- Source: Wu manifold
In mathematics, the Wu manifold is a 5-manifold defined as a quotient space of Lie groups appearing in the mathematical area of Lie theory. Due to its special properties it is of interest in algebraic topology, cobordism theory and spin geometry. The manifold was first studied and named after Wu Wenjun.
Definition
The special orthogonal group
SO
(
n
)
{\displaystyle \operatorname {SO} (n)}
embeds canonically in the special unitary group
SU
(
n
)
{\displaystyle \operatorname {SU} (n)}
. The orbit space:
W
:=
SU
(
3
)
/
SO
(
3
)
{\displaystyle W:=\operatorname {SU} (3)/\operatorname {SO} (3)}
is the Wu manifold.
Properties
W
{\displaystyle W}
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.
W
{\displaystyle W}
has the cohomology groups:
H
0
(
W
;
Z
2
)
=
Z
2
{\displaystyle H^{0}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}}
H
1
(
W
;
Z
2
)
=
1
{\displaystyle H^{1}(W;\mathbb {Z} _{2})=1}
H
2
(
W
;
Z
2
)
=
Z
2
{\displaystyle H^{2}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}}
H
3
(
W
;
Z
2
)
=
Z
2
{\displaystyle H^{3}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}}
H
4
(
W
;
Z
2
)
=
1
{\displaystyle H^{4}(W;\mathbb {Z} _{2})=1}
H
5
(
W
;
Z
2
)
=
Z
2
{\displaystyle H^{5}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}}
W
{\displaystyle W}
is a generator of the oriented cobordism ring
Ω
5
SO
≅
Z
2
{\displaystyle \Omega _{5}^{\operatorname {SO} }\cong \mathbb {Z} _{2}}
.
W
{\displaystyle W}
is a
Spin
h
{\displaystyle \operatorname {Spin} ^{h}}
manfold, which doesn't allow a
Spin
c
{\displaystyle \operatorname {Spin} ^{c}}
structure.
References
External links
Wu manifold on nLab
Kata Kunci Pencarian:
- Mazda RX-7
- Daftar masalah matematika yang belum terpecahkan
- Wu manifold
- 5-manifold
- Stiefel–Whitney class
- Rational homology sphere
- Fiber bundle
- Whitney embedding theorem
- Diffeology
- Pontryagin class
- Intersection form of a 4-manifold
- Michael Freedman
