We investigate the existence of an –space structure on the function space, , of based maps in the component of the trivial map between two pointed connected CW–complexes and . For that, we introduce the notion of –space and prove that we have an –space structure on if is an –space and is of Lusternik–Schnirelmann category less than or equal to . When we consider the rational homotopy type of nilpotent finite type CW–complexes, the existence of an –space structure can be easily detected on the minimal model and coincides with the differential length considered by Y Kotani. When is finite, using the Haefliger model for function spaces, we can prove that the rational cohomology of is free commutative if the rational cup length of is strictly less than the differential length of , generalizing a recent result of Y Kotani.
"$H$–space structure on pointed mapping spaces." Algebr. Geom. Topol. 5 (2) 713 - 724, 2005. https://doi.org/10.2140/agt.2005.5.713