Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this method, recall the proof that if H G is a subgroup of a finite group G then the order of H, denoted H, divides the order of G, denoted G. Consider the action of H on G by right translation. Each orbit is isomorphic as a set to H, and G is a disjoint union of these orbits. This leads to the formula G = G/H H, where G/H is the set of left cosets of H, i.e. the space of H-orbits. Definition 0.1. An action of G on S consists of a map Φ : G S S, such that if we write Φ(g, s) = gs then g 1 (g 2 s) = (g 1 g 2 )s and 1s = s. We also say S is a G-space. Definition 0.2. The orbit of s S is the set subset Gs S. The G-action is called transitive if S consists of a single G orbit. In this case S is also called a homogeneous space. The action is called free if whenever gs = hs we have g = h. Examples. Each orbit of a G-space is a homogeneous space. The coset space S = G/H is a homogeneous space, where G acts according to g(g 1 H) = (gg 1 )H. All homogeneous spaces look like a coset space. To see this, we make Definition 0.3. The isotropy group of s 0 S consists of the subgroup of G which does not move s 0 ; thus G s0 = {g G : gs 0 = s 0 }. and Definition 0.4. An equivariant map F : S X between G-spaces is a map such that F (gs) = gf (x). To G-spaces are equivalent if there is an invertible equivariant map between them. Theorem 0.5.. If G acts transitively on S, then the orbit map G S defined by g gs 0 induces an isomorphism G/G s0 S between G-spaces: Over to manifolds. The results for finite groups acting on (finite) sets carry over to manifolds. Begin by recalling what a group is in the category of manifolds: a group and a manifold in such a way that the two structures talk to each other. We call this a Lie group. What we mean when we say that the two structures talk to each other is that (definition of a Lie group!) the multiplication map G G G and the inversion map G G are both smooth maps. Definition 0.6. The Lie group G acts smoothly on the manifold S if it acts on S as a set, and if the map Φ : G S S which defines the action is smooth. We also say S is a smooth G-space. The definition of orbit, transitive and free actions remain the same. The definition of equivariant map also carries over, except, of course, we insist that the map is smooth. Our first goal is to understand the analogue of theorem 0.3, and its many representative examples. Example 1. The sphere is a homogeneous space for the group G = SO(3) of rotations. Indeed, G acts smoothly on R 3, and maps S 2 to itself, so it acts smoothly 1
2 on S 2. This action on the sphere is transitive. The isotropy group G N SO(3) of the north pole N S 2 is the group of rotations about the z-axis. It is diffeomorphic to a circle S 1 and is embedded in SO(3). The smooth analogue of theorem 0.3 then will assert that SO(3)/S 1 = S 2. To obtain a smooth version of theorem 0.3 we will need to put a manifold structure on the coset space G/H where H G. Not any subgroup H will work, but those that arise as isotropy groups for smooth proper actions will work. Subgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold. Theorem 0.8. A subgroup of G which is also an embedded submanifold is a closed Lie subgroup. This is Lee s prop. 8.30. Examples. A. Irrational flow on the torus. This gives R T 2 as a Lie subgroup which is not an embedded submanifold, and not a closed submanifold. The quotient space T 2 /R is not a manifold. Indeed, it is not even Hausdorff. The remaining examples are closed, embedded Lie subgroups of their ambient groups. B. By the IFT and the theorem O(n) = {A Gl(n, R) : A T A = Id.} is a closed Lie subgroup of Gl(n, R). To apply the IFT, consider the map X X T X from the n 2 dim mfd GL(n, R) to the ( ) n+1 2 dim linear space of symmetric n n matrix. Check that the identity is a critical value. C. U(n) = {A Gl(n, C) : A A = Id.