H–space structure on pointed mapping spaces

Félix, Yves; Tanre, Daniel

doi:10.2140/agt.2005.5.713

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$H$–space structure on pointed mapping spaces

Yves Felix and Daniel Tanré

Algebraic & Geometric Topology 5 (2005) 713–724

DOI: 10.2140/agt.2005.5.713

arXiv: math.AT/0507147

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

Mathematical Subject Classification 2000

Primary: 55R80, 55P62, 55T99

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Publication

Received: 13 February 2005

Revised: 18 April 2006

Accepted: 30 June 2005

Published: 5 July 2005

Authors

Yves Felix
Département de Mathématiques Université Catholique de Louvain 2, Chemin du Cyclotron 1348 Louvain-La-Neuve Belgium

Daniel Tanré
Département de Mathématiques UMR 8524 Université de Lille 1 59655 Villeneuve d’Ascq Cedex France

Volume 5, issue 2 (2005)