View source for Action of a group on a manifold

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.

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).

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.

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.

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 <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

<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>

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]]).

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]]].

====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>



====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>