Differentiable Manifolds

# Differentiable Manifolds

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B1.8

commentWhat 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.

B2 7

commentFor 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 $g=1$.

B4.4

commentDefine 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?

Briefly justify your answers.

B1.8

commentState 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.

B2.7

commentLet $M$ 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.]

B4.4

commentDefine 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?

B1.8

commentWhat 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.

B2.7

commentState, 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.

B4.4

commentState 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.

B1.8

commentDefine 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 $\mathbb{R}^{(n+1)^{2}}$.

B2.7

commentState 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 $H^{m}\left(S^{m}\right) \cong \mathbb{R}$.

B4.4

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