• # B1.8

What is a smooth vector bundle over a manifold $M$ ?

Assuming the existence of "bump functions", prove that every compact manifold embeds in some Euclidean space $\mathbb{R}^{n}$.

By choosing an inner product on $\mathbb{R}^{n}$, or otherwise, deduce that for any compact manifold $M$ there exists some vector bundle $\eta \rightarrow M$ such that the direct sum $T M \oplus \eta$ is isomorphic to a trivial vector bundle.

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• # B2 7

For each of the following assertions, either provide a proof or give and justify a counterexample.

[You may use, without proof, your knowledge of the de Rham cohomology of surfaces.]

(a) A smooth map $f: S^{2} \rightarrow T^{2}$ must have degree zero.

(b) An embedding $\varphi: S^{1} \rightarrow \Sigma_{g}$ extends to an embedding $\bar{\varphi}: D^{2} \rightarrow \Sigma_{g}$ if and only if the map

$\int_{\varphi\left(S^{1}\right)}: H^{1}\left(\Sigma_{g}\right) \rightarrow \mathbb{R}$

is the zero map.

(c) $\mathbb{R} \mathbb{P}^{1} \times \mathbb{R P}^{2}$ is orientable.

(d) The surface $\Sigma_{g}$ admits the structure of a Lie group if and only if $g=1$.

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• # B4.4

Define what it means for a manifold to be oriented, and define a volume form on an oriented manifold.

Prove carefully that, for a closed connected oriented manifold of dimension $n$, $H^{n}(M)=\mathbb{R}$.

[You may assume the existence of volume forms on an oriented manifold.]

If $M$ and $N$ are closed, connected, oriented manifolds of the same dimension, define the degree of a map $f: M \rightarrow N$.

If $f$ has degree $d>1$ and $y \in N$, can $f^{-1}(y)$ be

(i) infinite? (ii) a single point? (iii) empty?

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• # B1.8

State the Implicit Function Theorem and outline how it produces submanifolds of Euclidean spaces.

Show that the unitary group $U(n) \subset G L(n, \mathbb{C})$ is a smooth manifold and find its dimension.

Identify the tangent space to $U(n)$ at the identity matrix as a subspace of the space of $n \times n$ complex matrices.

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• # B2.7

Let $M$ and $N$ be smooth manifolds. If $\pi: M \times N \rightarrow M$ is the projection onto the first factor and $\pi^{*}$ is the map in cohomology induced by the pull-back map on differential forms, show that $\pi^{*}\left(H^{k}(M)\right)$ is a direct summand of $H^{k}(M \times N)$ for each $k \geqslant 0$.

Taking $H^{k}(M)$ to be zero for $k<0$ and $k>\operatorname{dim} M$, show that for $n \geqslant 1$ and all $k$

$H^{k}\left(M \times S^{n}\right) \cong H^{k}(M) \oplus H^{k-n}(M)$

[You might like to use induction in n.]

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• # B4.4

Define the 'pull-back' homomorphism of differential forms determined by the smooth map $f: M \rightarrow N$ and state its main properties.

If $\theta: W \rightarrow V$ is a diffeomorphism between open subsets of $\mathbb{R}^{m}$ with coordinates $x_{i}$ on $V$ and $y_{j}$ on $W$ and the $m$-form $\omega$ is equal to $f d x_{1} \wedge \ldots \wedge d x_{m}$ on $V$, state and prove the expression for $\theta^{*}(\omega)$ as a multiple of $d y_{1} \wedge \ldots \wedge d y_{m}$.

Define the integral of an $m$-form $\omega$ over an oriented $m$-manifold $M$ and prove that it is well-defined.

Show that the inclusion map $f: N \hookrightarrow M$, of an oriented $n$-submanifold $N$ (without boundary) into $M$, determines an element $\nu$ of $H_{n}(M) \cong \operatorname{Hom}\left(H^{n}(M), \mathbb{R}\right)$. If $M=N \times P$ and $f(x)=(x, p)$, for $x \in N$ and $p$ fixed in $P$, what is the relation between $\nu$ and $\pi^{*}\left(\left[\omega_{N}\right]\right)$, where $\left[\omega_{N}\right]$ is the fundamental cohomology class of $N$ and $\pi$ is the projection onto the first factor?

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• # B1.8

What is meant by a "bump function" on $\mathbb{R}^{n}$ ? If $U$ is an open subset of a manifold $M$, prove that there is a bump function on $M$ with support contained in $U$.

Prove the following.

(i) Given an open covering $\mathcal{U}$ of a compact manifold $M$, there is a partition of unity on $M$ subordinate to $\mathcal{U}$.

(ii) Every compact manifold may be embedded in some Euclidean space.

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• # B2.7

State, giving your reasons, whether the following are true or false.

(a) Diffeomorphic connected manifolds must have the same dimension.

(b) Every non-zero vector bundle has a nowhere-zero section.

(c) Every projective space admits a volume form.

(d) If a manifold $M$ has Euler characteristic zero, then $M$ is orientable.

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• # B4.4

State and prove Stokes' Theorem for compact oriented manifolds-with-boundary.

[You may assume results relating local forms on the manifold with those on its boundary provided you state them clearly.]

Deduce that every differentiable map of the unit ball in $\mathbb{R}^{n}$ to itself has a fixed point.

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• # B1.8

Define an immersion and an embedding of one manifold in another. State a necessary and sufficient condition for an immersion to be an embedding and prove its necessity.

Assuming the existence of "bump functions" on Euclidean spaces, state and prove a version of Whitney's embedding theorem.

Deduce that $\mathbb{R}^{n}$ embeds in $\mathbb{R}^{(n+1)^{2}}$.

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• # B2.7

State Stokes' Theorem.

Prove that, if $M^{m}$ is a compact connected manifold and $\Phi: U \rightarrow \mathbb{R}^{m}$ is a surjective chart on $M$, then for any $\omega \in \Omega^{m}(M)$ there is $\eta \in \Omega^{m-1}(M)$ such that $\operatorname{supp}(\omega+d \eta) \subseteq \Phi^{-1}\left(\mathbf{B}^{m}\right)$, where $\mathbf{B}^{m}$ is the unit ball in $\mathbb{R}^{m}$.

[You may assume that, if $\omega \in \Omega^{m}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\omega) \subseteq \mathbf{B}^{m}$ and $\int_{\mathbb{R}^{m}} \omega=0$, then $\exists \eta \in \Omega^{m-1}\left(\mathbb{R}^{m}\right)$ with $\operatorname{supp}(\eta) \subseteq \mathbf{B}^{m}$ such that $\left.d \eta=\omega .\right]$

By considering the $m$-form

$\omega=x_{1} d x_{2} \wedge \ldots \wedge d x_{m+1}+\cdots+x_{m+1} d x_{1} \wedge \ldots \wedge d x_{m}$

on $\mathbb{R}^{m+1}$, or otherwise, deduce that $H^{m}\left(S^{m}\right) \cong \mathbb{R}$.

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• # B4.4

Describe the Mayer-Vietoris exact sequence for forms on a manifold $M$ and show how to derive from it the Mayer-Vietoris exact sequence for the de Rham cohomology.

Calculate $H^{*}\left(\mathbb{R} \mathbb{P}^{n}\right)$.

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