H-space structure on pointed mapping spaces

Details H-space structure on pointed mapping spaces H-space structure on pointed mapping spaces

2005-07-07

arxiv.org/abs/math/0507147v1

Algebr. Geom. Topol. 5 (2005) 713-724

Mathematics

Algebraic Topology

Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-29.abs.html

Scientific paper

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.

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