Examples for open disc bundle which is not vector bundle

Question

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?

Answer 1

Given a disc bundle over a space $X$, there's the classifying map

$$ X \to BDiff(D^n)$$

$Diff(D^n)$ has as a subgroup $O_n$, the orthogonal group. The disc bundle over $X$ is a vector bundle if and only if the classifying map $X \to BDiff(D^n)$ factors (up to homotopy) through a map $X \to BO_n$, where the other map is the induced map $BO_n \to BDiff(D^n)$ induced from inclusion $O_n \to Diff(D^n)$.

There's a homotopy fiber sequence

$$PDiff(D^n) \to Diff(D^n) \to O_n$$

where $PDiff(D^n)$ is the subgroup of diffeomorphisms of $D^n$ which restrict to the identity on a hemi-disc. The idea is to think about restricting a diffeomorphism of $D^n$ to a half of the disc, and linearizing that embedding. The fiber is sometimes called the pseudo-isotopy diffeomorphism group of $D^{n-1}$. Moreover, this fiber sequence is a direct product (up to homotopy)

$$Diff(D^n) \times O_n \simeq PDiff(D^n)$$

So your question boils down to the existence of non-null maps $X \to B(PDiff(D^n))$.

The homotopy type of the pseudo-isotopy diffeomorphism group isn't terribly well known but there are things known about it. Cerf's theorem says $PDiff(D^n)$ is connected for $n$ large. A big result in this area is Kiyoshi Igusa's stability theorem, which tells you that certain stabilization maps $PDiff(D^n) \to PDiff(D^{n+1})$ induce isomorphisms on a range of homotopy groups, and the stable pseudo-isotopy diffeomorphism group is then relateable to K-theory. I don't know what the state of the art is in computing those homotopy groups, but people like John Rognes probably know.

$\pi_1 PDiff(D^n)$ has four elements for $n$ large. This is in Hatcher's 1978 Concordance spaces, higher himple homotopy theory and applications Proc. Symp. Pure. Math. 32. I haven't chased through the details but I believe "n large" in Hatcher's paper means Kiyoshi Igusa's stable range, which in this case I believe it means $n \geq 11$.

So your base space need only by $S^2$ provided the fiber dimension is high enough. Hatcher gives an explicit construction of related homotopy-classes in his paper although it's a fair bit of work to digest it.

edit: Looking at the Browder paper it appears you're interested in topological bundles where the fiber is a non-compact ball. In this case you're interested in the lifting property $X \to BHomeo(\mathbb R^n)$ for the map $BO_n \to BHomeo(\mathbb R^n)$.

There's quite a bit known about the inclusion $O_n \to Homeo(\mathbb R^n)$. The Cerf-Morlet Comparison Theorem says $Diff(D^n, fix \partial D^n)$ has the homotopy-type of $\Omega^{n+1}(PL_n/O_n)$. This says that diffeomorphisms of $D^n$ that fix the boundary in some sense measure the existence of $PL$ bundles that are not vector bundles. There are similar statements for topological bundles. There are non-trivial elements in the homotopy of $Diff(D^n fix \partial D^n)$ due to people like Igusa, Hatcher and Farrell. This would tell you that there are PL bundles with fiber $\mathbb R^n$ which are not smooth bundles. In this case the dimension of your base sphere would have to be larger, but order-of-magnitude a little less than $20$.

Answer 2

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.