1
With the definition of a vector bundle from class, show that the vector space operations define continuous maps:
+:E×BE→E×:R×E→E
Definition of vector bundle: need charts (U,ϕ) with ϕ:π−1(U)→U×Rn which when restricted to a fiber Fb yields an isomorphism Fb∼→Rn. What are these maps??
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2.
Suppose you are given the following data:
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Topological spaces B and F
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A set E and a map of sets π:E→B
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An open cover U={Ui} of B and for each i a bijection ϕ:π−1(Ui)→Ui×F so that π∘ϕi=π.
Give conditions on the maps ϕi so that there is a topology on E making ϕ:E→B into a fiber bundle with {(Ui,ϕi)} as an atlas.
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3.
An oriented n-dimensional vector bundle is a vector bundle π:E→B together with an orientation of each fiber Eb, so that these orientations are continuous in the following sense.
For each b∈B there is a chart (U,ϕ) with b∈U and ϕ:π−1(U)→U×Rn so that for all b′∈U, ϕ|Eb′:Eb′→Rn is orientation-preserving.
Show that given an oriented n-dimensional vector bundle there is an induced principal GL+(Rn)-bundle (the “bundle of oriented frames”), and conversely given a principal GL+(Rn)-bundle there is an induced oriented n-plane bundle.
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4.
A Riemannian metric on a vector bundle π:E→B is an inner product ⟨⋅,⋅⟩b on each fiber Eb of E, which is continuous in the sense that the induced map E⊕E=E×BE→R is continuous.
Show that given a Riemannian metric on a vector bundle, there is an induced principal O(n)-bundle (the “bundle of orthonormal frames”), and conversely given a principal O(n)-bundle there is an induced vector bundle with Riemannian metric.
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5.
What operation on principal O(n)-bundles corresponds to dualizing a vector bundle? What about the direct sum of vector bundle?
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6.
For nice spaces X (e.g. CW complexes) and abelian groups G, there is a canonical isomorphism ˇHi(X;G)≅Hi(X;G) between Čech and singular cohomology of X with coefficients in G.
A nice, readable proof can be found in Frank Warner’s Foundations of Differential Manifolds and Lie Groups, Chapter 5. In the rest of this problem, cohomology either means Čech cohomology or singular cohomology after applying this isomorphism.
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a: Let π:E→B be an n-dimensional vector bundle, or equivalently, a principal GL(n,R)-bundle, given by a Čech cocycle ϕ∈H1(B;GL(n,R)). Show that the sign of the determinant sgndet:GLn(R)→{±1}≅Z/2Z induces a map ˇH1(B;GL(n,R))→ˇH1(B;Z/2Z), and so ϕ induces an element w1(E)∈H1(B;Z/2Z).
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b: Compute w1 for the trivial line bundle (1-dimensional vector bundle) over the circle and for the Möbius band.
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c: Prove that (for nice spaces) a line bundle π:E→B is trivial if and only if w1(E)=0∈ H1(B;Z/2Z)
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7.
Show that the exact sequence of abelian topological groups
0→Z→R→S1=GL(1,C)→0
induces an exact sequence in Čech cohomology
ˇH1(B,Z)→˘H1(B,R)→ˇH1(B;S1)δ→ˇH2(B;Z)
Given a complex line bundle (principal GL(1,C)-bundle) π:E→B coming from the cocycle data ϕ∈H1(B;GL(1,C)), let c1(E)=δ(ϕ). Compute c1(E) for some complex line bundle over S2.
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