SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES

Transcription

1 SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES Definition 1. A fiber bundle with fiber F is a map p: E X with the property that any point x X has a neighbourhood U X for which there exists a homeomorphism over U, i.e. for which p ϕ U = π 2. ϕ U : F U p 1 (U) Remark 2. The projection π 2 : F X X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F. By definition, a fiber bundle is a map which is locally homeomorphic to a trivial bundle. The homeomorphism ϕ U in the definition is called a local trivialization of the bundle, or a trivialization over U. Example 3. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R S 1 is a covering projection with fiber Z. Cocyles. Let p: E X be a fiber bundle with fiber F. Then there exists an open cover U = {U i } of X and for each U i a trivializing homeomorphism ϕ i = ϕ Ui : F U i p 1 (U i ) over U i. On overlaps U ij = U i U j, we thus have maps over U ij ϕ j F U ij p 1 (U ij ) ϕ ij ϕ i F U ij ϕ ij = ϕ 1 i ϕ j, and these ϕ ij satisfy a cocycle condition { ϕij ϕ jk = ϕ ik ϕ ii = id. For a fixed x U ij, the map ϕ ij (, x) is a homeomorphism F F, and we can alternatively rewrite the ϕ ij as maps ϕ ij : U ij Homeo(F ), into the group of homeomorphisms of F into itself. If F is locally compact (as we will from now on assume), then φ ij is continuous iff ϕ ij is, and we will use these two points of view interchangeably, often simply writing ϕ ij for ϕ ij. In many examples, these ϕ ij land in a subgroup G of Homeo(F ), giving rise to the following definition. Definition 4. Let G Homeo(F ) be a group of homeomorphisms of F. (In other words, G is a group acting faithfully on F.) A fiber bundle p: E X with fiber F and structure group G is a fiber bundle for which there exists a trivialization for which the corresponding cocycle 1

2 2 SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES ϕ ij : U ij Homeo(F ) factors as U ij ϕ ij Homeo(F ) G These ψ ij then again satisfy the cocycle condition ψ ij ψ jk = ψ ik and ψ ii = id. ψ ij Example 5. Let V be a real or complex vector space, and let GL(V ) be the group of linear isomorphisms of V. A vector bundle over X is a fiber bundle with fiber V and structure group GL(V ). Another way of phrasing the same definition is: A vector bundle is a fiber bundle p: E X with fiber V for which each fiber p 1 (x) is a vector space, and for which each trivializing map F U p 1 (U) is linear on the fibers. (This implies that the fiberwise vector space structure is given by continuous maps E X E + E, X 0 E and R E E or C E E, see Exercises.) Definition 6. Let G be a topological group. A principal G-bundle is a fiber bundle p: P X with fiber G and group G, where G acts on itself by left translations. Thus, a principal G-bundle P X has trivializing maps ϕ i : G U i p 1 (U i ) such that the cocycle ϕ ij : G U ij G U ij is of the form ϕ ij (g, x) = (ψ ij (x) g, x) for a map ψ ij : U ij G. In the literature, one often encounters the following alternative definition of a principal bundle: Definition 6 (bis). Let G be a topological group. A principal G-bundle is a map p: P X equipped with a fiberwise right action of G on E, with the property that (i) The map P G P X P, (e, g) (e, e g) is a homeomorphism; (ii) P X has enough local sections. (By a fiberwise action we mean an action with the property that p(e g) = p(e), so that the map in (i) indeed lands in the fibered product P X P. Condition (ii) means that every point x X has a neighbourhood U x over which there exists a section s: U P of p, i.e. a map with p s = id. ) Let us try to see that these two definitions of principal bundle are equivalent. In one direction, given P X as in this last definition, the homeomorphism P G P X P in condition (i) gives raise to a division map δ : P X P G characterized by the identity e δ(e, e ) = e for any two e and e in the same fiber of p: P X. Thus, if s: U P is a local section, the map G U p 1 (U), (g, x) s(x) g is a homeomorphism, with inverse e (δ(sp(e), e), p(e)). So P X is a fiber bundle with fiber G. We still need to check that the structure group is G as well; more precisely, that the corresponding

