A fibre-restricted Gottlieb group and its rational realization problem

Yamaguchi, Toshihiro

doi:10.1016/j.topol.2015.05.066

Cited by 2 publications

(12 citation statements)

References 25 publications

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“…The identification given in Theorem 2.5 is natural with respect to maps induced by pull-backs of fibrations [17,Pro.1.5]. A map f :…”

Section: Derivations and Fibrewise Self-equivalencesmentioning

confidence: 99%

“…Given spaces X and Y , the product fibration implies the relation: [17,Lem.1.13]). In particular, Changing the degree of just one generator in Example 3.6 gives a more complicated example: We show the full poset of evaluation subgroups G * (ξ; X Q ) is isomorphic to P 9 × Z 2 where P 9 ⊆ Z 4 2 has Hasse diagram: P 9 (1, 1, 1, 1) r r r r r r r r r r…”

Section: The Evaluation Subgroups Of the Classifying Spacementioning

confidence: 99%

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The universal fibration with fibre X in rational homotopy theory

Lupton

Smith

2020

J. Homotopy Relat. Struct.

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Let X be a simply connected space with finite-dimensional rational homotopy groups. Let p∞ : U E → Baut 1 (X) be the universal fibration of simply connected spaces with fibre X. We give a DG Lie algebra model for the evaluation map ω : aut 1 (Baut 1 (X Q )) → Baut 1 (X Q ) expressed in terms of derivations of the relative Sullivan model of p∞. We deduce formulas for the rational Gottlieb group and for the evaluation subgroups of the classifying space Baut 1 (X Q ) as a consequence. We also prove that CP n Q cannot be realized as Baut 1 (X Q ) for n ≤ 4 and X with finite-dimensional rational homotopy groups. Introduction.Given a simply connected CW complex X of finite type, let aut 1 (X) denote the space of self-maps of X homotopic to the identity map. The group-like space aut 1 (X) has a classifying space Baut 1 (X). The space Baut 1 (X) appears as the base space of the universal example p ∞ : U E → Baut 1 (X) of a fibration of simply connected CW complexes with fibre of the homotopy type of X [13,2,9].The classifying space Baut 1 (X) offers a computational challenge in homotopy theory. When X is a finite complex, Baut 1 (X) is of CW type (albeit, generally infinite) and satisfies the localization identity Baut 1 (X P ) ≃ Baut 1 (X) P for any collection of primes by work of May [9, 10]. In rational homotopy theory, models for Baut 1 (X Q ) are due to Sullivan,12,15]. The study of the classifying space using these models is an area of continued activity (see, e.g., [8,17,16]).We say a space X is π-finite if X is a simply connected CW complex and dim π * (X Q ) < ∞. A π-finite space X has a finitely generated Sullivan minimal model ∧ (V ; d). If X is a π-finite space then Baut 1 (X Q ) is one also (Proposition 2.3, below). Consequently, we may iterate the classifying space construction for πfinite rational spaces. Our first result here describes the passage from Baut 1 (X Q ) to aut 1 (Baut 1 (X Q )) in the setting of derivations of Sullivan models. We describe this result briefly now, with fuller definitions in Section 2. The relative Sullivan model for the universal fibration p ∞ : U E → Baut 1 (X) with fibre X a π-finite space is an inclusion of DG algebras. We write this model throughout as:2010 Mathematics Subject Classification. Primary: 55P62 55R15; Secondary: 55P10.

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“…The identification given in Theorem 2.5 is natural with respect to maps induced by pull-backs of fibrations [17,Pro.1.5]. A map f :…”

Section: Derivations and Fibrewise Self-equivalencesmentioning

confidence: 99%

Section: The Evaluation Subgroups Of the Classifying Spacementioning

confidence: 99%

The universal fibration with fibre X in rational homotopy theory

Lupton

Smith

2020

J. Homotopy Relat. Struct.

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“…This approach is motivated by the previous work [33] of the first named author. Using the base point b we have the evaluation map ω : aut 1 X → X defined by ω(h) := h(b) for h ∈ aut 1 X , thus the above map π * (resξ) Q : π * (aut 1 p) Q → π * (aut 1 X ) Q can be extended to the following sequence:…”

mentioning

confidence: 99%

“…Here Im(π n (ω)) is a well-known group called the n-th Gottlieb group of X , denoted by G n (X ), introduced by Gottlieb [12,13]. In this sense, Im(π n (ω • resξ)) is named the n-th fiber-restricted Gottlieb group of X with respect to the fibration ξ and denoted by G ξ n (X ) in [33]. Two fibrations ξ 1 and ξ 2 are said to be Q-Gottlieb-group-equivalent provided that G [24,33], a certain poset on X is introduced by using the inclusion of the fiber-restricted Gottlieb groups.…”

mentioning

confidence: 99%

“…In this sense, Im(π n (ω • resξ)) is named the n-th fiber-restricted Gottlieb group of X with respect to the fibration ξ and denoted by G ξ n (X ) in [33]. Two fibrations ξ 1 and ξ 2 are said to be Q-Gottlieb-group-equivalent provided that G [24,33], a certain poset on X is introduced by using the inclusion of the fiber-restricted Gottlieb groups. This Q-Gottlieb-groupequivalence of fibrations is coarser because the Gottlieb group G n (X ) Q is identified as a subgroup of π n (aut 1 X )…”

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confidence: 99%

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Fiber-homotopy self-equivalences and a classification of fibrations in rational homotopy

Yamaguchi

Yokura

2016

J. Homotopy Relat. Struct.

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For a fibration ξ : X i → E p → B we consider the image of the rationalized homotopy group homomorphism π * (res ξ) Q : π * (aut 1 p) Q → π * (aut 1 X) Q obtained from the fibre-restricting map res ξ : aut 1 p → aut 1 X. Then we consider a finite type classification of fibrations ξ with fibre X and base B. In particular, we measure the size of it by a rational homotopical invariant "depth B X" when X are certain homogeneous spaces and B are spheres.

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A fibre-restricted Gottlieb group and its rational realization problem

Cited by 2 publications

References 25 publications

The universal fibration with fibre X in rational homotopy theory

The universal fibration with fibre X in rational homotopy theory

Fiber-homotopy self-equivalences and a classification of fibrations in rational homotopy

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