nLab orbit type

Idea

In discussion of topological G-spaces (such as in equivariant homotopy theory and/or equivariant differential topology) by an orbit type one means the isomorphism class of orbits of the equivariance group.

Since every orbit of a group $G$ (typically taken to be a topological group) is isomorphic, as a topological G-space, to a coset space $G/H$, and since these coset spaces, in turn, are isomorphic to each other precisely if the subgroups $H$ are conjugate to each other (see also at orbit category), the $G$-orbit types are often identified with the conjugacy classes $(H)$ of the subgroups of $G$.

Related concepts

• orbit

• orbit category

References

• Glen Bredon, Section I.4 of: Introduction to compact transformation groups, Academic Press 1972 (ISBN 9780080873596, pdf)