} is a closed Lie subgroup of Gl(n, C). To apply the IFT, consider the map X X X from the 2n 2 (real) dim mfd GL(n, C) to the n 2 dim linear space of Hermitian n n matrix. Check that the identity is a critical value. D. SL(n, R) or Sl(n, C). Write K for either field, R or C. By definition, SL(n, K) is the subgroup of Gl(n, K) consisting of matrices of determinant 1. The determinant map det : Gl(n, K) K is polynomial hence smooth. Show 1 K is a critical value. conclude SL(n, K) is a Lie subgroup of Gl(n, K). E. The intersection of two closed Lie subgroups is another Lie subgroup. In particular, SO(n) = SO(n) Sl(n, R) and SU(n) = U(n) Sl(n, C) are closed Lie subgroups of their Gl(n)s. Theorem 0.9. Let S be a homogeneous space for G. Then, for each s S, the isotropy subgroup G s G is a closed Lie subgroup of G. Half of proof: Fix s. Consider the orbit map θ s : G S; g gs. It is smooth, and hence continuous. G s = θ s 1 (s) and so is a closed subset of G. It is also a subgroup. If we knew that θ s were a submersion, we could apply the IFT to conclude that G s were an embedded submanifold. That θ s is a submersion will follow from:
3 Theorem 0.10 (Equivariant rank theorem). Suppose that F : X N is a smooth equivariant map between smooth G-spaces, and that X is homogeneous with respect to G. Then F has constant rank and its image F (X) is a an immersed submanifold. We put off the proof of the equivariant rank theorem for the moment. Let us see how it proves that θ s is a submersion. The orbit map θ s satisfies the hypothesis of the equivariant rank theorem, since it is a smooth equivariant map, and G itself is a homogeneous G-space. Thus the differential of θ s has constant rank. I claim this differential is everywhere onto. For if not, by the constant rank theorem (Theorem 7.13, p. 167 of Lee) the image θ s (G) would be an immersed submanifold of the homogeneous space S, which contradicst the fact that θ s (G) = S. (The latter follows from the definition of homogeneous space: Gs = S,) Remark. Instead of using the equivariant rank theorem, we could have quoted Theorem 0.11. (Big Theorem) A closed subgroup of a Lie group is Lie. (This theorem is not the Montgomery-Zippin theorem as I had previously said!) I prefer the route we took, and not to use this big gun. Proof of Equivariant rank theorem For X a G-space and g G write: L g : X X; L g (s) = gs := Φ(g, s). Because L 1 g = L g 1 we have that each L g is a diffeo of S, and the map g L g is a group homomorphism G Diff(X). In particular the pushforward L g, also written dl g (x) or dl g,x are all linear isomorphisms. The x X indicates the point at which the derivative is taken. Now suppose, as per the theorem, that F : X S is an equivariant map. Equivariance means that F L g = L g F for all g G. The first L g is from the action on X, the second from the action on S. From the chain rule then df gx = dl g,f (x) df x dl 1 g,gx. If, now, X is a homogeneous space, every point y X can be written as y = gx and so the dl g,gx s provide linear isomorphisms between T x X and each T y X. It follows from (*) that the map F has constant rank. We can now invoke Lee s rank theorem for manifolds (p. 167, Theorem 7.13). QED Consider again S a smooth homogeneous space for G. Let G s G be the isotropy subgroup of s S, which we now know is a a closed Lie subgroup of G. In preparation for showing that S is diffeomorphic to G/G s, we need a manifold structure on G/G s. We may as well consider the case H G of any closed Lie subgroup of G and show how to put a smooth manifold structure on G/H. For this, the notion of a slice to a group action is useful. We start out more abstractly. Suppose that two submanifolds X, Y Q intersect at a point x. The intersection of X and Y is said to be transverse if T x X + T x Y = T x Q. Example: Suppose that X Q is an embedded k-dimensional submanifold of the n-manifold Q. Let x 1,..., x k, x k+1,..., x n be slice coordinates for X. Let Y be the manifold coordinatized by x k+1,..., x n.. In other words: if we write φ : U R k R n k for the slice coordinates, then X U = φ 1 (R k 0) and Y = φ 1 (0 R n k. Then Y is an n k dimensional disc intersecting X transversally at the point φ 1 (0, 0).
