nLab Wu manifold

Contents

Context

Spheres

Manifolds and cobordisms

Contents

Definition

The Wu manifold is the coset space

WSU(3)/SO(3), 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

WW is a simply connected rational 5 5 -homology sphere (with non-trivial homology groups H 0(W)H_0(W)\cong\mathbb{Z}, H 2(W) 2H_2(W)\cong\mathbb{Z}_2 and H 5(W)H_5(W)\cong\mathbb{Z}); but it is not homotopy 5-sphere.

Proposition

WW has the cohomology groups:

H 0(W; 2) 2 H^0(W;\mathbb{Z}_2) \cong\mathbb{Z}_2
H 1(W; 2)1 H^1(W;\mathbb{Z}_2) \cong 1
H 2(W; 2) 2 H^2(W;\mathbb{Z}_2) \cong\mathbb{Z}_2
H 3(W; 2) 2 H^3(W;\mathbb{Z}_2) \cong\mathbb{Z}_2
H 4(W; 2)1 H^4(W;\mathbb{Z}_2) \cong 1
H 5(W; 2) 2 H^5(W;\mathbb{Z}_2) \cong\mathbb{Z}_2

(Crowley 11)

Proposition

WW is a generator of the oriented bordism ring Ω 5 SO\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:

Last revised on July 19, 2024 at 13:36:59. See the history of this page for a list of all contributions to it.