In algebraic topology, a rational homology n {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold with the same rational homology groups as the n {\displaystyle 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 {\displaystyle n} -sphere is an n {\displaystyle n} -dimensional manifold Σ {\displaystyle \Sigma } with the same rational homology groups as the n {\displaystyle n} -sphere S n {\displaystyle S^{n}} :
H k ( Σ , Q ) = H k ( S n , Q ) ≅ { Q ; k = 0 or k = n 1 ; otherwise . {\displaystyle H_{k}(\Sigma ,\mathbb {Q} )=H_{k}(S^{n},\mathbb {Q} )\cong {\begin{cases}\mathbb {Q} &;k=0{\text{ or }}k=n\\1&;{\text{otherwise}}\end{cases}}.}
Properties Every (integral) homology sphere is a rational homology sphere. Every simply connected rational homology n {\displaystyle n} -sphere with n ≤ 4 {\displaystyle n\leq 4} is homeomorphic to the n {\displaystyle n} -sphere.
Examples The n {\displaystyle n} -sphere S n {\displaystyle S^{n}} itself is obviously a rational homology n {\displaystyle n} -sphere. The pseudocircle (for which a weak homotopy equivalence from the circle exists) is a rational homotopy 1 {\displaystyle 1} -sphere, which is not a homotopy 1 {\displaystyle 1} -sphere. The Klein bottle has two dimensions, but has the same rational homology as the 1 {\displaystyle 1} -sphere as its (integral) homology groups are given by:
H 0 ( K ) ≅ Z {\displaystyle H_{0}(K)\cong \mathbb {Z} }
H 1 ( K ) ≅ Z ⊕ Z 2 {\displaystyle H_{1}(K)\cong \mathbb {Z} \oplus \mathbb {Z} _{2}}
H 2 ( K ) ≅ 1 {\displaystyle 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. The real projective space R P n {\displaystyle \mathbb {R} P^{n}} is a rational homology sphere for n {\displaystyle n} odd as its (integral) homology groups are given by:
H k ( R P n ) ≅ { Z ; k = 0 or k = n if odd Z 2 ; k odd , 0 < k < n 1 ; otherwise . {\displaystyle H_{k}(\mathbb {R} P^{n})\cong {\begin{cases}\mathbb {Z} &;k=0{\text{ or }}k=n{\text{ if odd}}\\\mathbb {Z} _{2}&;k{\text{ odd}},0<k<n\\1&;{\text{otherwise}}\end{cases}}.}
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