- Source: Rational homology sphere
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
)
≅
{
Z
;
k
=
0
or
k
=
n
1
;
otherwise
.
{\displaystyle H_{k}(\Sigma ,\mathbb {Q} )=H_{k}(S^{n},\mathbb {Q} )\cong {\begin{cases}\mathbb {Z} &;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
R
P
1
≅
S
1
{\displaystyle \mathbb {R} P^{1}\cong S^{1}}
is the sphere in particular.
The five-dimensional Wu manifold
W
=
SU
(
3
)
/
SO
(
3
)
{\displaystyle W=\operatorname {SU} (3)/\operatorname {SO} (3)}
is a simply connected rational homology sphere (with non-trivial homology groups
H
0
(
W
)
≅
Z
{\displaystyle H_{0}(W)\cong \mathbb {Z} }
,
H
2
(
W
)
≅
Z
2
{\displaystyle H_{2}(W)\cong \mathbb {Z} _{2}}
und
H
5
(
W
)
≅
Z
{\displaystyle H_{5}(W)\cong \mathbb {Z} }
), which is not a homotopy sphere.
See also
Rational homotopy sphere
Literature
Hatcher, Allen (2002), Algebraic Topology, Cambridge University Press, ISBN 0-521-79540-0
External links
rational homology sphere at the nLab
References
Kata Kunci Pencarian:
- Daftar masalah matematika yang belum terpecahkan
- Homology sphere
- Rational homology sphere
- Casson invariant
- Rational homotopy sphere
- Seifert–Weber space
- Simplicial sphere
- N-sphere
- Floer homology
- Rational homotopy theory
- Euler characteristic
