1
With the definition of a vector bundle from class, show that the vector space operations define continuous maps:
\begin{align*} &+: E { \underset{\scriptscriptstyle {B} }{\times} } E \rightarrow E \\ &\times: \mathbb{R} \times E \rightarrow E \end{align*}
Definition of vector bundle: need charts \((U, \phi)\) with \(\phi: \pi^{-1}(U) \to U\times{\mathbb{R}}^n\) which when restricted to a fiber \(F_b\) yields an isomorphism \(F_b { { \, \xrightarrow{\sim}\, }}{\mathbb{R}}^n\). 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 \(\pi: E \rightarrow B\)
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An open cover \(\mathcal{U}=\left\{U_{i}\right\}\) of \(B\) and for each \(i\) a bijection \(\phi: \pi^{-1}\left(U_{i}\right) \rightarrow U_{i} \times F\) so that \(\pi \circ \phi_{i}=\pi\).
Give conditions on the maps \(\phi_{i}\) so that there is a topology on \(E\) making \(\phi: E \rightarrow B\) into a fiber bundle with \(\left\{\left(U_{i}, \phi_{i}\right)\right\}\) as an atlas.
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3.
An oriented \(n\)-dimensional vector bundle is a vector bundle \(\pi: E \rightarrow B\) together with an orientation of each fiber \(E_{b}\), so that these orientations are continuous in the following sense.
For each \(b \in B\) there is a chart \((U, \phi)\) with \(b \in U\) and \(\phi: \pi^{-1}(U) \rightarrow U \times \mathbb{R}^{n}\) so that for all \(b^{\prime} \in U\), \begin{align*} \left.\phi\right|_{E_{b^{\prime}}}: E_{b^{\prime}} \rightarrow \mathbb{R}^{n} \end{align*} is orientation-preserving.
Show that given an oriented \(n\)-dimensional vector bundle there is an induced principal \(G L_{+}\left(\mathbb{R}^{n}\right)\)-bundle (the “bundle of oriented frames”), and conversely given a principal \(G L_{+}\left(\mathbb{R}^{n}\right)\)-bundle there is an induced oriented \(n\)-plane bundle.
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4.
A Riemannian metric on a vector bundle \(\pi: E \rightarrow B\) is an inner product \(\langle\cdot, \cdot\rangle_{b}\) on each fiber \(E_{b}\) of \(E\), which is continuous in the sense that the induced map \(E \oplus E=E \times_{B} E \rightarrow \mathbb{R}\) 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 \begin{align*} \check{H}^{i}(X ; G) \cong H^{i}(X ; G) \end{align*} 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 \(\pi: E \rightarrow B\) be an \(n\)-dimensional vector bundle, or equivalently, a principal \(G L(n, \mathbb{R})\)-bundle, given by a Čech cocycle \(\phi \in H^{1}(B ; G L(n, \mathbb{R}))\). Show that the sign of the determinant \begin{align*} \operatorname{sgn}\operatorname{det}: \operatorname{GL}_n(\mathbb{R}) \rightarrow\{\pm 1\} \cong \mathbb{Z} / 2 \mathbb{Z} \end{align*} induces a map \begin{align*} \check{H}^{1}(B ; G L(n, \mathbb{R})) \rightarrow \check{H}^{1}(B ; \mathbb{Z} / 2 \mathbb{Z}) ,\end{align*} and so \(\phi\) induces an element \(w_{1}(E) \in H^{1}(B ; \mathbb{Z} / 2 \mathbb{Z})\).
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b: Compute \(w_{1}\) 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 \(\pi: E \rightarrow B\) is trivial if and only if \(w_{1}(E)=0 \in\) \(H^{1}(B ; \mathbb{Z} / 2 \mathbb{Z})\)
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7.
Show that the exact sequence of abelian topological groups
\begin{align*} 0 \rightarrow \mathbb{Z} \rightarrow \mathbb{R} \rightarrow S^{1}=G L(1, \mathbb{C}) \rightarrow 0 \end{align*}
induces an exact sequence in Čech cohomology
\begin{align*} \check{H}^{1}(B, \mathbb{Z}) \rightarrow \breve{H}^{1}(B, \mathbb{R}) \rightarrow \check{H}^{1}\left(B ; S^{1}\right) \stackrel{\delta}{\rightarrow} \check{H}^{2}(B ; \mathbb{Z}) \end{align*}
Given a complex line bundle (principal \(G L(1, \mathbb{C})\)-bundle) \(\pi: E \rightarrow B\) coming from the cocycle data \(\phi \in H^{1}(B ; G L(1, \mathbb{C}))\), let \(c_{1}(E)=\delta(\phi)\). Compute \(c_{1}(E)\) for some complex line bundle over \(S^{2}\).
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