Sullivan's Minimal Models and Higher Order Whitehead Products

Andrews, Peter C.; Arkowitz, Martin

doi:10.4153/cjm-1978-083-6

Cited by 31 publications

(33 citation statements)

References 3 publications

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“…and so using the generators x, y, z we obtain the following formula for H * (BG λ ; Q) in the statement of Theorem 1.5 which is similar to the formula in the untwisted case: [AA,Theorem 5.4] in [AM, p. 1007]. Using the notation of [AA] In this way one sees that the higher Whitehead products in π * (G λ ) ⊗ Q can not be used exclusively to compute the ring structure in H * (BG λ ; Q).…”

Section: Remark 518 Making the Change Of Variablesmentioning

confidence: 86%

Compatible complex structures on symplectic rational ruled surfaces

Abreu¹,

Granja²,

Kitchloo³

2009

Duke Math. J.

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ABSTRACT. In this paper we study the topology of the space Iω of complex structures compatible with a fixed symplectic form ω, using the framework of Donaldson. By comparing our analysis of the space Iω with results of McDuff on the space Jω of compatible almost complex structures on rational ruled surfaces, we find that Iω is contractible in this case.We then apply this result to study the topology of the symplectomorphism group of a rational ruled surface, extending results of Abreu and McDuff.

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Section: Remark 518 Making the Change Of Variablesmentioning

confidence: 86%

Compatible complex structures on symplectic rational ruled surfaces

Abreu¹,

Granja²,

Kitchloo³

2009

Duke Math. J.

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“…Very little is known about the homotopy groups of Diff + (M ) (even for the 4-sphere) while in certain cases it has turned out that Symp(M, ω) is accessible. Using the method of pseudoholomorphic curves, Gromov [10] proved that the identity component of the group Symp(S 2 × S 2 , ω 1 ) is homotopy equivalent to to SO(3) × SO (3). Here ω λ denotes the product symplectic form in which the first sphere S 1 × {p} has area equal to λ and the second {p} × S 2 has area equal to 1.…”

Section: Overview Of Resultsmentioning

confidence: 99%

“…This means that the Whitehead product [α, α] is zero. Thus we can consider the higher order Whitehead product [α, α, α] ⊂ π 5 (M ) which is defined as follows (see [3] for details). Let W ⊂ S 2 ×S 2 ×S 2 denote the fat wedge, that is it consists of triples with at least one coordinate at the base point.…”

Section: The Evaluation Map and Whitehead Productsmentioning

confidence: 99%

Homotopy properties of Hamiltonian group actions

Kędra

McDuff

2004

Geom. Topol.

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Consider a Hamiltonian action of a compact Lie group G on a compact symplectic manifold (M, ω) and let G be a subgroup of the diffeomorphism group Diff M . We develop techniques to decide when the maps on rational homotopy and rational homology induced by the classifying map BG → BG are injective. For example, we extend Reznikov's result for complex projective space CP n to show that both in this case and the case of generalized flag manifolds the natural map H * (BSU (n + 1)) → H * (BG) is injective, where G denotes the group of all diffeomorphisms that act trivially on cohomology. We also show that if λ is a Hamiltonian circle action that contracts in G := Ham(M, ω) then there is an associated nonzero element in π 3 (G) that deloops to a nonzero element of H 4 (BG). This result (as well as many others) extends to c-symplectic manifolds (M, a), ie, 2n-manifolds with a class a ∈ H 2 (M ) such that a n = 0. The proofs are based on calculations of certain characteristic classes and elementary homotopy theory.

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“…We begin by recalling the result in [1] concerning rational higher order Whitehead products. Fix positive integers n 1 , ..., n r .…”

Section: A Geometric Interpretation Of a Detective Elementmentioning

confidence: 99%

“…With the aid of results in [1], we shall show that a nonzero element in a higher order Whitehead product set is detective; see Theorem 6.1.…”

Section: Introductionmentioning

confidence: 99%

A function space model approach to the rational evaluation subgroups

2007

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Abstract. Let f : U → X be a map from a connected nilpotent space U to a connected rational space X. The evaluation subgroup G * (U, X; f ), which is a generalization of the Gottlieb group of X, is investigated. The key device for the study is an explicit Sullivan model for the connected component containing f of the function space of maps from U to X, which is derived from the general theory of such a model due to Brown and Szczarba [5]. In particular, we show that non Gottlieb elements are detected by analyzing a Sullivan model for the map f and by looking at non-triviality of higher order Whitehead products in the homotopy group of X. The Gottlieb triviality of a fibration in the sense of Lupton and Smith [27] is also discussed from the function space model point of view. Moreover, we proceed to consideration of the evaluation subgroup of the fundamental group of a nilpotent space. In consequence, the first Gottlieb group of the total space of each S 1 -bundle over the n-dimensional torus is determined explicitly in the non-rational case.

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Sullivan's Minimal Models and Higher Order Whitehead Products

Cited by 31 publications

References 3 publications

Compatible complex structures on symplectic rational ruled surfaces

Compatible complex structures on symplectic rational ruled surfaces

Homotopy properties of Hamiltonian group actions

A function space model approach to the rational evaluation subgroups

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