#### Algebraic and Geometric Topology 5 (2005),
paper no. 29, pages 713-724.

## H-space structure on pointed mapping spaces

### Yves Felix and Daniel Tanre

**Abstract**.
We investigate the existence of an H-space structure on the function
space, F_*(X,Y,*), of based maps in the component of the trivial map
between two pointed connected CW-complexes X and Y. For that, we
introduce the notion of H(n)-space and prove that we have an H-space
structure on F_*(X,Y,*) if Y is an H(n)-space and X is of
Lusternik-Schnirelmann category less than or equal to n. When we
consider the rational homotopy type of nilpotent finite type
CW-complexes, the existence of an H(n)-space structure can be easily
detected on the minimal model and coincides with the differential
length considered by Y. Kotani. When X is finite, using the Haefliger
model for function spaces, we can prove that the rational cohomology
of F_*(X,Y,*) is free commutative if the rational cup length of X is
strictly less than the differential length of Y, generalizing a recent
result of Y. Kotani.
**Keywords**.
Mapping spaces, Haefliger model, Lusternik-Schnirelmann category

**AMS subject classification**.
Primary: 55R80, 55P62, 55T99.

**E-print:** `arXiv:math.AT/0507147`

**DOI:** 10.2140/agt.2005.5.713

Submitted: 13 February 2005.
Accepted: 30 June 2005.
Published: 5 July 2005.

Notes on file formats
Yves Felix Daniel Tanre

Departement de Mathematiques, Universite Catholique de Louvain

2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium

and

Departement de Mathematiques, UMR 8524, Universite de Lille 1

59655 Villeneuve d'Ascq Cedex, France

Email: felix@math.ucl.ac.be, Daniel.Tanre@univ-lille1.fr

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