In algebraic topology, a rational homology -sphere is an -dimensional manifold with the same rational homology groups as the -sphere. These serve, among other things, to understand which information the rational homology groups of a space can or cannot measure and which attenuations result from neglecting torsion in comparison to the (integral) homology groups of the space.
Definition A rational homology -sphere is an -dimensional manifold with the same rational homology groups as the -sphere :
Properties
Every (integral) homology sphere is a rational homology sphere. Every simply connected rational homology -sphere with is homeomorphic to the -sphere.
Examples
The -sphere itself is obviously a rational homology -sphere. The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy -sphere, which is not a homotopy -sphere. The Klein bottle has two dimensions, but has the same rational homology as the -sphere as its (integral) homology groups are given by:Hatcher 02, Example 2.47., p. 151
Hence it is not a rational homology sphere, but would be if the requirement to be of same dimension was dropped.
The real projective space is a rational homology sphere for odd as its (integral) homology groups are given by:Hatcher 02, Example 2.42, S. 144
is the sphere in particular.
The five-dimensional Wu manifold is a simply connected rational homology sphere (with non-trivial homology groups , und ), which is not a homotopy sphere.
See also
Rational homotopy sphere
Literature
External links
rational homology sphere at the nLab
References