An operadic model for a mapping space and its associated spectral sequence
type: Item
instance of: scholarly article
title: An operadic model for a mapping space and its associated spectral sequence
zbMATH Open document ID: 1123.55004
author: David Chataur
author: Katsuhiko Kuribayashi
published in: Journal of Pure and Applied Algebra
publication date: 2007-06-26T00:00:00Z
full work available at URL: http://hdl.handle.net/10091/257
review text: \textit{M. Mandel} has proved that the homotopy category of nilpotent, \(p\)-complete spaces of finite type is equivalent to a full subcategory of the homotopy category of algebras over an \(\overline{\mathbb{F}}_p\)-operad \({\mathcal E}\). Here \(\overline{\mathbb{F}}_p\) denotes the closure of the finite field \(\mathbb{F}_p\). In this paper the authors construct an \({\mathcal E}\)-algebra model for a mapping space \({\mathcal F}(X, Y)\) when \(X\) is a finite simplicial set, \(Y\) is a connected nilpotent simplicial set of finite type and the connectivity of \(Y\) is greater than or equal to the dimension of \(X\). In this case, they also construct a left-half plane spectral sequence with \(E_2\)-term expressed in terms of the left derived functor of the Lannes' division functor and converging to the cohomology of \({\mathcal F}(X, Y)\).
Mathematics Subject Classification ID: 55P48
Mathematics Subject Classification ID: 18D50
zbMATH DE Number: 5167547
zbMATH Keywords: operad
zbMATH Keywords: mapping space
reviewed by: Jean-Claude Thomas
MaRDI profile type: Publication
OpenAlex ID: W1967553470
cites work: Q3997914
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cites work: Combinatorial operad actions on cochains
cites work: Q4953194
cites work: A spectral sequence for string cohomology
cites work: On the Homology Spectral Sequence of a Cosimplicial Space
cites work: The homotopy spectral sequence of a space with coefficients in a ring
cites work: The rational homology of function spaces
cites work: On the rational homotopy type of function spaces
cites work: Generalized differential forms over an operand and algebraic models of fibrations
cites work: \(E_{\infty}\)-model for a mapping space
cites work: Rational Homotopy of the Space of Sections of a Nilpotent Bundle
cites work: Module derivations and cohomological splitting of adjoint bundles
cites work: Module derivations and the adjoint action of a finite loop space
cites work: Eilenberg-Moore spectral sequence calculation of function space cohomology
cites work: \(E_{\infty}\) algebras and \(p\)-adic homotopy theory
cites work: On the cohomology of the classifying space of the gauge group over some 4-complexes
cites work: Q4528987
cites work: Multivariable cochain operations and little 𝑛-cubes
cites work: Q5653373
cites work: Homotopical algebra
cites work: Steenrod operations in the Eilenberg-Moore spectral sequence
cites work: Q4875306
cites work: Homological Algebra and the Eilenberg-Moore Spectral Sequence
DOI: 10.1016/J.JPAA.2006.09.008
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description: scientific article; zbMATH DE number 5167547