Reference: Topological Transformation Groups by Deane Montgomery and Leo Zippin, Dover Publications, available Amazon India.

1.21 COSET SPACE of LOCAL GROUPS

DEFINITION. A closed subset H of a local group G is called a subgroup if a neighbourhood V of e exists such that is defined, V open and symmetric, and:

i) implies

ii) implies .

The subgroup is called invariant if a V exists satisfying (i) and (ii) and also

(iii) implies

LEMMA.

Let G be a local group, H a subgroup, V a neighbourhood as described above, and let W be a symmetric open neighbourhood of a e with .. Then for and is an equivalence relation and if and only if .

Proof:

The relation is reflexive since . It is symmetric since if then hence is defined and belongs to . Now implies . The relation is transitive because and implies by condition (2).

It can be seen that the sets where are the equivalence classes of this equivalence relation. They are called the local cosets and are the points of a coset space . We shall now define the topology for this space.

Let T denote the natural map of W onto defined by

Note that T (for local groups) is defined only on W. Let the open sets of the coset space be those which have an open set in W as inverse under T. Then T is open and continuous.

COROLLARY.

With the same assumptioins and notation as in the preceding Lemma, let H be invariant. Then is a local group when the product is defined in the natural way.

The product function is defined for by:

To prove that this is a single valued function (or a well-defined function) on cosets it needs to be shown that for and the product exists and is in H. The product is in and defines an element of V, This element can be seen to be in by the use of the associative law and condition (3) above; it must be verified that all indicated products are well-defined. Finally let be an open neighbourhood of e in G, with . It can be seen that the product of each pair of elements in the open set is defined as an element of . The remaining details can be verified by the reader. ( I have yet to ! :-))

THEOREM.

A local group G possesses such a family of neighbourhoods as is defined in Section 1.17 (previous blog). For every subgroup H of G and neighbourhood W of e (this neighbourhood chosen as above) the coset space is a Hausdorff space.

The proof is the same as in 1.16 and 1.17. (previous blog)

Remarks on the side: Some of the principal results of the succeeding chapters are valid for local groups as well as global groups. However, the consideration of local groups in each preliminary Lemma and Theorem is not feasible in a work of this kind. In the sequel we shall only occasionally need to make explicit mention of the local groups.

1.22 INVARIANT METRIC

The construction of metrics and invariant metrics in groups was carried out by Garrett Birkhoff and Kakutani independently.

A local group is called metric if some neighbourhood of the identity is metric.

THEOREM.

Let G be a topological group whose open sets at e have a countable basis. Then G is metrizable (Section 1.9) and moreover, there exists a metric which is right invariant.

(From Section 1.9 we reproduce the definition of metrizable: A space is called metrizable if a metric can be defined for it which induces in it the original topology.)

Let for be a countable basis for open sets at e and for each positive integer n let . The sets are monotonic decreasing and form a basis at e. Let for each dyadic rational , where , where be a family of neighbourhoods of e in G such as is constructed in Section 1.17 (previous blog).

Define a function

on G as follows:

(i) if and only if for every r.

(ii) otherwise where

For any set U and element a in G, if and only if . From this it follows that is right invariant:

(iii)

Next from the fact that is symmetric it follows that if and only if for the same r. It follows that f is symmetric

The sets are symmetric so that by (5) of section 1.17. Also and . It follows now that

if and only if

We next define the distance function:

*)

The definition shows that

and

If then and . Finally the triangle inequality:

The right invariance of the metric is shown as follows:

Finally we must show the equivalence of the original neighbourhoods of G and the sphere neighbourhoods of the metric. It is sufficient to verify this at a e because of the invariance of the metric on the one hand and the translation properties of G (Section 1.13 previous blog) on the other. Let denote the set of elements whose distance from a is less than . The fact that

shows that the metric spheres are neighbourhoods of e in the original topology. It remains to show that they form a basis at e. Let a neighbourhood O of e be given. Since the sets are a basis at e there is an integer k such that . Then if , and and , and finally

hence

This completes the proof.

1.23 METRIC HOMOGENEOUS SPACE

THEOREM.

Let H be a closed subgroup of a metric group G. Then G/H is metrizable.

Proof.

Let d(x,y) be a right invariant metric in G. Define as follows:

where

then for every , H contains elements a, b, c, d such that

This shows that D satisfies the triangle inequality. It can be verified that D is single-valued in G/H x G/H, that is , D is symmetric and that if and only if . Then D is a metric.

