Math 225A Discussion Session Week 5 Notes November 1, 2019 Per a request, we'll spend today's discussion reviewing some material on vector bundles. Much (but not all) of this will be a repeat of material from lecture. Another relatively readable source is Conlon's Differentiable Manifolds (particularly sections 3.3 and 3.4). Vector bundles Definition. Let M be a smooth n-manifold, E a smooth (n + k)-manifold, and π : E ! M a smooth map. We will call π : E ! M a vector bundle of rank k if: −1 (1) For each p 2 M, Ep := π (p) has the structure of a k-dimensional vector space over R. (2) There is a locally trivializing cover of M for E. That is, an open cover fUαgα2A of M with diffeomorphisms −1 k α : π (Uα) ! Uα × R −1 such that p1 ◦ α = πjπ (Uα), where p1 is projection onto the first factor. k (3) For each α 2 A and p 2 Uα, αjEp : Ep ! fpg × R is a vector space isomorphism. We call E the total space, M the base space, and call each (Uα; α) a trivializing neighborhood. k Remark. Notice that E = M × R is a vector bundle of rank k, with the obvious projection map and local trivialization. The definition of vector bundles is meant to generalize this. A vector bundle has the local appearance of a product, but may have nontrivial global topology. Let us define a notion of equivalence for vector bundles. Definition. Let πi : Ei ! M be a vector bundle of rank k, for i = 0; 1. A vector bundle isomor- phism between π0 : E0 ! M and π1 : E1 ! M is a diffeomorphism ': E1 ! E2 such that the following diagram commutes: ' E0 E1 ; π0 π1 M −1 −1 and such that ': π0 (p) ! π1 (p) is a vector space isomorphism, for each p 2 M. 1 1 Example. Let M = S and consider the trivial bundle E = S × R, with π given by projection @ onto the first factor. We have a vector bundle isomorphism ': E ! TM defined as follows. Let @θ be a nonvanishing vector field on S1. Then we can define @ '(p; c) = c j 2 T M: @θ p p Check that ' is indeed a vector bundle isomorphism. Note that tangent bundles aren't usually trivial | when TM is trivial, we say that M is parallelizable. 1 Cocycles As we said above, a vector bundle has the local appearance of a product | that is, just a trivial bundle | and each trivializing neighborhood gives us a way of identifying this product structure. But what happens where the trivializing neighborhoods overlap? Say we have p 2 Uα \ Uβ. Then k we can identify Ep with R using either α or β: k α β k fpg × R Ep fpg × R : −1 k k So we have a vector space isomorphism β ◦ α : fpg × R ! fpg × R , which we can consider as k an element αβ(p) 2 GL(k). This tells us how to take a vector in fpg×R , as identified by α, and −1 produce its coordinates in the trivialization β. Notice that, by construction, βα(p) = ( αβ(p)) . What happens on triple intersections? This time we have a diagram of the form Ep α γ β ; k k k R R R where we've suppressed the first coordinate in the second row. Now αγ(p) 2 GL(k) is obtained from −1 −1 −1 γ ◦ α = ( γ ◦ β ) ◦ ( β ◦ α ); so αγ(p) = βγ(p) αβ(p). Note that we recover the previous property of αβ from this one by taking γ = α. Definition. We call the data of the open cover fUαgα2A of M and the maps f αβ : Uα \ Uβ ! GL(k)g the structure cocycle for the vector bundle π : E ! M. More generally, a GL(k)-cocycle on M is an open cover fVαgα2A of M along with maps 'αβ : Vα \ Vβ ! GL(k) satisfying the cocycle condition 'αγ = 'βγ'αβ; for all α; β; γ 2 A. So we obtain a GL(k)-cocycle on M from any vector bundle of rank k on M (namely, the structure cocycle). Let us now consider the converse problem of constructing a vector bundle from a GL(k)-cocycle. Let the open cover of our GL(k)-cocycle be given by fVαgα2A, with transition maps given by 'αβ : Vα \ Vβ ! GL(k). Consider the disjoint union ~ a k E := Vα × R ; α2A and define an equivalence relation on E~ by k k (p; v) 2 Vα × R ∼ (p; w) 2 Vβ × R if 'αβ(p) · v = w: 2 (Check that this is an equivalence relation.) Of course we have a projection mapπ ~ : E~ ! M, k defined on each Vα × R as projection onto the first component. This plays nicely with ∼, and thus we have π : E ! M: It remains to