LECTURE NOTES IN TOPOLOGICAL GROUPS (2014) UPDATED 7.2.17 MICHAEL MEGRELISHVILI Contents 1. Lecture 1 1 2. Lecture 2 4 3. Lecture 3 8 4. Lecture 4 13 5. Lecture 5 17 6. Lectures 6 and 7 21 7. Lecture 8 27 8. Lecture 9 30 9. Lectures 10,11,12 34 References 43 1. Lecture 1 Definition 1.1. Let (G; m) be a group and τ be a topology on G. We say that (G; m; τ) (or, simply, G) is a topological group if the two basic operations m : G × G ! G; (x; y) 7! m(x; y) := xy and the inversion i : G ! G; x 7! x−1 are continuous. We say also that τ is a group topology on the group G. By TGr we denote the class of all topological groups. In most cases later we consider Hausdorff groups. Morphisms in TGr are continuous homomorphisms. For example, −1 id :(R; τdiscrete) ! R is a continuous homomorphism but not id . Algebraically id is of course an isomorphism. That is, isomorphism in Gr but not in TGr. Note that every group with the discrete topology is a topological group. Hence, Gr ⊂ T Gr. Trivial topology is also a group topology on every group. We say that a topological group is Hausdorff, compact, metrizable, separable etc. if the given topology on G satisfies the corresponding topological property. Remarks 1.2. Date: February, 2017. 1 2 (1) (a short version) 1.1 is equivalent to the following condition: G × G ! G; (x; y) 7! xy−1 is continuous. (2) (in terms of nbds) 1.1 is equivalent to the following conditions: (a) 8U 2 N(xy) 9V 2 N(x);W 2 N(y): VW ⊂ U (b) 8U 2 N(x−1) 9V 2 N(x): V −1 ⊂ U. Note that (b) is equivalent to (b') 8U 2 N(x−1) U −1 2 N(x). (3) (in terms of (generalized) sequences) If G is metrizable then 1.1 is equivalent to the following condition: −1 −1 xn ! x; yn ! y ) xnyn ! xy In general one may use the nets ( generalized sequences). Exercise 1.3. (1) Show that R2 and R are nonhomeomorphic topological spaces but algebraically these groups are isomorphic. (2) Show that the discrete spaces Z and Z × Z2 are homeomorphic but as groups they are not isomorphic. Definition 1.4. (1) In terms of Definition 1.1 we say that G is: paratopological group if m is continuous. (2) Let (S; m) be a semigroup and τ be a topology on S. We say that the semi- group S is: (a) topological semigroup if m : S × S ! S is continuous. So, a paratopolog- ical group is a topological semigroup. (b) semitopological if m is separately continuous. That is, if all left and right translations la : S ! S; x 7! ax; ra : S ! S; x 7! xa are continuous for every a 2 S. (c) right (left) topological if right (left) translations of S are continuous. Example 1.5. (1) (para but not topo) Let τs be the Sorgenfrey topology (standard topological base is f[a; b)g) on the group (R; +) of all reals. Then (R; τs; +) is a paratopological but not topological group. Hint: [0; 1) 2 τs but (−1; 0] 2= τs. (2) (semi but not para) For every group G the pair (G; τcof ) (with the cofinite topology) is a semitopological group which satisfies T1. It is a paratopological group iff G is finite. (3) (right but not left) For every topological space X consider the semigroup (XX ; ◦) of all selfmaps wrt product (=pointwise) topology. Then XX is right X X topological. * If X 2 T1 the teft translation lf : X ! X is continuous iff f 2 C(X; X). Note that if X is compact then XX is compact by the Tychonoff theorem. Theorem 1.6. Let S be a right topological semigroup. If S is compact then it contains at least one idempotent. 3 Proof. We have to show that there exists m 2 S such that m2 = m. By Zorn's Lemma (and compactness of S) thee exists a minimal compact sub- semigroup M ⊆ S (indeed, for any chain of compact subsemigroups the intersection is a nonempty compact subsemigroup (by the compactness) of S). Take arbitrary m 2 M. Our aim is to show that m2 = m. Consider the set Mm. Then Mm ⊆ M is a subsemigroup. Observe also that Mm is compact. By the minimality of M we necessarily have Mm = M. This implies that um = m for some u 2 S. Hence, the following set K := fx 2 M : xm = mg is nonempty. Moreover, K is a subsemigroup and compact (again use the continuity of right translations). Since K ⊆ M we necessarily have K = M. This implies that m 2 K. Therefore, m2 = m. 4 2. Lecture 2 Definition 2.1. A topological space X is said to be homogeneous if for every x; y 2 X there exists an