I am trying to understand the smooth classification of $n$-disk bundles over $S^n$. As vector bundles, these are classified by $\pi_{n-1}(SO(n))$ via the clutching construction but I am interested in their smooth type. For example, the map $\pi_{n-1}(SO(n)) \rightarrow \pi_{n-1}(Diff(D^n))$ might not be injective in which case two distinct vector bundles would be the same as smooth manifolds. What is known about the smooth type of these bundles? The homology of the total space is just $\mathbb{Z}$ in degree $0, n$ and the intersection form is determined by a single integer, which equals $HJ: \pi_{n-1}(SO(n)) \rightarrow \mathbb{Z}$, where $J: \pi_{n-1}(SO(n)) \rightarrow \pi_{2n-1}(S^n)$ and $H: \pi_{2n-1}(S^n) \rightarrow \mathbb{Z}$ is the Hopf invariant.

These $n$-disk bundles over $S^n$ also correspond to handlebodies with a single critical point of index 0 and n and are $(n-1)$-connected $2n$ manifolds as studied by Wall. Wall came up with an invariant of handlebody presentations; in the case of these disk bundles, Wall's invariant is just an element of $\pi_{n-1}(SO(n))$. However, as explained earlier, these don't seem to be really diffeomorphism invariants (although Wall calls them that..). The problem is that Wall's invariant is invariant under handleslides (change of basis) but he seems to ignore the possibility of birth-death moves that create cancelling $n-1, n$ handles and then more handleslides. Am I correct in thinking this?

Edit See my other question for the general case: What does Wall's classification classify?.

  • $\begingroup$ Do you know of an example of two non-isomorphic bundles with diffeomorphic total spaces? I know of examples of non-isomorphic bundles with homeomorphic total space: for linear $S^3$ bundles over $S^4$ with Euler class $1$, the total space is always homemorphic to $S^7$. Because the bundles are linear, they are sphere bundles inside vector bundles. One can use the Alexander trick to show the total spaces are homeomorphic. In addition, there are precisely 10 diffeomorphism types of total spaces occurring, but I don't know if the resulting vector bundles are diffeomorphic (the Alexander ... $\endgroup$ – Jason DeVito Aug 7 '17 at 1:19
  • $\begingroup$ trick is not necessarily smooth at the $0$-section). $\endgroup$ – Jason DeVito Aug 7 '17 at 1:19
  • $\begingroup$ @JasonDeVito That is precisely my question. I think the examples you give are non-isomorphic bundles that result in homeomorphic but not diffeomorphic total spaces; thank you for pointing this out. I am asking for non-isomorphic bundles that result in homeomorphic spaces. $\endgroup$ – user39598 Aug 7 '17 at 5:55

I can't find an example which is not behind a paywall, but in

R. De Sapio and G. Walschap, Diffeomorphism of total spaces and equivalence of bundles, J. Top., Volume 39, Issue 5, September 2000, Pages 921-929

One finds an argument (attributed to Haefliger and Levine) that there is an embedding of $S^{11}$ into $\mathbb{R}^{17}$ for which the normal bundle is not isomorphic to a product, but whose total space is diffeomorphic to the product. Later on in the same paper (pg 927, first paragraph), there are other related examples.

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  • $\begingroup$ Thank you! Could it be that this argument shows that Wall's framing invariants are actually diffeomorphism invariants? That is, suppose you have two 2n-manifolds built using handles of index n that have the homology and intersection form but different framings (up to handleslides which can change the framing), then are the manifolds actually non-diffeomorphic? $\endgroup$ – user39598 Aug 7 '17 at 18:55
  • $\begingroup$ I know nothing of handlebodies - perhaps you could ask that question separately on MSE? $\endgroup$ – Jason DeVito Aug 7 '17 at 20:06

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