Rational homotopy sphere

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 yields with the long exact sequence of homotopy groups 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

• at the nLab

References