autohomeomorphism h : X ! X (notation: h 2 H(X)) s.t. h(x) = y. Lemma 2.2. For every semitopological group the left (right) translations are home- omorphism. For every topological group the inversion map is a homeomorphism. −1 −1 −1 Proof. Observe that la = la−1 (ra = ra−1 ) and i = i. Proposition 2.3. Every semitopological group (hence, also, every topological, as a topological space) is homogeneous. Definition 2.4. A topological space (X; τ) is said to be of group type if there exists a group structure m on X such that (X; τ; m) is a topological group. Remarks 2.5. (1) [0; 1]n for every natural n is not homogeneous hence not of group type. (2) The Hilbert cube [0; 1]N is homogeneous (Keller). At the same time it has the fixed point property: every continuous map h : [0; 1]N ! [0; 1]N has a fixed point. It follows that the Hilbert cube is not of group type. Moreover, there is no structure of a left (right) topological group on it. (3) The Cantor set C ⊂ [0; 1] is of group type. Indeed, C is homeomorphic to the N topological space Z2 , which is a topological group. (4) The space of all irrational numbers RnQ is of group type being homeomorphic to ZN. (5) The Sorgenfrey line as a topological space is homogeneous but not of group type (up to a non-trivial theorem of Kakutani below: every Hausdorff topological group with the first countable property B1 is metrizable). Some examples of topological groups: (1) Every group in the discrete topology. (2) GLn(R) are locally compact metrizable topological group. It, as a metric 2 space, is embedded isometrically into the Euclidean metric space Rn . (3) The orthogonal group On(R) ⊂ GLn(R) is compact by Heine-Borel thm being a bounded and a closed subset in the metric space GLn(R). Note that O2(R) is isomorphic in TGr with the "circle group" T ≤ C∗. (4) TGr is closed under: subgroups, factor-groups, topological products, box products. (5) Every Euclidean space Rn and Tn the n-dimensional torus. (6) Every normed space (more generally, any linear topological space). (7) (Z; dp) the integers wrt the p-adic metric. It is a precompact group (totally bounded in its metric) and its completion is the compact topological group of all p-adic integers. (8) For every compact space K the group of all autohomeomorphisms H(K) en- dowed with the so-called compact-open topology (we define it later). (9) For every metric space (X; d) the group of all onto isometries Iso(X; d) ⊂ XX endowed with the pointwise topology inherited from XX . (10) For every Banach space (V; jj·jj) the group Isolin(V ) of all linear onto isometries V ! V endowed with the pointwise topology inherited from V V . For example, 5 n if V := R is the Euclidean space then Isolin(V ) = On(R) the orthogonal group. Note that, in contrast to the case of Rn, for infinite dimensional V the topological group Isolin(V ) as usual is not compact. Moreover, Teleman's theorems show that every Hausdorff topological group G is embedded into Isolin(V ) for suitable V and also into some H(K) for suitable compact space K. As we will see below even groups like Z and R cannot be embedded into compact groups. We examine the question which topological groups admit representations on good Banach spaces (like: Hilbert, reflexive, ...). For these purposes we give a necessary basic material for topological group theory. Including among others: first steps in uniform structures and uniformly continuous functions on groups. We touch also some questions from the van-Kampen Pontryagin duality theory for locally compact abelian groups. 6 2.1. First homework. Let G be a topological group. Exercise 2.6. Prove that (1) cl(A−1) = cl(A)−1 and cl(A)cl(B) ⊂ cl(AB) for every subsets A; B of G. (2) If H ≤ G is a subgroup then cl(H) ≤ G is also a subgroup. (3) If H ¢ G is a normal subgroup then cl(H) ¢ G is also a normal subgroup. (4) If G, in addition, is abelian and H ≤ G then cl(H) ≤ G is also an abelian subgroup. Give a counterexample if G is not Hausdorff. Exercise 2.7. Prove that the function n k1 k2 kn (1) G ! G; (x1; x2; ··· ; xn) 7! x1 x2 ··· xn is continuous for every given tuple n (k1; k2; ··· ; kn) 2 Z . (2) For every nbd U 2 N(e) of the identity e 2 G and every given natural n 2 N there exists V 2 N(e) such that V = V −1 and V n := VV ··· V ⊂ U.