nLab Wu manifold

Definition

The Wu manifold is the coset space

$$W \coloneqq SU(3)/SO(3) \,,$$

namely the quotient space of the Lie group SU(3) by the right multiplication action of its subgroup SO(3).

(Crowley 11, Debray)

Properties

Proposition

$W$ is a simply connected rational $5$-homology sphere (with non-trivial homology groups $H_0(W)\cong\mathbb{Z}$, $H_2(W)\cong\mathbb{Z}_2$ and $H_5(W)\cong\mathbb{Z}$); but it is not homotopy 5-sphere.

Proposition

$W$ has the cohomology groups:

$$H^0(W;\mathbb{Z}_2) \cong\mathbb{Z}_2$$

$$H^1(W;\mathbb{Z}_2) \cong 1$$

$$H^2(W;\mathbb{Z}_2) \cong\mathbb{Z}_2$$

$$H^3(W;\mathbb{Z}_2) \cong\mathbb{Z}_2$$

$$H^4(W;\mathbb{Z}_2) \cong 1$$

$$H^5(W;\mathbb{Z}_2) \cong\mathbb{Z}_2$$

(Crowley 11)

Proposition

$W$ is a generator of the oriented bordism ring $\Omega_5^{\operatorname{SO}}$.

(Crowley 11, Debray)

Proposition

The Wu manifold is a spinʰ manifold, which does not allow a spinᶜ structure.

(MO/304471)

The Wu manifold:

• does not allow a stable complex structure, therefore no almost contact structure and therefore no contact structure
• has a non-vanishing deRham invariant and therefore a non-vanishing symmetric signature

References

• Diarmuid Crowley, 5-manifolds: 1-connected, In: Bulletin of the Manifold Atlas. 2011, p. 49–55, pdf
• Arun Debray, Characteristic classes, pdf

See also:

• Wikipedia, Wu manifold