Use in The Study of Group Actions
The total space of a universal bundle is usually written EG. These spaces are of interest in their own right, despite typically being contractible. For example in defining the homotopy quotient or homotopy orbit space of a group action of G, in cases where the orbit space is pathological (in the sense of being a non-Hausdorff space, for example). The idea, if G acts on the space X, is to consider instead the action on
- Y = X×EG,
and corresponding quotient. See equivariant cohomology for more detailed discussion.
If EG is contractible then X and Y are homotopy equivalent spaces. But the diagonal action on Y, i.e. where G acts on both X and EG coordinates, may be well-behaved when the action on X is not.
Read more about this topic: Universal Bundle
Famous quotes containing the words study, group and/or actions:
“Parents who want a fresh point of view on their furniture are advised to drop down on all fours and accompany the nine or ten month old on his rounds. It is probably many years since you last studied the underside of a dining room chair. The ten month old will study this marvel with as much concentration and reverence as a tourist in the Cathedral of Chartres.”
—Selma H. Fraiberg (20th century)
“The conflict between the need to belong to a group and the need to be seen as unique and individual is the dominant struggle of adolescence.”
—Jeanne Elium (20th century)
“In history an additional result is commonly produced by human actions beyond that which they aim at and obtain—that which they immediately recognize and desire. They gratify their own interest; but something further is thereby accomplished, latent in the actions in question, though not present to their consciousness, and not included in their design.”
—Georg Wilhelm Friedrich Hegel (1770–1831)