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The best-studied case of the general concept of the action of a group on a space. A topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105501.png" /> acts on a space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105502.png" /> if to each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105503.png" /> there corresponds a homeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105504.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105505.png" /> (onto itself) satisfying the following conditions: 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105506.png" />; 2) for the unit element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105507.png" /> the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105508.png" /> is the identity homeomorphism; and 3) the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a0105509.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055010.png" /> is continuous. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055012.png" /> have supplementary structures, the actions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055013.png" /> which are compatible with such structures are of special interest; thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055014.png" /> is a differentiable manifold and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055015.png" /> is a Lie group, the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055016.png" /> is usually assumed to be differentiable.
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The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055017.png" /> is called the orbit (trajectory) of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055018.png" /> with respect to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055019.png" />; the orbit space is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055020.png" />, and is also called the quotient space of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055021.png" /> with respect to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055022.png" />. An important example is the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055023.png" /> is a Lie group and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055024.png" /> is a subgroup; then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055025.png" /> is the corresponding [[Homogeneous space|homogeneous space]]. Classical examples include the spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055026.png" />, the Grassmann manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055027.png" />, and the Stiefel manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055028.png" /> (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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055029.png" /> of fixed points is non-empty. A free action of a group is an action for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055030.png" /> follows if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055031.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055032.png" />. On the contrary, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055033.png" /> is a manifold if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055034.png" /> is a differentiable manifold and the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055035.png" /> is differentiable; this statement is valid for cohomology manifolds over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055036.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055037.png" /> as well (Smith's theorem).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055038.png" /> is a non-compact group, the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055039.png" /> is usually inseparable, and this is why a study of individual trajectories and their mutual locations is of interest. The group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055040.png" /> of real numbers acting on a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055041.png" /> 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.
+
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.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055042.png" /> is a compact group, it is known that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055043.png" /> is a manifold and if each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055045.png" />, acts non-trivially on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055046.png" /> (i.e. not according to the law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055047.png" />), then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055048.png" /> is a Lie group [[#References|[8]]]. Accordingly, the main interest in the action of a compact group is the action of a Lie group.
+
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).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055049.png" /> be a compact Lie group and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055050.png" /> be a compact cohomology manifold. The following results are typical. A finite number of orbit types exists in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055051.png" />, and the neighbourhoods of an orbit look like a direct product (the slice theorem); the relations between the cohomology structures of the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055053.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055054.png" /> are of interest.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055055.png" /> is a compact Lie group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055056.png" /> a differentiable manifold and if the action
+
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.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055057.png" /></td> </tr></table>
+
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.
  
is differentiable, then one naturally obtains the following equivalence relation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055058.png" /> if and only if it is possible to find an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055059.png" /> such that the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055060.png" /> has the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055061.png" /> and such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055063.png" />. If the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055064.png" /> acts freely, the equivalence classes can be found from the one-to-one correspondence with the bordisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055065.png" /> of the classifying space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055066.png" /> (cf. [[Bordism|Bordism]]).
+
If  $  G $
 +
is a compact Lie group, $  X $
 +
a differentiable manifold and if the action
  
Recent results (mid-1970s) mostly concern: 1) the determination of types of orbits with various supplementary assumptions concerning the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055067.png" /> and the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055068.png" /> ([[#References|[6]]]); 2) the classification of group actions; and 3) finding connections between global invariants of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055069.png" /> and local properties of the group actions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055070.png" /> in a neighbourhood of fixed points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055071.png" />. In solving these problems an important part is played by: methods of modern differential topology (e.g. surgery methods); <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055072.png" />-theory [[#References|[1]]], which is the analogue of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055073.png" />-theory for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055074.png" />-vector bundles; bordism and cobordism theories [[#References|[3]]]; and analytical methods of studying the action of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055075.png" /> based on the study of pseudo-differential operators in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055076.png" />-bundles [[#References|[2]]], [[#References|[7]]].
+
$$
 +
\phi :  G \times X  \rightarrow  X
 +
$$
 +
 
 +
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====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055077.png" />-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>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.F. Atiyah,  "<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010550/a01055077.png" />-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====
 
====Comments====
 
  
 
====References====
 
====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>
 
<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>

Latest revision as of 16:08, 1 April 2020


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. 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; 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 [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).

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 $([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 [1], which is the analogue of $ K $- theory for $ G $- vector bundles; bordism and cobordism theories [3]; and analytical methods of studying the action of the group $ G $ based on the study of pseudo-differential operators in $ G $- bundles [2], [7].

References

[1] M.F. Atiyah, "-theory: lectures" , Benjamin (1967)
[2] M.F. Atiyah, I.M. Singer, "The index of elliptic operators" Ann. of Math. (2) , 87 (1968) pp. 484–530
[3] 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
[4] P.E. Conner, E.E. Floyd, "Differentiable periodic maps" , Springer (1964)
[5] G. Bredon, "Introduction to compact transformation groups" , Acad. Press (1972)
[6] W.Y. Hsiang, "Cohomology theory of topological transformation groups" , Springer (1975)
[7] D.B. Zagier, "Equivariant Pontryagin classes and applications to orbit spaces" , Springer (1972)
[8] , Proc. conf. transformation groups , Springer (1968)
[9] , Proc. 2-nd conf. compact transformation groups , Springer (1972)

Comments

References

[a1] T. Petrie, J.D. Randall, "Transformation groups on manifolds" , M. Dekker (1984)
How to Cite This Entry:
Action of a group on a manifold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Action_of_a_group_on_a_manifold&oldid=14980
This article was adapted from an original article by A.V. Zarelua (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article