にもあるが、新たにまとめる。
Isotropy Group
Some elements of a group G acting on a space X may fix a point x. These group elements form a subgroup called the isotropy group, defined by
- Gx := {g∈G | gx = x}
When two points x and y are on the same group orbit, say y = gx, then the isotropy groups are conjugate subgroups. More precisely, Gy = gGxg-1. In fact, any subgroup conjugate to Gx occurs as an isotropy group Gy to some point y on the same orbit as x.
Stabilizer
Let G be a permutation group on a set Ω and x be an element of Ω. Then
- Gx = {g∈G | g(x) = x}
is called the stabilizer of x and consists of all the permutations of G that produce group fixed points in x, i.e., that send x to itself.
Effective Action
A group action G×X→X is effective if there are no trivial actions. In particular, this means that there is no element of the group (besides the identity element) which does nothing, leaving every point where it is. This can be expressed as ∩x∈XGx = {e}, where Gx is the isotropy group at x and e is the identity of G.
Free Action
A group action G×X→X is called free if, for all x in X, gx = x implies g = I (i.e., only the identity element fixes any x). In other words, G×X→X is free if the map G×X→X×X sending (g, x) to (a(g, x), x) is injective, so that a(g, x) = x implies g = I for all g, x. This means that all stabilizers are trivial. A group with free action is said to act freely.
The basic example of a free group action is the action of a group on itself by left multiplication L:G×G→G. As long as the group has more than the identity element, there is no element h which satisfies gh = h for all g.
An example of a free action which is not transitive is the action of S1 on S3 subset C2 by , which defines the Hopf map.
Proper Group Action
A group action of a topological group G on a topological space X is said to be a proper group action if the mapping
- G×X → X×X
- (g,x)|→ (gx,x)
is a proper map, i.e., inverses of compact sets are compact.
A proper action must have compact isotropy groups at all points of X.
Properly Discontinuous Action
Let G be a group and E a topological space on which G acts by homeomorphisms, that is there is a homomorphism ρ:G→Aut(E), where the latter denotes the group of self-homeomorphisms of E. The action is said to be properly discontinuous if each point e∈E has a neighborhood U with the property that all non trivial elements of G move U outside itself:
- ∀g∈G.( g≠id ⇒ gU∩U=∅ ).
For example, let p:E→X be a covering map, then the group of deck transformations of p acts properly discontinuously on E. Indeed if e∈E and D∈Aut(p) then one can take as U to be any neighborhood with the property that p(U) is evenly covered. The following shows that this is the only example:
Theorem. Assume that E is a connected and locally path connected Hausdorff space. If the group G acts properly discontinuously on E then the quotient map p:E→E/G is a covering map and Aut(p) = G.
定理
- XへのG作用が、真性不連続〈固有不連続〉のとき、X→X/G は被覆写像になる。逆も成立。
- Lie 群 G が completely regular space に自由に作用するとき、射影 X → X∕G は principal G-bundle になる 。
