuous) section over each U . Definition 1.1 ([8]). The topological complexity of a space X, TC(X), is defined as the genus of the end-points evaluation map ev : P (X) → X × X, where P (X) is the free path space X [0,1] with the compact-open topology.Let F (X, 2) ⊂ X × X denote the configuration space of ordered pairs of distinct points in X, and ev 1 : P 1 (X) → F (X, 2) be the restriction of the fibration ev. Thus P 1 (X) is the subspace of P (X) obtained by removing the free loops on X. The group Z/2 acts freely on both P 1 (X) and F (X, 2), by running a path backwards in

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“…The work back then, mostly represented by [4,5,6,8,10,11,12,13,14,25,26,32,37], focuses mainly on additive descriptions with field coefficients. The work of Kriz and Totaro ( [29,36]) in the early 90's, and of Felix-Tanré and Felix-Thomas ( [23,24]) in the 2000's settled much of the multiplicative structure (mostly with field coefficients) in the case where M is a projective algebraic variety and, more generally, if M is rationally formal (mostly with characteristic-zero coefficients). But the problem is still largely unsettled for more general manifolds and coefficients.…”

Section: Introductionmentioning

confidence: 99%

The cohomology ring away from 2 of configuration spaces on real projective spaces

González

Guzmán-Sáenz

Xicoténcatl

2015

Topology and its Applications

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Abstract. Let R be a commutative ring containing 1/2. We compute the R-cohomology ring of the configuration space Conf(RP m , k) of k ordered points in the m-dimensional real projective space RP m . The method uses the observation that the orbit configuration space of k ordered points in the m-dimensional sphere (with respect to the antipodal action) is a 2 k -fold covering of Conf(RP m , k). This implies that, for odd m, the Leray spectral sequence for the inclusion Conf(RP m , k) ⊂ (RP m ) k collapses after its first non-trivial differential, just as it does when RP m is replaced by a complex projective variety. The method also allows us to handle the R-cohomology ring of the configuration space of k ordered points in the punctured manifold RP m − ⋆. Lastly, we compute the LusternikSchnirelmann category and all of the higher topological complexities of some of the auxiliary orbit configuration spaces.MSC 2010: Primary 55R80, 55T10, 55M30.

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The Cohomology Algebra of Unordered Configuration Spaces

Cited by 20 publications

References 16 publications

A remarkable DGmodule model for configuration spaces

A remarkable DGmodule model for configuration spaces

Symmetric topological complexity as the first obstruction in Goodwillie’s Euclidean embedding tower for real projective spaces

The cohomology ring away from 2 of configuration spaces on real projective spaces

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