Rational homology sphere

(Redirected from Rational homology 3-sphere)

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

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A rational homology  -sphere is an  -dimensional manifold   with the same rational homology groups as the  -sphere  :

 

Properties

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  • Every (integral) homology sphere is a rational homology sphere.
  • Every simply connected rational homology  -sphere with   is homeomorphic to the  -sphere.

Examples

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  • 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:[1]
     
     
     
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:[2][3]
     
  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

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Literature

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  • Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
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References

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  1. ^ Hatcher 02, Example 2.47., p. 151
  2. ^ Hatcher 02, Example 2.42, S. 144
  3. ^ "Homology of real projective space". Retrieved 2024-01-30.