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 acts on a space
if to each
there corresponds a homeomorphism
of
(onto itself) satisfying the following conditions: 1)
; 2) for the unit element
the mapping
is the identity homeomorphism; and 3) the mapping
,
is continuous. If
and
have supplementary structures, the actions of
which are compatible with such structures are of special interest; thus, if
is a differentiable manifold and
is a Lie group, the mapping
is usually assumed to be differentiable.
The set is called the orbit (trajectory) of the point
with respect to the group
; the orbit space is denoted by
, and is also called the quotient space of the space
with respect to the group
. An important example is the case when
is a Lie group and
is a subgroup; then
is the corresponding homogeneous space. Classical examples include the spheres
, the Grassmann manifolds
, and the Stiefel manifolds
(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
of fixed points is non-empty. A free action of a group is an action for which
follows if
for any
. On the contrary,
is a manifold if
is a differentiable manifold and the action of
is differentiable; this statement is valid for cohomology manifolds over
for
as well (Smith's theorem).
If is a non-compact group, the space
is usually inseparable, and this is why a study of individual trajectories and their mutual locations is of interest. The group
of real numbers acting on a differentiable manifold
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 is a compact group, it is known that if
is a manifold and if each
,
, acts non-trivially on
(i.e. not according to the law
), then
is a Lie group [8]. Accordingly, the main interest in the action of a compact group is the action of a Lie group.
Let be a compact Lie group and let
be a compact cohomology manifold. The following results are typical. A finite number of orbit types exists in
, and the neighbourhoods of an orbit look like a direct product (the slice theorem); the relations between the cohomology structures of the spaces
,
and
are of interest.
If is a compact Lie group,
a differentiable manifold and if the action
![]() |
is differentiable, then one naturally obtains the following equivalence relation: if and only if it is possible to find an
such that the boundary
has the form
and such that
,
. If the group
acts freely, the equivalence classes can be found from the one-to-one correspondence with the bordisms
of the classifying space
(cf. Bordism).
Recent results (mid-1970s) mostly concern: 1) the determination of types of orbits with various supplementary assumptions concerning the group and the manifold
([6]); 2) the classification of group actions; and 3) finding connections between global invariants of the manifold
and local properties of the group actions of
in a neighbourhood of fixed points of
. In solving these problems an important part is played by: methods of modern differential topology (e.g. surgery methods);
-theory [1], which is the analogue of
-theory for
-vector bundles; bordism and cobordism theories [3]; and analytical methods of studying the action of the group
based on the study of pseudo-differential operators in
-bundles [2], [7].
References
[1] | M.F. Atiyah, "![]() |
[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) |
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=45017