Lecture 1 This lecture gives the essential definitions of fiber and principal bun- dles as well as the construction of associated bundles. Any propositions will have relatively easy proofs which will be left as exercises. We start with the standard definition of a fiber bundle. All manifolds will be smooth. Definition 1. A fiber bundle of manifolds consists of manifolds F , E and B along with a map π : E → B such that for every b ∈ B there −1 exists a neighborhood Ub and a diffeomorphism φb : Ub × F → π (Ub) such that proj1 = π ◦φb where proj1 is the projection to the first factor. Examples include vector bundles, principal bundles and, by Ehres- mann’s Theorem, any smooth submersion with π proper. Exercise 1.1. Give an example to show that if π is a submersion but is not proper, then π : E → B is not necessarily a fiber bundle. Now we set up some general notation that will be in force through- out this lecture. Let G be a topological group and C a category whose objects are CW complexes plus extra structure. I.e. we take C to be a category with a faithful functor from C to the category of CW complexes and continuous maps. A G-representation in C is an ob- ject A of C and a continuous group homomorphism ψ : G → Aut(A) (here Aut(A) will be given the subspace topology of the compact open topology on End(A)). With this in mind, we define: Definition 2. i) A G-bundle with fiber A in C is a fiber bundle π : E → B with a right G action and with fiber F ≈ A as a G-space. ii) A G-bundle is a principal bundle if it is a G-bundle with fiber G in G − sp. Here G − sp is the category of right G-spaces and G acts on itself by right multiplication. Examples: i) The trivial bundle B × G → B is clearly a principal bundle. Exercise 1.2. Prove that a principal bundle π : P → B is equivalent to the trivial bundle iff it has a global section. A global section is a continuous map s : B → P such that π ◦ s = idB. (for definition of equivalent bundles, see definition later in Lecture 1) ) ii) The Hopf Bundle is obtained by letting S1 = {z ∈ C : |z| = 1} act on the 3-sphere S3 ⊂ C2 by multiplication. Exercise 1.3. Show that the quotient π : S3 → S3/S1 is a principal S1-bundle (or U(1) = S1) and that the base manifold is diffeomorphic to S2. 1 2 iii) A general example occurs when H < G is a closed subgroup for a Lie group G. Then π : G → G/H is a principal H bundle. A useful example of this is G = O(n) and H = O(n − k) × O(k). The base manifold of the bundle is just the real Grassmannian. We will need to have the category of G-bundles with fibers in C. In particular, we need a definition of morphisms between these bundles. Definition 3. Let ξ = (π : E → B) and ξ0 = (π0 : E0 → B0) be G-bundles with fibers in C. A map (f, fˆ) from ξ to ξ0 consists of a commutative diagram of continuous maps: fˆ E −→ E0 ↓ ↓ f B −→ B0 such that fˆ restricts to C morphisms on trivializations of the fibers. 0 Denote the set of such maps HomCB(ξ, ξ ) and note that this defines a category CB of G-bundles with fibers in C. We will list several examples of categories that one may study in a moment. For now the most obvious examples are C = V ectR or C = V ectC with groups GL(n, R) and GL(n, C) respectively. In these ˆ ˆ −1 0 −1 cases the maps (f, f) are such that f|π−1(b) : π (b) → (π ) (f(b)) is a linear map for all b. The notion of equivalent G-bundles is stronger than one might initially think. In particular, one only allows the base map to be the identity. Definition 4. An equivalence of G-bundles ξ = (π : E → B) and 0 0 0 ˆ ˆ ξ = (π : E → B) is a bundle map (IdB, f) such that f restricts to an isomorphism on every fiber. One of the reasons for studying principle bundles instead of vector bundles is that they encode the symmetry of the fiber in a convenient way. Vector bundles often are decorated with extra structures to specify the symmetry, or representation of the group. In a very general setting, we can go from principal bundles to vector bundles using the object A ∈ C and the representation G → Aut(A). Definition 5. Let ξ = (ρ : P → B) be a principal G-bundle and A ∈ C. The associated bundle of P with fiber A has total space P ×G A = P × A/ ∼ where (pg, a) ∼ (p, ga) for all g ∈ G. The map π : P ×G A → B is just π(p, a) = ρ(p) which is clearly constant on equivalence classes. 3 Exercise 1.4. i) Show that P ×G A is a G-bundle with fiber A and action g(p, a) = (gp, a) in some trivialized neighborhood of P over B. ˆ 0 ii) Show that if (f, f) ∈ HomCB(ξ, ξ ) is a map of principal bundles, ˆ then it induces a map (fA, fA) of the associated bundles. iii) Conclude that one can define the functor Assoc : PB → CBA from principal bundles to their associated bundles with fiber A. To indicate the importance of principal bundles and their abundant manifestations in geometry via associated bundles, we give the fol- lowing lists of categories C and groups G. Objects in the categories carrying non-trivial G representations should be apparent. Category C Group G real vector spaces GL(n, R) real oriented vector spaces SL(n, R) real vector spaces with positive definite metric O(n) real oriented vector spaces with positive definite metric SO(n) (oriented) complex vector spaces (SL(n, C)) GL(n, C) (oriented ) complex vector spaces with hermitian metric (U(n)) SU(n) real symplectic vector spaces Sp(2n, R) Differentiable Manifolds Diff(M) Differentiable Manifolds up to isotopy MCG(M) The last entry MCG(M) is the mapping class group of a manifold M. The associated bundle construction takes as input a principal bundle with group G above and outputs a vector bundle with fibers in C. In the case of differentiable manifolds, we obtain start with a principal Diff(M) bundle and end up with a fiber bundle with fiber M..