Affine bundle of sections of a surjective morphism of vector bundles?

Question

Everything that follows is over the real numbers, in the $C^p$ category for $1\leq p\leq \infty$.

Fix an affine bundle of class and consider a fiber-subbundle whose fibers are convex. A partition of unity can be used to average local sections of the subbundle (which exist because it's assumed to be locally trivial) into a global section.

This proves for instance the existence of vector bundle metrics.

Exercise 5.7 of this document suggests using this same result to prove that every surjective morphism of vector bundles has a section (without vector bundle metrics). The outline is to construct an affine bundle whose sections are sections of the vector bundle surjection, and apply the above.

I don't understand how to define this affine bundle of sections/splittings. I also don't understand how I might show it is a fiber bundle: for one, I have no idea why sections/splittings should exist even locally! In the $C^p$ category for $p\geq 1$ this is a consequence of the constant rank theorem, which I consider to be heavier artillery than existence of vector bundle metrics.

Question. What is the affine bundle whose sections are splittings of a given surjective morphism of vector bundles? Why is it a fiber bundle?

Answer 1

This answer is probably incomplete, it only addresses the question raised in my comment.

So let $X$ be a $C^p$ manifold, let $p_0 \in X$. Let us consider a $C^p$ map $u: (p,v) \in X \times \mathbb{R}^n \longmapsto u_p(v) \in \mathbb{R}^p$ such that each $u_p$ is linear and surjective.

In particular, $u_{p_0}: \mathbb{R}^n \rightarrow \mathbb{R}^p$ is linear surjective.

Let $v_{p+1},\ldots,v_n$ be a basis of its kernel. This is a free family of $n-p$ vectors, so there are indices $1 \leq i_{p+1} < \ldots < i_n \leq n$ such that the matrix $(x_{i_j}(v_k))_{p < j,k \leq n}$ is invertible.

Let, for every $p \in X$, $v_p:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be such that $v_p(y)=(u_p(y),x_{i_{p+1}}(y),\ldots,x_{i_n}(y))$, so that $(p,y) \longmapsto v_p(y)$ is $C^p$ and $v_{p_0}$ is invertible.

In particular, $\det{v_{p_0}} \neq 0$. Let $U=\{p \in X,\, \det{v_p} \neq 0$: it is a neighborhood of $p_0$ in $X$. Let $i:\mathbb{R}^p \rightarrow \mathbb{R}^n$ be such that $i(x)=(x,0,\ldots,0)$.

Then $(p,y) \in U \times \mathbb{R}^p \longmapsto v_p^{-1}(i(y))$ is a $C^p$ local section of $u$ around $p_0$.

Answer 2

I apologize for reviving an old post, but I'm currently trying to go through this exercise so I thought I would post how I was thinking about it. I think it helps to use the full data of the SES of vector bundles, not just that you have a surjective bundle map. I will use the notation established in the notes. Let $\{U_\gamma\}$ be an open cover of Y over which $E', E, $and $E''$ are all trivial with trivializations $\phi_\gamma',\phi_\gamma,\phi''_\gamma$, respectively. For all $y\in Y$, let $A_y$ be the set of splittings for the SES of vector spaces $$ 0\to E'_y\xrightarrow{i_y} E_y\xrightarrow{j_y} E''_y\to 0.$$ Following the argument in the notes, this is an affine space over Hom$(E''_y,E'_y)$. Even better, we don't have to worry about this being empty because every short exact sequence of vector spaces splits so this is indeed an affine space. You can then use the local trivializations to topologize $$A = \bigsqcup_{y\in Y} A_y$$ as follows. First topologize $\pi^{-1}(U_\gamma)$ (here $\pi:A\to Y$ is the canonical projection) by requiring the natural map $$\pi^{-1}(U_\gamma) = \bigsqcup_{y\in U_\gamma}A_y\to U_\gamma\times S(\mathbb{R}^n,\mathbb{R}^w)\subset U_\gamma\times \textrm{Hom}(\mathbb{R}^n,\mathbb{R}^w)$$ to be a homomorphism. Here $S(\mathbb{R}^n,\mathbb{R}^w)$ is the space of splittings of the SES $$0\to \mathbb{R}^m\xrightarrow{\hat{i}}\mathbb{R}^w\xrightarrow{\hat{j}}\mathbb{R}^n\to 0,$$ where $\hat{i},\hat{j}$ are induced by $i$ and $j$ via $\phi_\gamma,\phi'_\gamma,\phi''_\gamma$. Lastly, topologize all of $A$ by letting $V\subset A$ be open if and only if $V\cap \pi^{-1}(U_\gamma)$ is open for all $U_\gamma$. The resulting $\pi:A\to Y$ is then the required affine bundle whose section, should they exist (part (ii)), are splittings of the SES of vector bundles.