Geodesic
H–space structure on pointed mapping spaces
Félix, Yves1 ; Tanre, Daniel2
1Département de Mathématiques, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium
2Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France
Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 713-724
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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.

DOI : 10.2140/agt.2005.5.713
Keywords: mapping spaces, Haefliger model, Lusternik–Schnirelmann category

Félix, Yves  1   ; Tanre, Daniel  2

1 Département de Mathématiques, Université Catholique de Louvain, 2, Chemin du Cyclotron, 1348 Louvain-La-Neuve, Belgium
2 Département de Mathématiques, UMR 8524, Université de Lille 1, 59655 Villeneuve d’Ascq Cedex, France 
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Félix, Yves; Tanre, Daniel. H–space structure on pointed mapping spaces. Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 713-724. doi: 10.2140/agt.2005.5.713

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