THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 9

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

February 11, 1998

New Bundles From Old Bundles II

Given two bundles

E = (∐ U_{α} × ℝ^{n})/{g_{αβ}}

and

F = (∐ U_{α} × ℝ^{n})/{h_{αβ}}.

Claim. There exists a bundle Hom(E, F) such that rank = n · m and Hom(E, F)_{b} = Hom(E_{b}, F_{b}).

Proof: There are two points of view.

(1) For vector spaces V and W, Hom(V, W) ≅ W ⊗ V^{∗}.

Exercise 1. Show that Hom(V, W) ≅ W ⊗ V^{∗}.

We can define

Hom(E, F) := F ⊗ E^{∗}

which has the properties: (i) rank nm, (ii) fiber (F ⊗E^{∗})_{b} = F_{b}⊗E_{b}^{∗}, and (iii) transition functions h_{αβ}⊗g_{αβ}^{t}^{−1}. (2) More directly: We can identify Hom(E, F) by looking at its sections.

Exercise 2. Show that s ∈ Γ(Hom(E, F)) ⟺ s : E → F is a bundle map.

Using local trivializations, bundle maps can give local descriptions

s_{α} : U_{α} → Hom(ℝ^{n}, ℝ^{m})

with

(∗)  s_{β} = h_{βα}s_{α}g_{βα}^{−1}.

\underline{Question:} For which bundle is (∗) the description of a section?

\underline{Answer:} Take principal bundles

P_{F} = ∐ U_{α} × GL(m)/{h_{αβ}}

and

P_{E} = ∐ U_{α} × GL(n)/{g_{αβ}}.

Typeset by AMS-TEX

Define

P = P_{F} × P_{E} = ∐ U_{α} × (GL(m) × GL(n))/{h_{αβ} × g_{αβ}}.

Take the representation of GL(m) × GL(n) on Hom(ℝ^{n}, ℝ^{m}) given by

ρ : GL(m) × GL(n) ⟶^{Ad} GL(Hom(ℝ^{n}, ℝ^{m})),

that is, such that ρ(A, B)(U) = A ∘ U ∘ B^{−1}. Then the bundle we want is

Hom(E, F) ≡ P ×_{Ad} Hom(ℝ^{n}, ℝ^{m}).

Exercise 3. Show that sections of this are given by (∗).

Exercise 4. Show that Hom(E, F) ≅ E^{∗} ⊗ F!

Remark. The case E = F is an important special case. In this case, Hom(E, E) = End(E).

Exercise 5. Show that End(E) = P_{E} ×_{Ad} Mat_{n}, where

Ad : GL(n) ⟶ GL(Mat_{n})
A ⟼ (U ↦ A ∘ U ∘ A^{−1}).

Note. If we define Aut(E) to be the bundle of isomorphisms of E such that Aut(E)|_{p} = Aut(E_{p}), then Aut(E) = P_{E} ×_{Ad} GL(n) is a bundle of groups, but not quite a principal bundle (transition functions are not quite by left action of structure group on fibers). If 𝔊 is the space of sections of Aut(E), then 𝔊 can be given a group structure using fiberwise identifications Aut(E|_{P}) ≃ GL(n). The group 𝔊 is the gauge group of E.

Kernel, Image, and Cokernel Bundles

Given a bundle map

E  \overline{}^{f} F
[figure]
B

Define

(Ker(f))_{b} := Ker(f_{b} : E_{b} → F_{b})
(Im(f))_{b} := Im(f_{b} : E_{b} → F_{b})
(Coker(f))_{b} := F_{b}/(Im(f)_{b})

\underline{Questions:}

(1) Is ∪_{b∈B}(Ker(f))_{b} := Ker(f) a bundle over B?(i.e. a subbundle of E.)

(2) Is ∪_{b∈B}(Im(f))_{b} := Im(f) a subbundle of F?

(3) What about ∪_{b∈B}(Coker(f))_{b} := Coker(f)?

\underline{Answers:} In general, the answers are \underline{not} YES. For example, take trivial bundles:

ℝ^{n} × ℝ^{m}  \overline{}^{f} ℝ^{n} × ℝ^{m}
[figure]
ℝ^{n}

and define f(x) = x · I_{n}. (It is certainly a bundle map.) Then

(Ker(f))_{x} = \{ \begin{matrix}0, & for x ≠ 0 \\ ℝ^{n}, & for x = 0. \end{matrix}

So fiber dimensions can jump!

Theorem 1. If Rank f_{b} is \underline{constant}, then Ker f, Im f, and Coker f are bundles.

Note. The proof uses “Constant Rank Theorem” for bundle maps. It amounts to showing that trivializations for E and F can be chosen so on each U_{α},

E_{b}⟶^{≃^{Ψα}}ℝ^{n} = ℝ^{k} ⊕ ℝ^{n−k},

where ℝ^{k} = Ker(f_{α}), and

F_{b}⟶^{≃^{Φα}}ℝ^{m} = ℝ^{i} ⊕ ℝ^{m−i},

where ℝ^{i} = Im(f_{α}). Then, on U_{α} ∩ U_{β}, we have the commutative diagram (w.r.t. given decompositions of ℝ^{n}):

ℝ^{k} ⊕ ℝ^{n−k}  \overline{}^{f_{α}} ℝ^{i} ⊕ ℝ^{m−i}
[figure: g_{βα}; h_{βα}]
ℝ^{k} ⊕ ℝ^{n−k}  \overline{}_{f_{β}} ℝ^{i} ⊕ ℝ^{m−i}

with

g_{βα} = ( \begin{matrix}k_{βα} & Λ_{βα} \\ 0 & C_{βα} \end{matrix} )

h_{βα} = ( \begin{matrix}i_{βα} & Π_{βα} \\ 0 & D_{βα} \end{matrix} )

Note. {k_{βα}} are transition functions for Ker f. Similarly for Im f and Coker f.

Exercise 6. Using \underline{orthonormal frames} (assuming E and F have matrices), we can assume g_{βα} ∈ O(n) and k_{βα} ∈ O(n). Show that it follows that Λ_{βα} = 0 and Π_{βα} = 0.

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801

E-mail address: bradlow@math.uiuc.edu