Rational homotopy sphere explained

In algebraic topology, a rational homotopy

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

n -sphere is an n -dimensional manifold \Sigma with the same rational homotopy groups as the n -sphere S ^{n} :\pi _{k(\Sigma) ⊗ Q =\pi} ^{n) ⊗ Q \cong\begin{cases} Z} ^{k(S} & ;k = n if n even \\ Z & ;k = n, 2 n -1 if n odd \\ 1 & ; otherwise \end{cases}.

Properties

• Every (integral) homotopy sphere is a rational homotopy sphere.

Examples

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

• The Poincaré homology sphere is a rational homology

3 -sphere in particular.\R P ^{n} is a rational homotopy sphere for all n> 0 . The fiber bundle S ^{0 →} S ^{n → RP} ^{n} ^[1] yields with the long exact sequence of homotopy groups^[2] that \pi ^{n)} _{k(S} for k> 1 and n> 0 as well as \pi _{1(\R} P ^{1) =Z} and \pi _{1(\R} P ^{n) = Z} _{2} for n> 1 ,^[3] which vanishes after rationalization. \R P ^{1\cong} S ^{1} is the sphere in particular.

See also

• Rational homology sphere

External links

• rational homotopy sphere at the nLab

References

1. Hatcher 02, Example 4.44., p. 377
2. Hatcher 02, Theorem 4.41., p. 376
3. Web site: Homotopy of real projective space . 2024-01-31 . en.