Fibrewise stable rational homotopy

Félix, Yves; Murillo, Aniceto; Tanré, Daniel

doi:10.1112/jtopol/jtq023

Cited by 5 publications

(7 citation statements)

References 13 publications

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“…It is easy to check that the geometrical realization of (Λ(x, y), d) has the homotopy type of S 0 . Moreover, consider as in [13] the CDGA model of S 0 given by Qα ⊕ Qβ with α and β idempotent elements of degree 0, α 2 = α, β 2 = β, with αβ = 0. Note that the identity in this algebra is α + β.…”

Section: The Maurer-cartan Functor and Cdga Morphismsmentioning

confidence: 99%

Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

Buijs

Murillo

2013

Advances in Mathematics

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Section: The Maurer-cartan Functor and Cdga Morphismsmentioning

confidence: 99%

Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

Buijs

Murillo

2013

Advances in Mathematics

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“…An interesting consequence of Theorem 1.5 is the classification of rational homotopy classes of fibrewise stable maps in terms of Ext and Coext groups. This allows for the computation of rationalised twisted cohomology purely in terms of algebraic data, subsuming and contextualising previous results obtained using Sullivan's approach to rational homotopy theory [FMT10]. Under some additional hypotheses, rational homotopy classes of fibrewise stable maps can be computed either by means of a hyper-Ext or a coalgebraic twisted Atiyah-Hirzebruch spectral sequence.…”

Section: Introductionmentioning

confidence: 71%

Strict algebraic models for rational parametrised spectra I

Braunack-Mayer

2019

Preprint

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Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space X there is a dg Lie algebra Λ X and a (coassociative cocommutative) dg coalgebra C X that model the rational homotopy type. In this article, we prove that the rational homotopy type of an X-parametrised spectrum is completely encoded by a Λ X -representation and also by a C X -comodule. The correspondence between rational parametrised spectra and algebraic data is obtained by means of symmetric monoidal equivalences of homotopy categories that vary pseudofunctorially in the simply-connected parameter space X.Our results establish a comprehensive dictionary enabling the translation of topological constructions into homological algebra using Lie representations and comodules, and conversely. For example, the fibrewise smash product of parametrised spectra is encoded by the derived tensor product of dg Lie representations and also by the derived cotensor product of dg comodules. As an application, we obtain algebraic descriptions of rational homotopy classes of fibrewise stable maps, providing new tools for the study of spaces of sections.

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“…Extending classical results of Eilenberg and Moore, we show that Theorem 4.20 identifies the fibrewise smash product of X-spectra with the derived tensor product of A-modules in rational homotopy theory. Generalising the main result of [FMT10], we show that rational homotopy classes of fibrewise stable maps between X-spectra are computed as Ext-groups of A-modules and we use this result to derive spectral sequences for computing fibrewise stable maps.…”

Section: Introductionmentioning

confidence: 77%

“…We make this change for two related reasons: upper indices in homotopy theory really ought to be "cohomologically graded", and we wish to be consistent with existing notation (e.g. in [CJ98,FMT10]).…”

Section: Notation and Conventionsmentioning

confidence: 99%

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Strict algebraic models for rational parametrised spectra II

Braunack-Mayer¹

2020

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In this article, we extend Sullivan's PL de Rham theory to obtain simple algebraic models for the rational homotopy theory of parametrised spectra. This simplifies and complements the results of Strict algebraic models for rational parametrised spectra I, which are based on Quillen's rational homotopy theory.According to Sullivan, the rational homotopy type of a nilpotent space X with finite Betti numbers is completely determined by a commutative differential graded algebra A modelling the cup product on rational cohomology. In this article we extend this correspondence between topology and algebra to parametrised stable homotopy theory: for a space X corresponding to the cdga A, we prove an equivalence between specific rational homotopy categories for parametrised spectra over X and for differential graded A-modules. While not full, the rational homotopy categories we consider contain a large class of parametrised spectra. The simplicity of the approach that we develop enables direct calculations in parametrised stable homotopy theory using differential graded modules.To illustrate the usefulness of our approach, we build a comprehensive dictionary of algebraic translations of topological constructions; providing algebraic models for base change functors, fibrewise stabilisations, parametrised Postnikov sections, fibrewise smash products, and complexes of fibrewise stable maps.

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Fibrewise stable rational homotopy

Cited by 5 publications

References 13 publications

Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

Algebraic models of non-connected spaces and homotopy theory ofL∞algebras

Strict algebraic models for rational parametrised spectra I

Strict algebraic models for rational parametrised spectra II

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