Defn: G is a topological group if 1.) (G, ∗) is a group 2.) G is a topological space. 3.) ∗ : G × G → G, ∗(g_{1}, g_{2}) = g_{1} ∗ g_{2}, and In : G → G, In(g) = g^{−1} are both continuous functions.

Defn: G is a Lie group if 1.) G is a topological group 2.) G is a smooth manifold. 3.) ∗ and In are smooth functions.

Ex: (R, +), (R−{0}, ·), (C−{0}, ·), (S^{1}, ·) where S^{1} ⊂ C, (Z, +), (Z_{p}, +), (Gl(n, R), matrix multiplication) are Lie groups. For G_{1}, G_{2} lie groups, G_{1} × G_{2} is a lie group. Defn: G = group, X = set. G acts on X (on the left) if ∃σ : G × X → X such that

1.) σ(e, x) = x ∀x ∈ X

2.) σ(g_{1}, σ(g_{2}, x)) = σ(g_{1}g_{2}, x)

Notation: σ(g, x) = gx. Thus 1) ex = x; 2) g_{1}(g_{2}x) = (g_{1}g_{2})(x). If G is a topological group and X is a topological space, then we require σ to be continuous. If G is a Lie group and X is a smooth manifold, then we require σ to be smooth.

Defn: The orbit of x ∈ X = G(x) = {y ∈ X | ∃g such that y = gx} Note: 1.) x ∈ G(x) 2.) If G(x) ∩ G(y) ≠ ∅, then G(x) = G(y) Thus we can use an action of G to partition X into disjoint subsets. Hence the action of G on X can be used to define an equivalence relation on X: x ∼ y iff y ∈ G(x) iff ∃g such that

y = gx. X/G = X/ ∼.

If X is a topological space, then X/G = X/ ∼ is a topological space with the quotient topology. When is X/G = X/ ∼ a manifold? Ex: G = (Z, +), M = R, σ(n, x) = n + x. M/G = Ex: G = (Z × Z, +), M = R^{2}, σ((n, m), (x, y)) = (n + x, m + y). M/G = Ex: G = (Z_{2}, +), M = S^{n}, σ(0, x) = x, σ(1, x) = −x, . M/G = Defn: The action of G on X is free if gx = x implies g = e. Thm 1.3.9: If M is a smooth n-manifold, and G is a finite Lie group acting freely on M, then M/G is a smooth nmanifold. Also, p : M → M/G is smooth. Cor: Defn: G is a discrete group if

0.) G is a group. 1.) G is countable 2.) G has the discrete topology

Note a discrete group is a Lie group. Defn: The action of G on M is properly discontinuous if ∀x ∈ M, ∃U^{open} such that x ∈ U and U ∩ gU = ∅ ∀g ∈ G. Ex: (Z, +) acting on R^{1} where σ(n, x) = n + x. Thm 1.3.2: M smooth n-manifold, G discrete group acting properly discontinuously on M implies M/G is a smooth n-manifold. Also, p : M → M/G is smooth.