# 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, "-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)