Algebraic & Geometric Topology

$H$–space structure on pointed mapping spaces

Yves Félix and Daniel Tanre

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We investigate the existence of an H–space structure on the function space, (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 (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 (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

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

Received: 13 February 2005
Revised: 18 April 2006
Accepted: 30 June 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55R80: Discriminantal varieties, configuration spaces 55P62: Rational homotopy theory 55T99: None of the above, but in this section

mapping spaces Haefliger model Lusternik–Schnirelmann category


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.

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