Journal of Homotopy and Related Structures

, Volume 11, Issue 2, pp 309–332 | Cite as

Homotopy transfer and rational models for mapping spaces

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Abstract

By using homotopy transfer techniques in the context of rational homotopy theory, we show that if \(C\) is a coalgebra model of a space \(X\), then the \(A_\infty \)-coalgebra structure in \(H_*(X;\mathbb {Q})\cong H_*(C)\) induced by the higher Massey coproducts provides the construction of the Quillen minimal model of \(X\). We also describe an explicit \(L_\infty \)-structure on the complex of linear maps \({{\mathrm{Hom}}}(H_*(X; \mathbb {Q}), \pi _*(\Omega Y)\otimes \mathbb {Q})\), where \(X\) is a finite nilpotent CW-complex and \(Y\) is a nilpotent CW-complex of finite type, modeling the rational homotopy type of the mapping space \(\text {map}(X, Y)\). As an application we give conditions on the source and target in order to detect rational \(H\)-space structures on the components.

Keywords

Rational homotopy \(L_\infty \)-algebra Mapping space 

Mathematics Subject Classification

Primary 55P62 Secondary 54C35 

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Copyright information

© Tbilisi Centre for Mathematical Sciences 2015

Authors and Affiliations

  1. 1.Departamento de Álgebra, Geometría y TopologíaUniversidad de MálagaMálagaSpain
  2. 2.Institute for Mathematics, Astrophysics, and Particle PhysicsRadboud Universiteit NijmegenNijmegenThe Netherlands

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