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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