nLab
biquotient
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Contents
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:

In rational homotopy theory :

Vitali Kapovitch, A note on rational homotopy of biquotients (pdf )
Created on April 27, 2019 at 11:37:53.
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