The set yH belongs to the set if . Therefore any neighbourhood of xH in the original topology includes a metric neighbourhood. The converse can also be shown. Therefore the metric D induces the topology of G/H.

1.24 CONNECTEDNESS

A subset M of a topological space S is called connected if whenever where A and B are open relative to M and are not empty, it follows that is not empty.

LEMMA.

If M is connected and A is a subset which is both open and closed relative to M, then , or A is the empty set.

This follows from the definition.

LEMMA.

If G is a topological group and H is a subgroup which is open, then H is also closed.

Since H is open each coset of H is open. The compliment of H is the union of cosets of H. Hence, the complement of H is open so H must be closed.

THEOREM.

If G is a connected group and W is an open neighbourhood of e, then .

The set is an open subgroup of G. Therefore H is a closed subgroup, and then it follows from the connectedness of G that .

A union of an arbitrary number of connected subsets is connected, provided every two of the sets have a point in common. It follows that to each point there is uniquely associated a maximal connected subset of M containing x; these maximal connected subsets are called components of S. The closure of a component is connected — therefore the component is closed. A space is called totally-disconnected when each point if a component.

LEMMA.

If M is connected and f is a continuous map of M into a space N then is connected.

If and

Then and

COROLLARY.

If G is a connected group and H is a closed subgroup, then G/H is a connected space.

DEFINITION.

A space is called normal if any two disjoint closed sets are contained in disjoined open sets; two sets are disjoined (disjunct, mutually separated) if they have no common points.

LEMMA.

A compact Hausdorff space is normal.

The proof is straight forward and is omitted.

LEMMA.

Let S be a compact Hausdorff space and x a point in S. Let Q be a set of indices . If is the set of all compact open subsets containing x, then is the x-component of S.

Let

where

and X and Y are closed. There exist open sets and such that . Hence,

*)

and there is a finite set of indices such that *) continues to hold if the intersection os restricted to . Let where . Then, A is compact and open and .

Finally, since

it follows that and are compact open sets only one of which can contain x. Since the system where is maximal, one of and is in , say and therefore ; hence, Y is empty.

This shows that is connected. It is now clear that Cis the component containing x.

COROLLARY.

Let S be compact Hausdorff space, C be a component of S, let F be a closed set and suppose that . Then there is a compact open set , , .

It follows from the hypothesis that there is a finite set such that

where

Let

.

This is the set required by the corollary.

1.25 THE IDENTITY COMPONENT

THEOREM.

Let g be a topological group and the component of G containing the identity. Then is closed invariant subgroup and the factor group is totally disconnected.

If M is a connected subset of G, is connected, Mx and xM where are connected (section 1.13). It follows that the identity component is a group and that the cosets are also components. We have already remarked that components are closed sests. Then is a closed subgroup. Since is connected and contains the identity e for every , it is clear that is invariant.

Now let M denote a connected subset of . Let T be the natural map of G onto . We shall show that is connected, thus proving that it is in a coset of . Suppose that

where A and B are relatively open (and closed ) in and is empty. Then . Since there exists an open set, say U, in G, such that and since T is an open set, say U, in G, such that and since T is an open map, the set is open relatively to M. This is true also of . Since M is connected it follows that is not empty. Let . Since is a connected subset of G,

leads to a contradiction.

A subgroup of a group G is called central if each of its elements communtes with every element in the whole group. Subgroups H and are called conjugate.

THEOREM.

If G is a connected group and H is an invariant totally disconnected subgroup then H is central.

Let x be an element of H and consider the map of G into H depending on x:

where .

Since H contains no connected set with more than one point, the image of G which contains x, must coincide with x. This completes the proof.

1.26 TRANSFORMATION GROUPS

We shall soon confine our attention to groups which are locally compact, and we shall be particularly interested in the transformation groups of locally-euclidean spaces. For the time being, we continue to study the more general phenomenon.

DEFINITION I.

Let M denote a Hausdorff space and G a topological group each element of which is a homeomorphism of M onto itself:

(1) ; , and

The pair , or sometimes G itself will be called a topological transformation group if for every pair of elements of G, and every x of M,

(2) ;

and if is continuous simultaneously in and .

From (2) and from the fact that each g is one-one on M, it follows that for every

(3) ; e the identity in G.