check that this is a vector bundle of rank k on M. Notice that 0 1 −1 a k π (p) = @ fpg × R A = ∼ : Vα3p k Each copy of fpg × R in the disjoint union has a natural vector space structure, and ∼ identifies all of these as a single vector space. Checking that we have a locally trivializing cover is left to you. 1 2 2 2 Example. Consider the following pair of GL(1)-cocycles on M = S = fx + y = 1g ⊂ R . First, take the open sets 1 Uα = Uγ = f(x; y) 2 S jy > −g and 1 Uβ = Uδ = f(x; y) 2 S jy < g; for some small > 0. We will construct one GL(1)-cocycle with open sets Uα and Uβ, and another with open sets Uγ and Uδ. We define 'αα;'αβ;'βα; and 'ββ to all be identically 1. Then the vector bundle constructed from the cocycle fUα;Uβ; 'αα;'αβ;'βα;'ββg is certainly trivial | it consists of two vector bundles over intervals (necessarily trivial) glued together trivially. On the other hand, we can define 'γδ according to x ' (x; y) = ; γδ jxj defining 'γγ;'δγ, and 'δδ in the obvious way. Then the vector bundle constructed from fUγ;Uδ; 'γγ;'γδ;'δγ;'δδg is isomorphic to the M¨obiusband. This time we take a pair of line bundles over the interval and glue them together with a twist. We will see below that this vector bundle is not isomorphic to 1 S × R. We can now identify the collection of vector bundles of rank k over M with the collection of GL(k)-cocycles on M, once we identify cocycles which lead to the same vector bundle. Definition. We say that a pair of GL(k)-cocycles on M is equivalent if there is a common GL(k)- cocycle on M in which both are contained. (i.e., the common GL(k)-cocycle contains every set and map which appears in each of the first two.) Note that in this (somewhat strange) formulation, the open sets in a GL(k)-cocycle need not be distinct. Here's a lemma which is left for you to prove: 3 Lemma 1. Equivalence of GL(k)-cocycles on M is an equivalence relation. Hint: Show that if two GL(k)-cocycles on M contain a common GL(k)-cocycle, then they are contained in a common GL(k)-cocycle. Here's another fact that's not so difficult to prove: Theorem 2. The isomorphism class of a vector bundle constructed from a GL(k)-cocycle depends only on the equivalence class of the cocycle. From this it follows that there is a canonical bijective correspondence between isomorphism classes of vector bundles of rank k on M and equivalence classes of GL(k)-cocycles on M. Example. Consider the GL(1)-cocycles on S1 constructed in the previous example. We suspect that the nontrivial GL(1)-cocycle determines a nontrivial vector bundle, but how do we prove this? One thing we can do is this: if the bundle is trivial, then the GL(1)-cocycle is equivalent to the trivial GL(1)-cocycle defined above. In particular, the two GL(1)-cocycles are contained in a common cocycle. If this is the case, then the sign of 'αγ must be constant, since Uα \ Uγ is connected. Similarly, the sign of 'βδ is constant. The cocycle condition tells us that 'αγ = 'βγ'αβ = 'δγ'βδ'αβ: But then 'αγ(1; 0) = 'δγ(1; 0)'βδ(1; 0)'αβ(1; 0) = 'βδ(1; 0); while 'αγ(−1; 0) = 'δγ(−1; 0)'βδ(−1; 0)'αβ(−1; 0) = −'βδ(−1; 0); contradicting our assertion that the signs of 'αγ and 'βδ are constant. So these GL(1)-cocycles are inequivalent, and thus the corresponding vector bundles are not isomorphic. Perhaps less trivial is checking that every GL(1)-cocycle on S1 is equivalent to one of these two. Reduction of the structure group The GL(k)-cocycle (equivalence class) associated to a rank k vector bundle π : E ! M provides open sets of M over which E is trivial, as well as transition data telling us how to patch together the various local pieces. The map 'αβ : Uα \ Uβ ! GL(k) should be thought of as a \change of coordinates," telling us how to translate from the coordinates on Ep given by α into those given by β, for each p 2 Uα \ Uβ. The fact that 'αβ takes values in GL(k) corresponds to our requirement that the change of coordinates maps (on the fibers Ep) be vector space isomorphisms. We can sometimes ask more of our vector bundles, insisting that this transition data be valued in some subgroup of GL(k).