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Appendix a Topological Groups and Lie Groups

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Appendix a Topological Groups and Lie Groups Appendix A Topological Groups and Lie Groups This appendix studies topological groups, and also Lie groups which are special topological groups as well as manifolds with some compatibility conditions. The concept of a topological group arose through the work of Felix Klein (1849–1925) and Marius Sophus Lie (1842–1899). One of the concrete concepts of the the- ory of topological groups is the concept of Lie groups named after Sophus Lie. The concept of Lie groups arose in mathematics through the study of continuous transformations, which constitute in a natural way topological manifolds. Topo- logical groups occupy a vast territory in topology and geometry. The theory of topological groups first arose in the theory of Lie groups which carry differential structures and they form the most important class of topological groups. For exam- ple, GL (n, R), GL (n, C), GL (n, H), SL (n, R), SL (n, C), O(n, R), U(n, C), SL (n, H) are some important classical Lie Groups. Sophus Lie first systematically investigated groups of transformations and developed his theory of transformation groups to solve his integration problems. David Hilbert (1862–1943) presented to the International Congress of Mathe- maticians, 1900 (ICM 1900) in Paris a series of 23 research projects. He stated in this lecture that his Fifth Problem is linked to Sophus Lie theory of transformation groups, i.e., Lie groups act as groups of transformations on manifolds. A translation of Hilbert’s fifth problem says “It is well-known that Lie with the aid of the concept of continuous groups of transformations, had set up a system of geometrical axioms and, from the standpoint of his theory of groups has proved that this system of axioms suffices for geometry”. For this appendix, the books Bredon (1993), Chevelly (1957), Pontragin (1939), Sorani (1969), Switzer (1975) and some others are referred in Bibliography. A.1 Topological Groups: Definitions and Examples This section introduces the concept of topological groups with illustrative examples. A topological group is simply a combination of two fundamental concepts: group and topological space and hence the axiomatization of the concept of topological groups © Springer India 2016 569 M.R. Adhikari, Basic Algebraic Topology and its Applications, DOI 10.1007/978-81-322-2843-1 570 Appendix A: Topological Groups and Lie Groups is a natural procedure. O. Schreior (1901–1929) gave a formal definition of modern concept of topological groups in 1925 and F. Leja (1885–1979) in 1927 in terms of topological spaces. Topological groups are groups in algebraic sense together with continuous group operations. This means that the topology of a topological group must be compatible with its group structure. Definition A.1.1 A topological group G is a Hausdorff topological space together with a group multiplication such that TG(1) group multiplication m : G × G → G,(x, y) → xy is continuous; TG(2) group inversion inv : G → G, x → x−1 is continuous. The continuity in TG(1) and TG(2) means that the topology of G must be compatible with the group structure of G. The conditions TG(1) and TG(2) are equivalent to the single condition that the map G × G → G,(x, y) → xy−1 is continuous. Remark A.1.2 Some authors do not assume ‘Hausdorff property’ for a topological group. Example A.1.3 Rn (under usual addition) and S1 ={z ∈ C :|z|=1} (under usual multiplication of complex numbers) are important examples of topological groups. Wenow describe some classical topological groups GL(n, R), SL(n, R), O(n, R), SO(n, R) and their complex analogues. Definition A.1.4 (General linear group)GL(n, R) is the set of all n × n non- singular matrices with entries in R. It is a group under usual multiplication of matri- ces, called general linear group over R. Definition A.1.5 (Special linear group)SL(n, R) defined by SL(n, R) ={A ∈ GL(n, R) : det A = 1} is a subgroup of GL(n, R). Definition A.1.6 (Orthogonal group)O(n, R) defined by O(n, R) = {A ∈ GL(n, R) : AAt = I } is a subgroup of GL(n, R). Definition A.1.7 (Special orthogonal group)SO(n, R) defined by SO(n, R) = {A ∈ O(n, R) : det A = 1} is a group. Theorem A.1.8 The general real linear group GL (n, R) of all invertible n × n matrices over R is a topological group. This group is neither compact nor connected. Proof Let Mn(R) be the