Bundles with large rank

Question

Suppose $E\to M$ is a vector bundle of rank $k$ bundle over an n-dimensional manifold. If $k>n$, is there a splitting $E\cong F\oplus G$ where the rank of $F$ is $n$ and the rank of $G$ is $k-n$? If so, is there always a splitting such that $G$ is trivial?

Answer 1

Here's a sketch of how this works.

1. Let $G_k$ be the set of $k$-dimensional linear subspaces of $\mathbb{R}^{\infty}$, the $\mathbb{R}$-vector space of countably infinite dimension, and $E_k\subseteq G_k\times\mathbb{R}^{\infty}$ the set of pairs $(V,x)$ with $x\in V$. The projection $p_k\colon E_k\rightarrow G_k$ on the first factor has $p^{-1}(V)=\{V\}\times V$ for every $V\in G_k$, so the fibers naturally carry vector space structures. If you appropriately topologize these spaces, $p_k$ is a vector bundle of rank $k$, called the tautological bundle over $G_k$.

2. The bundle $E_k\rightarrow G_k$ is "universal" in the following sense. For any paracompact space $X$ (in particular, any manifold), there is a bijection $[X,G_k]\rightarrow\{\text{isomorphism classes of vector bundles over $X$}\}$ given by $[f]\mapsto f^{\ast}E_k$. In particular, the pullback depends up to isomorphism only on the homotopy class of $f$. If $E\rightarrow X$ is a vector bundle of rank $k$ and $f\colon X\rightarrow G_k$ a map (unique up to homotopy) such that $f^{\ast}E_k\cong E$, then we say that $f$ classifies $E$.

3. Let $s\colon\mathbb{R}^{\infty}\rightarrow\mathbb{R}^{\infty}$ be the shift map $(x_1,x_2,\dotsc)\mapsto(0,x_1,\dotsc)$. This yields a map $G_k\rightarrow G_{k+1}$ taking $V\mapsto \mathbb{R}e_1\oplus s(V)$. Iterating this, we obtain maps $i_{kn}\colon G_k\rightarrow G_n$ for $k\le n$. It is a good exercise to check that $i_{kn}^{\ast}E_n\cong(G_k\times\mathbb{R}^{n-k})\oplus E_k$ and conclude that if a map $f\colon X\rightarrow G_k$ classifies a bundle $E\rightarrow X$, then the composite $i_{kn}f\colon X\rightarrow G_n$ classifies $(X\times\mathbb{R}^{n-k})\oplus E\rightarrow X$.

4. Thus, putting 2. and 3. together, the question of whether a bundle $E\rightarrow X$ of rank $k$ splits as $E\cong(X\times\mathbb{R}^{k-n})\oplus F$ with $F\rightarrow X$ of rank $n$ (I'm sorry for switching the indices at this point) is equivalent to the question of whether a map $f\colon X\rightarrow G_k$ classifying $E$ factors through $i_{nk}$ up to homotopy, in other words if it represents an element in the image of $[X,G_n]\rightarrow[X,G_k]$.

5. The map $G_k\rightarrow G_{k+1}$ is fiber-homotopy equivalent to the sphere bundle of the universal bundle $E_{k+1}\rightarrow G_{k+1}$. Then, using the long exact sequence of this bundle, it follows that $G_k\rightarrow G_{k+1}$ induces isomorphisms on $\pi_n$ for $n<k$ and a surjection on $\pi_k$, i.e. it is $k$-connected.

6. If $X$ is a CW-complex of dimension $\le n$ and $f\colon Y\rightarrow Z$ is an $n$-connected map, then $[X,Y]\rightarrow[X,Z]$ is a surjection. This is a standard cell-by-cell induction argument.

7. If $M$ is an $n$-dimensional manifold, it has the homotopy type of a CW-complex of dimension $\le n$, so putting 2. and 6. together, the result follows.