nLab Aloff-Wallach space

Idea

Aloff-Wallach spaces (also Aloff-Wallach manifold or Aloff-Wallach quotients) are special 7-manifolds obtained as quotient spaces of the third special unitary group SU(3) by various free group actions of the first unitary group U(1). A similar construction using a biquotient space of different actions results in the related Eschenburg spaces. Both are related to the more well-known lens spaces and are of interest in Riemannian geometry.

Definition

Let $p,q\in\mathbb{Z}$ be two integers. U(1) acts on SU(3) by:

$$z*A =A diag(z^p,z^q,z^{-(p+q)}).$$

If the action is free, then the quotient space $W_{p,q}\coloneqq SU(3)/U(1)$ is a smooth 7-manifold, called Aloff-Wallach space.

References

Named after the original discussion in:

• Simon Aloff, Nolan Wallach, An infinite family of distinct 7–manifolds admitting positively curved Riemannian structures, Bull. Amer. Math. Soc. 81, pp. 93–97 (1975) [doi:10.1090/S0002-9904-1975-13649-4]