nLab biquotient

Idea

In group theory, but particularly in Lie group-theory, the term “biquotient” tends to mean the quotient space of a topological group or Lie group $G$ by the action of two subgroups $H_1, H_2 \subset G$, hence by the action of their direct product group $H_1 \times H_2$, one factor regarded as acting by group multiplication from the left, the other (more precisely: its opposite) acting by multiplication from the right.

This is typically and suggestively denoted as

$$H_1 \backslash G / H_2 \;\coloneqq\; G/( H_1 \times H_2^{op} ) \;\coloneqq\; G/( g \sim h_1 \cdot g \cdot h_2 \vert h_i \in H_i ) \,.$$

Another way to think of a biquotient is as a double coset space, see there for more.

Typically extra conditions are imposed on $H_1, H_2 \subset G$, such as that $H_i \subset G$ are closed subgroups and notably that the induced action of $H_1$ on the single quotient space/coset space $G/H_2$ of the other, is still free.

More generally, one can consider biquotients of $G$ by subgroups of the direct product group (e.g. Kapovitch).

Examples

Gromoll-Meyer sphere

The Gromoll-Meyer sphere – an exotic 7-sphere – arises as the biquotient of Sp(2) by two copies of Sp(1).

References

For instance:

• Burt Totaro, Cheeger manifolds and the classification of biquotients (arXiv:math/0210247)

In rational homotopy theory:

• Vitali Kapovitch, A note on rational homotopy of biquotients (pdf)