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
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:
- :
- :
- :
- :
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:
- :
- :
is the sphere in particular.
See also
Literature
External links
References