4 Theorem 0.12. If H G is a closed Lie subgroup, then G/H admits a unique manifold structure such that G G/H is a submersion. Proof. H G is an embedded submanifold. Let Y be a transverse n k disc, as above. Step 1. There is a nbhd V H of e and a perhaps smaller n k dimensional transverse Y to H at e such that Y V G sending (y, h) to yh is a diffeo of Y V onto a nbhd of e in G. The differential of this map at (e, e) is a linear isomorphism. Step 2. Taking Y small enough, we can insure that whenever y = ỹh with y, ỹ Y and h H, then h = e. This step requires that H is closed. Proof. If not, there exists sequences y n, ỹ n e and h n such that ỹ n = y n h n. By continuity, yn 1 ỹ n e. This implies that h n e in G. Now use that H is embedded, so that the topology on H agrees with the topology it inherits from G. (To see how this could go wrong, think on the irrational line immersed in the torus.) So h n is eventually in the nbhd V H of step 1. In step 1 we saw that Y V G was a diffeo onto its image. This means that ỹ n = y n h n implies that h n = e and ỹ n = y n. QED HERE is where we see the meaning of the word slice : Y slices across each coset gh at most once. Step 3. Y H = {yh : y Y, h H} is an open nbhd of Y. Proof. Y V = U is an open nbhd of e in G, The maps R h (g) = gh are diffeos, and Y H = h H R h (U). Step 4. The composition Y G G/H is a local homeo. Proof. It is continuous. Step 2 asserts that it is one-to-one. Because the projection is open, it is an open mapping. Step 5.Cover G/H by the translates gy of Y. The compositions gy G G/H are local homeos. Step 6. With the charts of step 5, the overlaps are smooth. Proof. Suppose that g 1 Y g 2 Y. Let gh pr(g 1 Y ) pr(g 2 Y ). (We write pr : G G/H for the canonical projection.) Then we have: gh = g 1 y 1 H = g 2 y 2 H. Some algebra yields the relation y 1 = g1 1 g 2y 2, so that the overlap map is given by left multiplication by g 1 1 g 2, a diffeo. QED Theorem 0.13. If S is a smooth homogeneous space for G, then S is G-equivariantly diffeomorphic to the coset space G/H where H = G s is the isotropy group of a fixed (but arbitrary) s S. Proof. From the above, we have smooth submersions: G G/H; G S. From general group theory, the map G S factors through to define a set-theoretic map G/H S which is a G-equivariant set-theoretic isomorphism. We now use: Theorem 0.14 (Submersion theorem). If f : M N is a submersion, and if g : M Y is a map which is constant along the fibers of f, then the induced map F : N Y is smooth, and the rank of the differential at the point f(m) equals the rank of g at n.
5 Proof. See Lee. This theorem is a version of his theorems 7.17, 7.18. It follows from this submersion theorem, that the induced map G/H S is smooth, and by the equivariant rank theorem, its differential is onto. Consequently it is a submersion onto its image. We have already seen that it is 1-1 and onto. The roles of G/H and S can be reversed to see that the inverse map is also smooth. QED. Examples. R n as Aff(n)/Gl(n) S n = SO(n + 1)/SO(n) = O(n + 1)/O(n) CP n = U(n + 1)/U(1) U(n) S 2n+1 = U(n + 1)/U(1) RP n = Gl(n + 1)/H COMPUTE H... and a huge variety more!! More on G G/H. this is the simplest nontrivial egs of a principal bundle Eg: G = SU(2) = S 3. H = S 1. Then G/H = S 2 and G G/H is the famous HOpf fibration. In general G G/H is a principal H bundle.... define...?? NEXT??? should cover: some Lie algebras some vector bundles; fiber bundles; principal bundles; normal bundles;... promised Frobenius integrability theorem and Chow non-integrabiliyt theorem applications of our differentiable manifolds technology to topology: proof of fund thm of algebra. hairy ball thm. index thms. smooth brower fixed pt thms...