Smooth quotient bundle

Question

Let $E \rightarrow M$ be a smooth vector bundle.

This link gives a construction of quotient bundle for a subbundle. $E' \subseteq E$.

We define an equivalence relation $\sim$ on $E$ by $v_x \sim v_y$ if and only if $x=y$ and $v_x-v_y \in E'_x$ and define $E/\sim \rightarrow M$ to be our quotient bundle.

My question is, how does this construction also give smooth bundle?

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My attempt, the problem is thus to prove we can give a smooth structure to $E/\sim$.

Transition function formulation: This is equivalent to finding an open cover with local trivializations $$ \{\phi_U:E/\sim |_U \rightarrow U \times \Bbb R^k \}$$ such that the transition functions of any two on $U \cap V$ are smooth map from $$ U \cap V \rightarrow GL_k(\Bbb R)$$

Using the transition functions of $E$, $E'$.

Let us choose transition functions, form $E'$ and $E$, on open sets

$$ \phi': U \cap V \rightarrow GL_m(\Bbb R)$$ $$ \phi: U \cap V \rightarrow GL_n(\Bbb R)$$ where $E$ is bundle of rank $n$, $E'$ is of rank $m$. But how does this induce a well defined smooth map on the quotient bundle?

Answer 1

You have to construct compatible trivializations of the bundles $E'$ and $E$, which is slightly complicated to express in terms of transition functions. The easilest way to describe this in my opinion is in terms of local frames. Choose a local frame for $E'$ defined on $U$. Shrinking $U$, you can extend this to a local frame for $E$. (In a point $x\in U$, extend the values of your frame which form a basis for $E'_x$ to a basis for $E_x$ and then extend the additional elements to local smooth sections of $E$.) Then you get a local trivialization $E|_U\to U\times\mathbb R^n$ which restricts to a trivialzation $E'|_U\to U\times\mathbb R^k$. From this, you get a trivialization from the quotient bundle to $\mathbb R^n/\mathbb R^k\cong\mathbb R^{n-k}$.

In terms of transition functions, you can use the above construction to show that you can arrange things in such a way that the transtition functions for $E$ have values in block matrices of the form $\begin{pmatrix} A & B \\ 0 & C\end{pmatrix}$ such that the $A$-component gives you transition functions for $E'$. In this setting, the $C$-block gives you the transition functions for $E/E'$.

Answer 2

We will appeal to Godemont's theorem, which states that if $X$ is a smooth manifold and $R$ is an equivalence relation on $X$, then $X/R$ has the structure of a smooth manifold such that $X\rightarrow X/R$ becomes a smooth submersion if and only if $R\subseteq X\times X$ is a smoooth submanifold, and $pr_1:R\rightarrow X$ is a submersion. I can't find the exact reference I was looking for at the moment for this theorem, but it is a standard result. I know Bourbaki has a statement (its in French). A good general reference is section 4 of Lee's book Introduction to Smooth Manifolds.

To this end we consider the fibred product $E\times_ME=\{(e_1,e_2)\in E\times E\mid \pi(e_1)=\pi(e_2)\}$, which is the pullback of $\pi$ along itself. This space is a smooth submanifold of the product $E\times E$. Likewise we consider $E\times_ME'=\{(e,e')\in E\times E'\mid \pi(e)=\pi'(e')\}$, which, under the assumption that $E'\subseteq E$ is a smooth submanifold, is itself a smooth submanifold $E\times_ME'\subseteq E\times_ME$.

We however, will consider a non-standard embedding of this manifold. Since $E$ is a vector bundle it has a fibrewise sum $\mu:E\times_ME\rightarrow E$, which restricts to the standard vector space addition $(e_1,e_2)\mapsto e_1+e_2$ on each fibre $E_p\times E_p$, and we use this map to define

$$\Theta:E\times_ME'\rightarrow E\times_ME,\qquad \Theta(e,e')=(e,e+e').$$

It is clear that this map is a diffeomorphism onto its image, and that its image $\Theta(E\times_ME'):=\mathcal{R}$ is precisely the graph of the relation $e_1\sim e_2$ iff $\pi(e_1)=\pi(e_2)$ and $e_2-e_1\in E'_{\pi(e_1)}$. Then $\mathcal{R}\subseteq E\times_ME\subseteq E\times E$ is a smooth submanifold, the projection $pr_1:\mathcal{R}\rightarrow E$ is smooth, and we can appeal to Godemont's critereon to get a smooth structure on the quotient $E/E'$ such that the canonical map $q:E\rightarrow E/E'$ becomes a smooth submersion. This then induces from the map $\pi$ a submersive projection $\rho:E/E'\rightarrow M$ satisfying $\pi=\rho\circ q$. Submersions always have local sections, so $E/E'$ is a locally trivial fibre bundle over $M$. In fact, if $s$ is any local section of $\pi$, then $q\circ s$ is a local section of $\rho$.

Now from here it is not difficult to see that the fibrewise linear structure on $E$ transfers to $E/E'$ in such a way as to make $\rho$ into a topological vector bundle, although ee have not yet shown that the induced operations of fiberwise addition and scalar multiplication are smooth. However this is not hard to do using the fact that $q$ is a submersion. In fact it is not difficult to see that $q\times_Mq:E\times_ME\rightarrow E'\times_ME'$ is a submersion, and so induces the smooth fibrewise addition $\mu':(E/E')\times_M(E/E')\rightarrow E/E'$ uniquely from $\mu$. Similarly, the smoothness of the fibrewise scalar multiplicaiton $\mathbb{K}\times E/E'\rightarrow E/E'$ follows.

To construct charts on $E/E'$ we use the previous remark on local sections to construct fibrewise linear local trivialisations $(U,\varphi)$, $U\subseteq M$, of $\rho$ in such a way that the restriction $q|_U$ becomes the quotient $U\times\mathbb{K}^s\rightarrow U\times\mathbb{K}^{s-r}$ locally modelling the vector space quotients $E_x\rightarrow E_x/E'_x$.