View source for Action of a group on a manifold

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The best-studied case of the general concept of the action of a group on a space. A topological group  $  G $
acts on a space  $  X $
if to each  $  g \in G $
there corresponds a homeomorphism  $  \phi _ {g} $
of  $  X $(
onto itself) satisfying the following conditions: 1)  $  \phi _ {g} \cdot \phi _ {h} = \phi _ {gh} $; 
2) for the unit element  $  e \in G $
the mapping  $  \phi _ {e} $
is the identity homeomorphism; and 3) the mapping  $  \phi :  G \times X \rightarrow X $, 
$  \phi (g, x) = \phi _ {g} (x) $
is continuous. If  $  X $
and  $  G $
have supplementary structures, the actions of  $  G $
which are compatible with such structures are of special interest; thus, if  $  X $
is a differentiable manifold and  $  G $
is a Lie group, the mapping  $  \phi $
is usually assumed to be differentiable.

The set  $  \{ \phi _ {g} ( x _ {0} ) \} _ {g \in G }  $
is called the orbit (trajectory) of the point  $  x _ {0} \in X $
with respect to the group  $  G $; 
the orbit space is denoted by  $  X/G $, 
and is also called the quotient space of the space  $  X $
with respect to the group  $  G $. 
An important example is the case when  $  X $
is a Lie group and  $  G $
is a subgroup; then  $  X/G $
is the corresponding [[Homogeneous space|homogeneous space]]. Classical examples include the spheres  $  S  ^ {n-1} = \textrm{ O } (n) / \textrm{ O } (n-1) $, 
the Grassmann manifolds  $  \textrm{ O } (n) / ( \textrm{ O } (m) \times \textrm{ O } (n-m) ) $, 
and the Stiefel manifolds  $  \textrm{ O } (n) / \textrm{ O } (m) $(
cf. [[Grassmann manifold|Grassmann manifold]]; [[Stiefel manifold|Stiefel manifold]]). Here, the orbit space is a manifold. This is usually not the case if the action of the group is not free, e.g. if the set  $  X  ^ {G} $
of fixed points is non-empty. A free action of a group is an action for which  $  g=e $
follows if  $  gx=x $
for any  $  x \in X $. 
On the contrary,  $  X  ^ {G} $
is a manifold if  $  X $
is a differentiable manifold and the action of  $  G $
is differentiable; this statement is valid for cohomology manifolds over  $  \mathbf Z _ {p} $
for  $  G = \mathbf Z _ {p} $
as well (Smith's theorem).

If  $  G $
is a non-compact group, the space  $  X/G $
is usually inseparable, and this is why a study of individual trajectories and their mutual locations is of interest. The group  $  G = \mathbf R $
of real numbers acting on a differentiable manifold  $  X $
in a differentiable manner is a classical example. The study of such dynamical systems, which in terms of local coordinates is equivalent to the study of systems of ordinary differential equations, usually involves analytical methods.

If  $  G $
is a compact group, it is known that if  $  X $
is a manifold and if each  $  g \in G $, 
$  g \neq e $, 
acts non-trivially on  $  X $(
i.e. not according to the law  $  (g, x) \rightarrow x $), 
then  $  G $
is a Lie group [[#References|[8]]]. Accordingly, the main interest in the action of a compact group is the action of a Lie group.

Let  $  G $
be a compact Lie group and let  $  X $
be a compact cohomology manifold. The following results are typical. A finite number of orbit types exists in  $  X $, 
and the neighbourhoods of an orbit look like a direct product (the slice theorem); the relations between the cohomology structures of the spaces  $  X $, 
$  X/G $
and  $  X  ^ {G} $
are of interest.

If  $  G $
is a compact Lie group,  $  X $
a differentiable manifold and if the action

$$ 
\phi :  G \times X  \rightarrow  X
$$

is differentiable, then one naturally obtains the following equivalence relation:  $  (X, \phi ) \sim ( X ^ { \prime } , \phi  ^  \prime  ) $
if and only if it is possible to find an  $  ( X ^ { \prime\prime } , \phi  ^ {\prime\prime} ) $
such that the boundary  $  \partial  X ^ { \prime\prime } $
has the form  $  \partial  X ^ { \prime\prime } = X \cup X ^ { \prime } $
and such that  $  \phi  ^ {\prime\prime} \mid  _ {X} = \phi $, 
$  \phi  ^ {\prime\prime} \mid  _ {X ^ { \prime }  } = \phi  ^  \prime  $. 
If the group  $  G $
acts freely, the equivalence classes can be found from the one-to-one correspondence with the bordisms  $  \Omega _ {*} ( B _ {G} ) $
of the classifying space  $  B _ {G} $(
cf. [[Bordism|Bordism]]).

Recent results (mid-1970s) mostly concern: 1) the determination of types of orbits with various supplementary assumptions concerning the group  $  G $
and the manifold  $  X $([[#References|[6]]]); 2) the classification of group actions; and 3) finding connections between global invariants of the manifold  $  X $
and local properties of the group actions of  $  G $
in a neighbourhood of fixed points of  $  X  ^ {G} $. 
In solving these problems an important part is played by: methods of modern differential topology (e.g. surgery methods);  $  K _ {G} $-
theory [[#References|[1]]], which is the analogue of  $  K $-
theory for  $  G $-
vector bundles; bordism and cobordism theories [[#References|[3]]]; and analytical methods of studying the action of the group  $  G $
based on the study of pseudo-differential operators in  $  G $-
bundles [[#References|[2]]], [[#References|[7]]].

====References====
<table>
<TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,   "$K$-theory: lectures" , Benjamin  (1967)</TD></TR>
<TR><TD valign="top">[2]</TD> <TD valign="top">  M.F. Atiyah,   I.M. Singer,   "The index of elliptic operators"  ''Ann. of Math. (2)'' , '''87'''  (1968)  pp. 484–530</TD></TR>
<TR><TD valign="top">[3]</TD> <TD valign="top">  V.M. Bukhshtaber,   A.S. Mishchenko,   S.P. Novikov,   "Formal groups and their role in the apparatus of algebraic topology"  ''Russian Math. Surveys'' , '''26'''  (1971)  pp. 63–90  ''Uspekhi Mat. Nauk'' , '''26''' :  2  (1971)  pp. 131–154</TD></TR>
<TR><TD valign="top">[4]</TD> <TD valign="top">  P.E. Conner,   E.E. Floyd,   "Differentiable periodic maps" , Springer  (1964)</TD></TR>
<TR><TD valign="top">[5]</TD> <TD valign="top">  G. Bredon,   "Introduction to compact transformation groups" , Acad. Press  (1972)</TD></TR>
<TR><TD valign="top">[6]</TD> <TD valign="top">  W.Y. Hsiang,   "Cohomology theory of topological transformation groups" , Springer  (1975)</TD></TR>
<TR><TD valign="top">[7]</TD> <TD valign="top">  D.B. Zagier,   "Equivariant Pontryagin classes and applications to orbit spaces" , Springer  (1972)</TD></TR>
<TR><TD valign="top">[8]</TD> <TD valign="top"> , ''Proc. conf. transformation groups'' , Springer  (1968)</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> , ''Proc. 2-nd conf. compact transformation groups'' , Springer  (1972)</TD></TR>
</table>

====Comments====

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  T. Petrie,   J.D. Randall,   "Transformation groups on manifolds" , M. Dekker  (1984)</TD></TR></table>