THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 10

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

February 13, 1998

Pull Back Constructions

Given a vector bundle and a smooth map f on manifolds:

E
π
X  \overline{}_{f} B

we can define a bundle f^{∗}(E) → X by the requirement that the diagram commutes:

[figure: f^{∗}(E); f̂; E; p; ∘; π; X; f; B]

where f̂ is an isomorphisms on fibers. Thus f^{∗}(E)|_{x} ≅ E|_{f(x)}.

\underline{The construction:} Define the total space

f^{∗}(E) = {(x, e) ∈ X × E|f(x) = π(e)} ⊂ X × E.

Then p : f^{∗}(E) → X is given by (x, e) ↦ x. So p^{−1}(x) = {(x, e)|π(e) = f(x)} = E_{f(x)}.

For local triviality of f^{∗}(E): Let E|_{U_{α}} ⟶^{≃^{Ψ}} U × ℝ^{n} be the local trivialization and V = f^{−1}(U) ⊂ X.

Claim.

\begin{matrix}• f^{∗}(E)|_{V} ⊂ V × E|_{U} \\ • f^{∗}(E)|_{V} ≅ V × ℝ^{n}(using Ψ) \end{matrix}

Exercise 1. Prove the above claim.

Exercise 2. Show that if E = B × ℝ^{n}, then f^{∗}(E) = X × ℝ^{n}.

For transition functions: If {V_{α}} is an open cover for X such that {U_{α} = f(V_{α})} is a cover of B and such that bundles are locally trivial, then the transition functions related by

g_{αβ}^{f}(x) = (f^{∗}g_{αβ})(x) = g_{αβ}(f(x)).

Typeset by AMS-TEX

Exercise 3. Prove the above equations. Note. There is no corresponding construction for maps f : B → X, that is, given

E

π
B  \overline{}_{f} X

we cannot define a bundle f_{∗}(E) → X by f_{∗}(E)|_{x} = E|_{f^{−1}(x)} (for example, if f is not 1 − 1, then the failure of this recipe is clear)! Good Exercise 4 (Eugene Lerman). Consider

E
π
E  \overline{}_{π} B

Show that

(a) π^{∗}(E) is a bundle over B.

(b) π^{∗}(E) ≅ E ⊕ E. Exercise 5. Suppose that we have a bundle map between two bundles

E′ \overline{}^{f̂} E
[figure: p; π]
B′  \overline{}_{f} B

such that f̂ is an isomorphism on fibers. By the above construction applied to E

π
B′  \overline{}_{f} B

we have a pullback f^{∗}(E). Show that f^{∗}(E) ≅ E′.

Corollory. f^{∗}(E) is uniquely defined! Theorem 1. (Key Result): Suppose we have

E
π
X ⟶^{f_{0}} ; ⟶_{f_{1}} B

with f_{0} ≃ f_{1} (homotopic), then f_{0}^{∗}(E) ≅ f_{1}^{∗}(E). (That is, homotopic maps produce isomorphic pullback bundles.) Proof: (Bott & Tu) Suppose f_{t}^{∗}(E) ≅ F, for some f and t ∈ [0, 1].

Claim. We can find I_{ϵ} = (t − ϵ, t + ϵ) such that f_{s}^{∗}(t) ≅ F, ∀s ∈ I_{ϵ}.

Then we can cover [0, 1] with finite number of intervals. Hence f_{0}^{∗}(E) ≅ F ≅ f_{1}^{∗}(E). □ Proof of Claim: Look at bundles on X × I:

\begin{matrix}F & π^{∗}(F) f^{∗}(E) & E \\ ^{p_{F}} & [figure] & ^{p_{E}} \\ X \underline{}_{π} & X × I & \overline{}_{f} B \end{matrix}

Then we can get Hom(π^{∗}(F), f^{∗}(E)) on X × I and Iso(π^{∗}(F), f^{∗}(E)) ⊂ Hom(π^{∗}(F), f^{∗}(E)). If f_{t}^{∗}(E) ≅ F, then Hom(π^{∗}(F), f^{∗}(E))|_{X×{t}} has a section, say,

σ : X × {t} → Iso(π^{∗}(F), f^{∗}(E))|_{X×{t}} ⊂ Hom(π^{∗}(F), f^{∗}(E))|_{X×{t}},

where Hom(π^{∗}(F), f^{∗}(E))|_{X×{t}} = Hom(π^{∗}(F)|_{X×{t}}, f^{∗}(E)|_{X×{t}}).

To prove our claim, we must show

(a) σ extends to X × I_{ϵ}.

(b) The extension remains in Iso ⊂ Hom. For (a), we have the proposition:

Proposition. For any vector bundle V → X × I, if σ : X × {t} → V |_{X×{t}} is a smooth section, then σ can extend to X × I_{ϵ}, for some ϵ.

Proof: Temporarily omitted. □ But then, (b) follows easily, since by continuity σ will remain in Iso on a suitably small strip I_{ϵ}. □

Corollory. If B is contractible, then E → B is trivial.

Exercise 6. Using the above corollary to analogue bundles over S^{n}.

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801 E-mail address: bradlow@math.uiuc.edu