TITLE: Examples for open disc bundle which is not vector bundle
QUESTION [8 upvotes]: William Browder showed in "Open and closed disc bundles", Ann. of Math. (2) 83 (1966), 218-230 that there are open disc bundles over some complex which cannot be isomorphic to a vector bundle.
My question is: Is there any related examples?
For example: What is the least dimensions for the base complex and the dimension of the fiber?
Or is there any explicit example for this complex?

REPLY [3 votes]: Some remarks which amount in some way to an answer.
(1) Let  $\text{Diff}(D^n)$ be the diffeomorphisms of $D^n$ which restrict to the identity on
the boundary.
When $n\gg k$ is large, $\pi_k(\text{Diff}(D^n))\otimes \Bbb Q$ was computed
by Farrell and Hsiang. The answer is that this is $\cong \Bbb Q$ when $k = 4j-1$ and $n$ is odd. In every other case this group is trivial.
(2) Let $\text{Diff}(D^n,S^{n-1})$ be the group of diffeomorphisms of $D^n$ which aren't necessarily the identity on the boundary of $D^n$. 
Then as in Ryan's answer there's a homotopy fiber
sequence
$$
C(S^{n-1}) \to \text{Diff}(D^n,S^{n-1}) \to O(n)
$$
where $C(S^{n-1})$ is the concordance space of the $(n-1)$-sphere. This is the group of
diffeomorphisms of $S^{n-1} \times [0,1]$ which restrict to the identity on 
$S^{n-1} \times 0$.
As Ryan notes
$$
\pi_k(\text{Diff}(D^n,S^{n-1})) \cong \pi_k(C(S^{n-1})) \times  \pi_k(O(n)) .
$$
Again, when $n\gg k$, we can compute $\pi_k(O(n)) \cong \pi_i(O)$ by Bott periodicity.
Rationally, it's given by $\Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$
and trivial otherwise.
Also $ \pi_k(C(S^{n-1}))\otimes \Bbb Q$ can be computed when $n \gg k$. It's also 
$\cong \Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$ and trivial otherwise.
So we get, for $n \gg k$, that $\pi_k(\text{Diff}(D^n,S^{n-1})) \otimes \Bbb Q$ is
isomorphic to $\Bbb Q \oplus \Bbb Q$ in dimensions congruent to $3 \text{ mod } 4$ and trivial otherwise.