## Algebraic & Geometric Topology

### $H$–space structure on pointed mapping spaces

#### 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.

#### Article information

Source
Algebr. Geom. Topol., Volume 5, Number 2 (2005), 713-724.

Dates
Revised: 18 April 2006
Accepted: 30 June 2005
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.agt/1513796427

Digital Object Identifier
doi:10.2140/agt.2005.5.713

Mathematical Reviews number (MathSciNet)
MR2153111

Zentralblatt MATH identifier
1082.55006

#### Citation

Félix, Yves; Tanre, Daniel. $H$–space structure on pointed mapping spaces. Algebr. Geom. Topol. 5 (2005), no. 2, 713--724. doi:10.2140/agt.2005.5.713. https://projecteuclid.org/euclid.agt/1513796427

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