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 $\operatorname {SO} (n)$ embeds canonically in the special unitary group $\operatorname {SU} (n)$ . The orbit space:

W:=\operatorname {SU} (3)/\operatorname {SO} (3)

is the Wu manifold.^[1]^[2]

$W$ is a simply connected rational homology sphere (with non-trivial homology groups $H_{0}(W)\cong \mathbb {Z}$ , $H_{2}(W)\cong \mathbb {Z} _{2}$ und $H_{5}(W)\cong \mathbb {Z}$ ), which is not a homotopy sphere.
$W$ has the cohomology groups:^[1]
$H^{0}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}$

$H^{1}(W;\mathbb {Z} _{2})=1$

$H^{2}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}$

$H^{3}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}$

$H^{4}(W;\mathbb {Z} _{2})=1$

$H^{5}(W;\mathbb {Z} _{2})=\mathbb {Z} _{2}$
$W$ is a generator of the oriented cobordism ring $\Omega _{5}^{\operatorname {SO} }\cong \mathbb {Z} _{2}$ .^[1]^[2]
$W$ is a $\operatorname {Spin} ^{h}$ manfold, which doesn't allow a $\operatorname {Spin} ^{c}$ structure.

^ ^a ^b ^c Diamuid Crowley (2011). "5-manifolds: 1-connected" (PDF). Bulletin of the Manifold Atlas. pp. 49–55. Retrieved 2024-06-10.
^ ^a ^b Debray, Arun. "Characteristic classes" (PDF). Archived from the original (PDF) on 2021-01-07. Retrieved 2024-06-09.