# The rational homotopy Lie algebra of function spaces

@article{Martn2008TheRH, title={The rational homotopy Lie algebra of function spaces}, author={Urtzi Buijs Mart{\'i}n and Aniceto Murillo Mas}, journal={Commentarii Mathematici Helvetici}, year={2008}, volume={83}, pages={723-739} }

In this paper we fully describe the rational homotopy Lie algebra of any component of a given (free or pointed) function space. Also, we characterize higher order Whitehead products on these spaces. From this, we deduce the existence of H-structures on a given component of a pointed mapping space F*(X,Y;f) between rational spaces, assuming the cone length of X is smaller than the order of any non trivial generalized Whitehead product in p*(Y)

## 26 Citations

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## References

SHOWING 1-10 OF 18 REFERENCES

Note on the rational cohomology of the function space of based maps

- Mathematics
- 2004

In this paper, for a formal, path connected, finite-dimensional CW-complex X of finite type and a q-connected space Y of finite type with q > dimX, we determine the necessary and sucient condition…

Rational Homotopy Theory

- Mathematics
- 2000

Sullivan Model and Rationalization of a Non-Simply Connected Space Homotopy Lie Algebra of a Space and Fundamental Group of the Rationalization, Model of a Fibration Holonomy Operation in a Fibration…

On the rational homotopy type of function spaces

- Mathematics
- 1997

The main result of this paper is the construction of a minimal model for the function space F(X,Y) of continuous functions from a finite type, finite dimensional space X to a finite type, nilpotent…

Rational category of the space of sections of a nilpotent bundle

- Mathematics
- 1990

Denote by ζ :F →E→pB a nilpotent fibration whereF is a 1-connected space of finite category andB a finite c.w. complex with non trivial rational cohomology. In this note we compute the rational…

Rationalized evaluation subgroups of a map I: Sullivan models, derivations and G-sequences

- Mathematics
- 2007

Rank of the fundamental group of any component of a function space

- Mathematics
- 2005

We compute the rank of the fundamental group of any connected component of the space map(X, Y) for X and Y connected, nilpotent CW complexes of finite type with X finite. For the component…

Sullivan's Minimal Models and Higher Order Whitehead Products

- MathematicsCanadian Journal of Mathematics
- 1978

The theory of minimal models, as developed by Sullivan [6; 8; 16] gives a method of computing the rational homotopy groups of a space X (that is, the homotopy groups of X tensored with the additive…

RATIONAL HOMOTOPY OF THE SPACE OF SECTIONS OF A NILPOTENT BUNDLE

- Mathematics
- 1982

We show that an algebraic construction proposed by Sullivan is indeed a model for the rational homotopy type of the space of sections of a nilpotent bundle. In his paper Uhomologie des espaces…

Basic constructions in rational homotopy theory of function spaces

- Mathematics
- 2006

Moyennant le foncteur de realisation de Bousfield-Gugenheim et a l'aide du modele de Brown Szczarba d'un espace de fonctions comme point de depart, on decrit les objets basiques et les applications…

Sur l'homotopie rationnelle des espaces fonctionnels

- Mathematics
- 1986

AbstractLet X be a nilpotent space such that it exists k⩾1 with Hp (X,ℚ) = 0 p > k and Hk (X,ℚ) ≠ 0, let Y be a (m−1)-connected space with m⩾k+2, then the rational homotopy Lie algebra of YX (resp.…