nLab
biquotient

Contents

Context

Group Theory

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

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 GG by the action of two subgroups H 1,H 2GH_1, H_2 \subset G, hence by the action of their direct product group H 1×H 2H_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\G/H 2G/(H 1×H 2 op)G/(gh 1gh 2|h iH i). 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 2GH_1, H_2 \subset G, such as that H iGH_i \subset G are closed subgroups and notably that the induced action of H 1H_1 on the single quotient space/coset space G/H 2G/H_2 of the other, is still free.

More generally, one can consider biquotients of GG 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)

Created on April 27, 2019 at 11:37:53. See the history of this page for a list of all contributions to it.