Volume 5, issue 2 (2005)

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$H$–space structure on pointed mapping spaces

Yves Felix and Daniel Tanré

Algebraic & Geometric Topology 5 (2005) 713–724

arXiv: math.AT/0507147


We investigate the existence of an H–space structure on the function space, (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 (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 (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.

mapping spaces, Haefliger model, Lusternik–Schnirelmann category
Mathematical Subject Classification 2000
Primary: 55R80, 55P62, 55T99
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Received: 13 February 2005
Revised: 18 April 2006
Accepted: 30 June 2005
Published: 5 July 2005
Yves Felix
Département de Mathématiques
Université Catholique de Louvain
2, Chemin du Cyclotron
1348 Louvain-La-Neuve
Daniel Tanré
Département de Mathématiques
UMR 8524
Université de Lille 1
59655 Villeneuve d’Ascq Cedex