If e is the only element in G which leaves all of M fixed, that is if e is the only element satisfying (3) for all x, then G is called effective.

DEFINITION I’

A pair (G,M) will be called a local transformation group if all conditions of the preceding Definition are fulfilled excepting only that G is assumed to be a local group, and condition (2) holds whenever is defined.

Some of the remarks below apply both to local and global case for the most part, except when noted.

Let G be a transformation group, , and suppose that

Since M is a Hausdorff space, there is a neighbourhood Y of y not containing x. By the definition of the transformation group, there is a neighbourhood U of g such that for , . Therefore, , and it follows that the set of elements of G which leave x fixed is closed. Therefore this set is a closed subgroup of G. We shall denote it by . Similarly, the set of elements of G leaving fixed every point of M is a closed subgroup of .

THEOREM

Let be a transformation group, and let K be the closed subgroup of G leaving all of M fixed. Then K is invariant and G/K is an effective transformation group of M under the action:

where .

For , , we must always have: . This shows the invariance of K. If and a neighbourhood Y of y is given, there is a neighbourhood U of g and X of x such that , imply . But then . From this it is clear that is continuous simultaneously in gK and x. It can be seen that

It follows also that for every x implies , equivalently .

1.28.1

If G is a transformation group of M and h is a homeomorphism of M onto itself, then the homeomorphisms:

of M onto itself form a transformation group which is said to be topologically equivalent to G.

1.26.2

A topological group G can be regarded as a transformation group on itself as space in several ways, in particular by associating with .

- (left translation)
- (right translation)
- (conjugation, taking of transforms.

Also G is a transformation group of a left coset space G/H where H is a closed subgroup, by - In cases (1) and (2) G is effective. In case (4) if for every then for every . Then is an invariant subgroup of G, and G/K is effective on G/H. Of course, K is a subgroup of H which depends on H as well as G and which may be trivial.

1.26.3

Further examples of transformation groups are given below, proofs are omitted. (PS: I will supply the proofs in a later blog, most probably after this blog):

- Let be the real n x n matrices where with . Let be the space of n real variables . Then G is a transformation group of whose elements are where
- With G as above, let in be the (n-1)-sphere defined by . Let G act on as follows: where and is as above. Here the effective group is G/Kn where Kn consists of scalar matrices: where is the Kronecker delta and h is positive.
- Let G be the group of two by two real matrices with determinant one and let it act on as in (1).
- Let G be the group of two by two real matrices of determinant one and let G act on itself by inner automorphisms. As a space G is the product of a circle and a plane. One parameter groups fill a neighbourhood of the identity and in the large are closed sets which are either circles or lines. No two of them cross and they are permuted by G.
- For fixed integers m, n let G be the circle group acting on as follows: ; ; ; . This can also be viewed as a transformation group on the unit sphere in since it leaves invariant. It is known that the simple closed curves swept out by points of are linked.
- A quasi-relation in in a fixed cylindrical coordinate system is a group of homeomorphisms depending on a positive continuous function , where , , F bounded on compact sets in and given by (*) for all real t.

Each point which is not a fixed point moves in a circle about the z-axis. The period of a moving point, that is the least positive t for which it is left fixed by (*) above varies continuously. If h is an arbitrary homeomorphism of upon itself, then defines a topological quasi-rotation group. As we shall mention later, these groups can be characterized abstractly.

1.26.4

The transformation group G is called transitive on M if for every there is at least one such that . As remarked earlier every topological group G is transitive on G/H. where H is a closed subgroup.

THEOREM.

Let be a topological transformation group which is transitive on M. Then the groups of stability where are conjugate and for any one of them is mapped in a continuous one-one way onto M by the map:

Let be given with . If , . It follows that and are conjugate.

Now let x be fixed. If , for every and this shows that the map

defined in the theorem is one-one. It maps onto M because G is transitive.

Let T be the natural map . Then maps G onto M .

This map is continuous in G by the definition of a transformation group. Let U be open in M. Then is open in G and is open in , since F is an open map. This shows that is continuous as well as one-one. In the cases of most interest it will turn out that is a homeomorphism.

1.27 LOCALLY EUCLIDEAN SPACES

We have used , where to denote euclidean n-space, with real coordinates . It is the topological class of spaces homeomorphic to which we have in mind, rather than the space endowed with a standard euclidean metric. By the Brouwer theorem, an open subset of and an open subset of cannot be homeomorphic if . This shows that the possibility of the one-one bicontinuous coordinatization of is a topological property. This number n is a topological invariant and is called the dimension.