3 SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES 3 cocycle is given by left translation. To this end, let s i : U i P and s j : U j P be two local sections, with trivializations ϕ i : G U i p 1 (U i ) and ϕ j : G U j p 1 (U j ) as above, i.e. ϕ i (g, x) = s i (x) g and ϕ j (g, x) = s j (x) g. The corresponding cocycle ϕ ij : G U ij G U ij is defined by the equation ϕ i (ϕ ij (g, x)) = ϕ j (g, x) or equivalently, s i (x) π 1 ϕ ij (g, x) = s j (x) g, so we find that π 1 ϕ ij (g, x) = δ(s i (x), s j (x)) g. In other words, the cocycle in the first definition of principal bundle can be read off from the local sections as ψ ij (x) = δ(s i (x), s j (x)). Now let us examine the other direction, and consider a principal bundle P X as in the first definition, with ϕ i : G U i p 1 (U i ) and ψ ij : U ij G as described there. Then clearly condition (ii) of the second definition is satisfied; in fact, any fiber bundle obviously has enough local sections. For condition (i), we first need to define an action of G on P. For e P, choose i with p(e) U i, and let (1) e g = ϕ i (π 1 (ϕ 1 i (e)) g, p(e)). Or in other words, the action is completely determined by (2) ϕ i (h, x) g = ϕ i (hg, x) This definition does not depend on the choice of i, because if x U ij and ϕ i (h, x) = e = ϕ j (k, x), then h = ψ ij (x) k and hence ϕ i (hg, x) = ϕ j (kg, x). To check that P G P X P, (e, g) (e, e g), is a homeomorphism, it suffices to check this over each open set U i X; i.e. to check that each map p 1 (U i ) G p 1 (U i ) Ui p 1 (U i ) is a homeomorphism. But via the trivializing maps ϕ i, this comes down to checking that the map (G U i ) G α (G U i ) Ui (G U i ) defined by α((h, x), g) = ((h, x), (hg, x)) is a homeomorphism, and this is obviously the case. This completes the proof that the two definitions of principal G-bundle are equivalent. Our next aim is to examine how the principal bundle can be reconstructed form its cocycle. Given P X, ϕ i and ψ ij as in the first definition, the ϕ i together define a map ϕ: G U i P on the disjoint union of the G U i. This map is an open surjection, but it is not injective. In fact, if we write elements of the sum as triples (i, g, x), then ϕ(i, g, x) = ϕ(j, h, y) iff x = y and g = ψ ij (x) h. Thus, if we define an equivalence relation on G U i by (i, g, x) (j, h, y) iff x = y and g = ψ ij (x) h, then the map ϕ becomes a homeomorphism over X, ( ) (3) G Ui / P. This homeomorphism respects the right G-action, defined on the left by [i, g, x] h = [i, gh, x] where the square brackets denote the equivalence classes. In this way, up to a G-equivariant

4 4 SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES homeomorphism (3), we have reconstructed the principal G-bundle from the cover {U i }, the group G and the cocycle {ψ ij : U ij G}! Of course, given the principal bundle P, the cocycle P was not at all unique. Let us fix the cover U = {U i }. (The variation in the cover will be discussed in the Exercises). Suppose {ϕ i } and {ϕ i } are two choices of trivialization over this cover U, defining cocycles ϕ ij and ϕ ij : U ij G, respectively. Suppose also that these two trivializations define the same right G-action on P. Then for any x U i and for the unit 1 G, there is a unique λ i (x) G for which and hence, for any g G ϕ i(1, x) = ϕ i (λ i (x), x), ϕ i(g, x) = ϕ i(1, x) g = ϕ i (λ i (x), x) g = ϕ i (λ i (x) g, x). This gives for each i a map λ i : U i G with ϕ i (g, x) = ϕ i(λ i (x) g, x) as above. For two indices i and j, we then find the following relation between these λ s and the cocycles: for x U ij and g G, we have on the one hand ϕ j(g, x) = ϕ j (λ j (x) g, x) = ϕ i (ψ ij (x)λ j (x) g, x) while on the other hand, ϕ j(g, x) = ϕ i(ψ ij(x) g, x) = ϕ i (λ i (x)ψ ij(x) g, x). Thus, for any x U ij, ψ ij (x)λ j (x) = λ i (x)ψ ij (x); or briefly (4) ψ ij λ j = λ i ψ ij. Conversely, suppose {ψ ij } and {ψ ij } are two cocycles related by maps λ i as in (4). Then there is an obvious homeomorphism between the two reconstructions (3) of a principal bundle, ( ) ( ) (5) Λ: G Ui / G Ui / Here is the equivalence relation given by ψ ij as in (3), and is the similar relation defined by the ψ ij. If we write the equivalence classes as [i, g, x] and [i, g, x] respectively, then Λ is defined by (6) Λ([j, h, x] ) = [j, λ j (x) h, x]. This is well-defined, because for two representatives (j, h, x) and (i, ψ ij (x) h, x) of the same -class, we have Λ([j, h, x] ) = [j, λ j (x) h, x] (by def) = [i, ψ ij (x)λ j (x)h, x] (by ) = [i, λ i (x)ψ ij(x)h, x] (by (4)) = Λ([i, ψ ij(x)h, x]). (by def)

5 SECTION 9: FIBER BUNDLES AND PRINCIPAL BUNDLES 5 This shows that Λ in (5), (6) is well-defined. It is obviously a homeomorphism, because it has an inverse defined in the same way (see Exercises). It also respects the G-action, as defined on the left of (3). What all these constructions discussed up to now really prove is a correspondence between principal bundles and cocycles, for any group G. A precise formulation uses the following definitions. Definition 7. Two principal G-bundles p: P X and q : Q X are said to be equivalent if there exists a homeomorphism α: P Q over X which respects the G-actions (so qα(e) = p(e) and α(e g) = α(e) g). Definition 8. Let U = {U i } be an open cover of X. A cocycle on U with values in G is a family {ψ ij : U ij G} satisfying the cocycle condition ψ ij ψ jk = ψ ik. Two cocycles {ψ ij } and {ψ ij } are called equivalent if there are maps λ i : U i G for which ψ ij λ j = λ i ψ ij on each U ij. The set of equivalence classes of principal G-bundles is denoted H 1 (U, G). Then what we have proved is the following. Theorem 9. There is a bijective correspondence between principal G-bundles over X trivialized by the cover U and elements of H 1 (U, G). Remark 10. H 1 (U, G) is a first example of cohomology with non-abelian coefficients. If G is abelian, then H 1 (U, G) is a group rather than just a set, and the definitions coincide with the ones in Section 7.