In algebraic topology, a rational homotopy -sphere is an -dimensional manifold with the same rational homotopy groups as the -sphere. These serve, among other things, to understand which information the rational homotopy groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homotopy groups of the space.
Definition A rational homotopy -sphere is an -dimensional manifold with the same rational homotopy groups as the -sphere :
Properties
Every (integral) homotopy sphere is a rational homotopy sphere.
Examples
The -sphere itself is obviously a rational homotopy -sphere. The Poincaré homology sphere is a rational homology -sphere in particular. The real projective space is a rational homotopy sphere for all . The fiber bundle Hatcher 02, Example 4.44., p. 377 yields with the long exact sequence of homotopy groupsHatcher 02, Theorem 4.41., p. 376 that for and as well as and for , which vanishes after rationalization. is the sphere in particular.
See also
Rational homology sphere
Literature
External links
rational homotopy sphere at the nLab
References