Rational homology sphere explained

In algebraic topology, a rational homology

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

n -sphere is an n -dimensional manifold \Sigma with the same rational homology groups as the n -sphere S ^{n} :H _{k(\Sigma,Q) =H} ^{n, Q ) \cong\begin{cases} Z} ^{k(S} & ;k =0 or k = n \\ 1 & ; otherwise \end{cases}.

Properties

• Every (integral) homology sphere is a rational homology sphere.
• Every simply connected rational homology

n -sphere with n\leq 4 is homeomorphic to the n -sphere.

Examples

S ^{n} itself is obviously a rational homology n -sphere.

• The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy

1 -sphere, which is not a homotopy 1 -sphere.

• The Klein bottle has two dimensions, but has the same rational homology as the 1 -sphere as its (integral) homology groups are given by:^[1]

H _{0(K) \congZ} H _{1(K) \congZ ⊕ Z} _{2} H _{2(K) \cong} 1

Hence it is not a rational homology sphere, but would be if the requirement to be of same dimension was dropped.

\R P ^{n} is a rational homology sphere for n odd as its (integral) homology groups are given by:^{[2] [3]} H _{k(\R} P ^{n) \cong\begin{cases} Z} & ;k =0 or k = n ifodd \\ Z _{2} & ;k odd , 0 <k<n \\ 1 & ; otherwise \end{cases}. \R P ^{1\cong} S ^{1} is the sphere in particular.W = \operatorname{SU}( 3 )/\operatorname{SO}( 3 ) is a simply connected rational homology sphere (with non-trivial homology groups H _{0(W)\congZ} , H _{2(W)\congZ} _{2} und H _{5(W)\congZ} ), which is not a homotopy sphere.

See also

• Rational homotopy sphere

External links

• rational homology sphere at the nLab

References

1. Hatcher 02, Example 2.47., p. 151
2. Hatcher 02, Example 2.42, S. 144
3. Web site: Homology of real projective space . 2024-01-30 . en.