THE GEOMETRY OF VECTOR BUNDLES AND AN INTRODUCTION TO GAUGE THEORY LECTURE 37

Professor Steven Bradlow Class Notes From Math 433 University of Illinois at Urbana-Champaign

April 29, 1998

Čech Cohomology and Orientability

Question: Given a vector bundle E = ∐ U_{α} × ℝ^{n}/g_{αβ}, g_{αβ} ∈ O(n). When can local frames be chosen such that g_{αβ} ∈ SO(n) ?

\underline{Last time}: Say we can find local gauge transformations λ_{α} : U_{α} → O(n) such that under

E|_{U_{α}} ⟶^{≃^{ψα}} U_{α} × ℝ^{n} ⟶^{1×λ_{α}} U_{α} × ℝ^{n} ,

new transition functions g̃_{αβ} = λ_{α} ∘ g_{αβ} ∘ λ_{β}^{−1} satisfy

det g̃_{αβ} = 1 .

We can reformulate this using

\begin{aligned}det(g_{αβ}) & = e^{iπu_{αβ}} \\ det(λ_{α}) & = e^{iπv_{α}} \end{aligned} u_{αβ} , v_{α} ∈ \{0, 1\} = ℤ_{2} .

Then

the cocycle condition ⟹ u_{αβ} + u_{βγ} + u_{γα} = 0  (closed) ,
and the condition det g̃_{αβ} = 1 ⟹ u_{αβ} = v_{β} − v_{α}  (exact) .

Cohomology Intepretation (using Čech cohomology with ℤ_{2} coefficients)

• Fix “suitable” cover {U_{α}}_{α∈I} for B, where “suitable” means all intersections are contractible and connected.

• Define a 0-cochain to be

g^{(0)} = { g_{α} ∈ ℤ_{2} | α ∈ I } (so g^{(0)} defines a map g^{(0)} : I → ℤ_{2}) .

• Define a 1-cochain to be

g^{(1)} = { g_{αβ} ∈ ℤ_{2} | α ≠ β, U_{α} ∩ U_{β} ≠ ∅, g_{αβ} = −g_{βα}(= g_{βα} in ℤ_{2} !) }.

Typeset by AMS-TEX

• Define a j-cochain to be

g^{(j)} = { g_{α_{0}···α_{j}} ∈ ℤ_{2} | ∀i ≠ j, α_{i} ≠ α_{j}, ⋂_{i=0}^{j} U_{α_{i}} ≠ ∅, g_{α_{0}···α_{j}} antisymmetric w.r.t. indices } .

• Define Č^{(j)} = Group of all j-cochains with group operation g^{(j)}+f^{(j)} = { (g+f)_{α_{0}···α_{j}} = g_{α_{0}···α_{j}} +f_{α_{0}···α_{j}} } .

• Define δ : Č^{(j)} → Č^{(j+1)} by

(δg^{(j)})_{α_{0}···α_{j+1}} := ∑_{i=0}^{j+1}(−1)^{i}g_{α_{0}···α̂_{i}···α_{j+1}}^{(j)} ,

where the “ˆ ” on α̂_{i} means “delete from the list of indices”. For example,

δ : C^{(0)} → C^{(1)}  (δg^{(0)})_{αβ} = g_{β}^{(0)} − g_{α}^{(0)}
δ : C^{(1)} → C^{(2)}  (δg^{(1)})_{αβγ} = g_{βα}^{(1)} − g_{αγ}^{(1)} + g_{αβ}^{(1)} .

Key Facts: In

Č^{(0)} ⟶^{δ^{(0)}} Č^{(1)} ⟶^{δ^{(1)}} Č^{(2)} ⟶^{δ^{(2)}} · · · .

(a) δ is a group homomorphism.

(b) δ^{2} = 0. Exercise 1. Prove the above key facts. Thus Im δ^{(p)} ⊆ Ker δ^{(p+1)} and we can form

Ȟ^{(p)} (\{U_{α}\}_{α∈I}, ℤ_{2}) = \begin{matrix}Ker & δ^{(p)} \\ Im & δ^{(p−1)} \end{matrix} .

Facts:

(1) Take Ȟ^{(0)} = Ker δ^{(0)}.

(2) For suitable covers, this result is independent of {U_{α}}_{α∈I}.

(3) Ȟ^{(p)}(B, ℤ_{2}) ≅ H^{p}(B, ℤ_{2}).

Back to {u_{αβ}} and {v_{α}}_{α∈I}

• Clearly {u_{αβ}} defines a Čech 1-cochain.

• u_{αβ} + u_{βγ} + u_{γα} = 0 says δ^{(1)}({u_{αβ}}) = 0. (since g_{αβ} = g_{βα}^{−1}, we get u_{γα} = −u_{αγ} etc.)

• Hence {u_{αβ}} defines elements in H^{(1)}(B, ℤ_{2}) = H^{1}(B, ℤ). [This is called w_{1}(E), the first Stiefel-Whitney class.]

• {v_{α}}_{α∈I} defines v^{(0)} ∈ Č^{(0)} and u_{αβ} = v_{β} − v_{α} says {u_{αβ}} = δv^{(0)}, i.e., w_{1}(E) = 0. Conclusion. E is orientable ⟹ w_{1}(E) .

Remark. The converse holds too:

Consider a vector bundle E = ∐ U_{α} × ℝ^{n}/g_{αβ}. Define the Čech cochain u^{(1)} = {u_{αβ}} as above, i.e., det g_{αβ} = e^{iπu_{αβ}}. As above, this defines w_{1}(E) = [u^{(1)}] ∈ H^{1}(B, ℤ_{2}).

Say w_{1}(E) = 0, i.e., [{u_{αβ}}] = 0 ∈ Ȟ^{(1)}(B, ℤ_{2}). It follows that we can find {v_{α}} such that u_{αβ} = v_{β} −v_{α}. Claim. Given {v_{α}} with v_{α} : U_{α} → ℤ_{2}, we can find λ_{α} : U_{α} → O(n) such that det λ_{α} = e^{iπv_{α}}. Exercise 2. Prove the above claim.

It follows from this, using λ_{α} to adjust local trivialization, that we get g̃_{αβ} ∈ SO(n). That is, E is orientable.

We have thus proved: Proposition. E is orientable ⟺ w_{1}(E) = 0 .

Higher Stiefel Whitney Classes and Obstructions

• Will see that w_{1}(E) is not the only ℤ_{2}-characteristic class. Hence w_{2}(E) ∈ H^{2}(B, ℤ_{2}) etc.

• w_{2}(E) is also an obstruction — to the existence of a Spin structure.

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

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