#### 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
##### Abstract

We investigate the existence of an $H$–space structure on the function space, ${\mathsc{ℱ}}_{\ast }\left(X,Y,\ast \right)$, 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 $H\left(n\right)$–space and prove that we have an $H$–space structure on ${\mathsc{ℱ}}_{\ast }\left(X,Y,\ast \right)$ if $Y$ is an $H\left(n\right)$–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 $H\left(n\right)$–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 ${\mathsc{ℱ}}_{\ast }\left(X,Y,\ast \right)$ 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