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 $ 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, "$K$-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)
[a1]	T. Petrie, J.D. Randall, "Transformation groups on manifolds" , M. Dekker (1984)