In algebraic topology, a rational homotopy n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homotopy groups as the n {\displaystyle 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 {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold Σ {\displaystyle \Sigma } with the same rational homotopy groups as the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} :
π k ( Σ ) ⊗ Q = π k ( S n ) ⊗ Q ≅ { Z ; k = n if n even Z ; k = n , 2 n − 1 if n odd 1 ; otherwise . {\displaystyle \pi _{k}(\Sigma )\otimes \mathbb {Q} =\pi _{k}(S^{n})\otimes \mathbb {Q} \cong {\begin{cases}\mathbb {Z} &;k=n{\text{ if }}n{\text{ even}}\\\mathbb {Z} &;k=n,2n-1{\text{ if }}n{\text{ odd}}\\1&;{\text{otherwise}}\end{cases}}.}
Properties Every (integral) homotopy sphere is a rational homotopy sphere.
Examples The n {\displaystyle n} -sphere S n {\displaystyle S^{n}} itself is obviously a rational homotopy n {\displaystyle n} -sphere. The Poincaré homology sphere is a rational homology 3 {\displaystyle 3} -sphere in particular. The real projective space R P n {\displaystyle \mathbb {R} P^{n}} is a rational homotopy sphere for all n > 0 {\displaystyle n>0} . The fiber bundle S 0 → S n → R P n {\displaystyle S^{0}\rightarrow S^{n}\rightarrow \mathbb {R} P^{n}} yields with the long exact sequence of homotopy groups that π k ( R P n ) ≅ π k ( S n ) {\displaystyle \pi _{k}(\mathbb {R} P^{n})\cong \pi _{k}(S^{n})} for k > 1 {\displaystyle k>1} and n > 0 {\displaystyle n>0} as well as π 1 ( R P 1 ) = Z {\displaystyle \pi _{1}(\mathbb {R} P^{1})=\mathbb {Z} } and π 1 ( R P n ) = Z 2 {\displaystyle \pi _{1}(\mathbb {R} P^{n})=\mathbb {Z} _{2}} for n > 1 {\displaystyle n>1} , which vanishes after rationalization. R P 1 ≅ S 1 {\displaystyle \mathbb {R} P^{1}\cong S^{1}} is the sphere in particular.
See also Rational homology sphere
Literature Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
External links rational homotopy sphere at the nLab
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