set of all n × n real matrices. Let A = (aij) ∈ Mn(R).We n2 can identify Mn(R) with the Euclidean space R by the mapping n2 f : Mn(R) → R ,(aij) → (a11, a12,...,a1n, a21, a22,...,a2n,...,an1, an2,...,ann). Appendix A: Topological Groups and Lie Groups 571 This identification defines a topology on Mn(R) such that the matrix multiplication m : Mn(R) × Mn(R) → Mn(R) is continuous. Let A = (aij) and B = (bij) ∈ Mn(R). Then the ijth entry in the product m(A, B) n 1 1 1 is aikbkj.AsMn(R) has the topology of the product space R × R × ...,×R k=1 (n2 copies), and for each pair of integers i, j satisfying 1 ≤ i, j ≤ n,wehavea 1 projection pij : Mn(R) → R , which sends a matrix A to its ijth entry. Then m is continuous if and only if the composite maps m pij 1 Mn(R) × Mn(R) −→ Mn(R) −→ R n are continuous. But pijm(A, B) = aikbkj, which is a polynomial in entries of A k=1 and B. Hence the composite maps pij ◦ m are continuous. GL (n, R) topologized as a subspace of the topological space Mn(R) is such that the matrix multiplication GL (n, R) × GL (n, R) → GL (n, R), (A, B) → AB is continuous. We next claim that the inverse map inv : GL (n, R) → GL (n, R), A → A−1 is continuous. The map inv : GL (n, R) → GL (n, R) ⊂ R1 × R1 × ...× R1(n2 copies) is continuous if and only if all the composite maps inv p jk GL (n, R) −→ GL (n, R) −→ R1, 1 ≤ j, k ≤ n are continuous. But each composite map p jk ◦ inv sends a matrix A to the jkth element of A−1, which is (1/ det A) (kjth cofactor of A), where det A = 0 ∀ A ∈ GL (n, R). Hence the composite maps p jk ◦ inv are continuous. Consequently, GL (n, R) is a topological group. The group GL (n, R) is not compact: Clearly, GL (n, R) is the inverse image of nonzero real numbers under the determinant function det : Mn(R) → R. 572 Appendix A: Topological Groups and Lie Groups The determinant function is continuous, since it is just a polynomial in the matrix co- efficient. Hence the inverse image of {0} is a closed subset of Mn(R). Its complement is the set of all nonsingular n × n real matrices is an open subset of Mn(R). Hence GL (n, R) is not compact. The group GL (n, R) is not connected: Clearly, the matrices with positive and negative determinants give a partition of GL (n, R) into two disjoint nonempty open sets. Hence GL (n, R) is not connected. ❑ Definition A.1.9 GL (n, C) is the set of all n × n nonsingular matrices with complex entries. It is a group under usual multiplication of matrices, called the general complex linear group. Theorem A.1.10 GL (n, C) is a topological group. It is not compact. Proof Every element A ∈ GL (n, C) is a nonsingular linear transformation of Cn n 2n over C.If{z1, z2,...,zn} is a basis of C , then {x1, y1,...,xn, yn} is a basis of R , where zi = xi + iyi . Every element A ∈ GL (n, C) determines a linear transforma- tion A˜ ∈ GL (2n, R) into a subset of GL (n, R). Since GL (n, C) is an open subset of a Euclidean space, it is not compact. ❑ Corollary A.1.11 The set U(n, C) ={A ∈ GL (n, C) : AA∗ = I }, is a compact subgroup of GL (n, C), where A∗ denotes the transpose of the complex conjugate of A. 2 Remark A.1.12 dimC GL (n, C) = n . Definition A.1.13 A homomorphism f : G → H between two topological groups G and H is a continuous map such that f is a group homomorphism. An isomorphism f : G → H between two topological groups is a homeomorphism and is also a group homomorphism between G and H. Example A.1.14 The special orthogonal group SO (2, R) and the circle group S1 are isomorphic topological groups under an isomorphism f of topological groups given by cos θ − sin θ f : SO (2, R) → S1, → eiθ. sin θ cos θ Remark A.1.15 For quaternionic analogue see the sympletic group SU(n, H) = {A ∈ GL (n, H) : AA∗ = I } (Ex. 10 of Sect. A.4). A.2 Actions of Topological Groups and Orbit Spaces This section introduces the concept of actions of topological groups and studies some important orbit spaces (thus obtained) with an eye to compute their funda- mental groups. Real and complex projective spaces, torus, Klein bottle, lens spaces, Appendix A: Topological Groups and Lie Groups 573 and figure-eight are important objects in geometry and topology and they can be represented as orbit spaces. Definition A.2.1 Let G be a topological group with identity element e and X a topological space. An action of G on X is a continuous map σ : G × X → X, with the image of (g, x) being denoted by gx such that (i) (gh)x = g(hx); (ii) ex = x, ∀ g, h ∈ G and ∀ x ∈ X.
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