The term locally euclidean is used to describe a topological space E of fixed dimension n each point of which has a neighbourhood that is homeomorphic to an open set in . The simplest examples of such spaces are the open subsets of . If a locally euclidean space is connected, it is called a MANIFOLD. For example, the spheres of all dimensions, the ordinary torus, the cylinder, etc. are manifolds.

A locally euclidean space can be covered by a certain number (not necessarily finite) of open sets each homeomorphic to an open set in ; let us call such sets each with a fixed homeomorphism, a coordinate neighbourhood. The circle can be covered by two (or more) coordinate neighbourhoods, the two dimensional sphere by two or more. However, to describe classical euclidean space one uses the entire family of those coordinate systems which are related to each other by orthogonal transformations. Similarly to define affine n-dimensional space, one uses a larger family, namely, all coordinate systems which are affinely related to each other.

We describe a topological manifold one could use in it the family of all coordinate neighbourhoods. However, if for some purpose a restricted class of coordinate neighbourhoods covering the manifold is specified, then one can speak of admissible coordinate systems. In general, where two coordinate neighbourhoods overlap, the coordinate systems will not be found to be in any simple relation. It may happen that the admissible coordinate have been so selected that in every region of overlap of two such systems the two sets of coordinates are related by functions which are differentiable or analytic.

A manifold is said to be a differentiable manifold and to have a differentiable structure of class where r is greater than or equal to 1, if there is given a covering family of coordinate neighbourhoods in such a way that where any two of the neighbourhoods overlap the coordinate transformation in both directions is given by n functions with continuous, partial derivatives of order r. A manifold may have essentially different differentiable structures (Milnor). A manifold need not possess a differentiable structure (Kervaire, Smale). The n-sphere (with the possibe exception of n=3) has only a finite number of such structures (see Kervaire-Milnor).

In the same way a manifold is said to be a real analytic manifold and to have a real analytic structure if there is given a covering family of coordinate neighbourhoods in such a way that where any two overlap the coordinate transformation in both directions is given by n functions which are real analytic, that is in some neighbourhood of each point of the overlap they can be expanded in power series.

The definition of a complex analytic manifold and structure is similar to the above. Such a manifold of course has an even number of real dimensions; it is automatically real analytic. However, there are many real analytic manifolds of even dimension which cannot be given a complex analytic structure; thus the existence of a complex analytic structure is a much stronger property than the existence of a differentiable or even real analytic structure.

1.27.1

Suppose that M is an n-dimensional manifold and that x and y are points of M belonging to a set U which is homeomorphic to an open n-sphere. Then it is not difficult to describe a homeomorphism of M onto itself which keeps fixed all points of M not inside of U, and which maps x onto y. Using this and using the connectedness of M, and being given an arbitrary pair of points x and y of M one can find a homeomorphism of M onto itself mapping x on y. The details are not presented here.

1.27.2

Suppose that M and N are manifolds and that there is given a continuous map f of M onto . The map f is called a covering map and M is said to cover if the following conditions are satisfied:

a) for each y in there is an open neighbourhood V of y such that is the union of disjoined open sets where there is a for each and

b) f is a homeomorphism of onto V for each x in . For each , each is called a covering point. It is clear that M and are of the same dimension, and it is clear that each point of is an isolated point of (each point is a relatively open subset) so that is a discrete set.

By way of example, let be the ordinary torus with momentarily convenient coordinates u, v: and let M be the ordinary plane with real coordinates x and y. Define the map by

if and only if and This pair of manifolds can also be regarded as an example of a group M covering a factor-group . Thus let M now denote the two-dimensional vector space and let and denote two independent vectors in . Let D be the countable, discrete subgroup of consisting of the linear combinations of and with integral coefficients. Finally, let now denote the toral group .

A more general example is the following:

If G is any connected locally euclidean group and H is a discrete subgroup of G then G covers the coset-space G/H under the natural map .

A manifold is called simply connected if whenever it is covered by a manifold M, the covering map is a homeomorphism (in that case is single-valued). Euclidean spaces of all dimensions and the sphere-spaces of dimension greater than one are simply connected manifolds; the one-dimensional sphere (circumference of a circle) and more generally the toral spaces of all dimensions are not simply connected.

Cheers,

Nalin Pithwa