In http://www.maths.ed.ac.uk/~aar/papers/n-1con2n.pdf, Wall studied $(2n, n)$-handlebodies, ie. 2n-manifolds that have a handle presentation with only 0 and n handles. Given a presentation, he combines the homology, intersection form, and framing into an algebraic invariant which he calls a pre-n-space. This is a free abelian group $H= \mathbb{Z}^n$ with maps $H \otimes H \rightarrow \mathbb{Z}, H \rightarrow \pi_{n-1}(SO(n))$ being the intersection form and framing, respectively. There are some relations between these maps. Handle-slides of the handlebody presentations correspond to change of basis of the pre-n-spaces and so there is an isomorphism $$ \mbox{pre-n spaces}/\mbox{change of basis} \rightarrow \mbox{(2n,n)-handlebodies}/\mbox{handle-slides}. $$ But Wall seems to be claiming something stronger- that is an isomorphism $$ \mbox{pre-n spaces}/\mbox{change of basis} \rightarrow \mbox{(2n,n)-handlebodies}/\mbox{diffeomorphism}. $$ The right-hand-side is equivalently $\mbox{(2n,n)-handlebodies}/\mbox{handle-slides and birth-death moves}$; birth-death moves create cancelling handles of index $k, k+1$. Why is there such an isomorphism? In particular, is it possible that two different pre-n-spaces result in diffeomorphic manifolds? Maybe I am misinterpreting Wall's result, but this is what I would expect for the result to be called a classification. Maybe there is a way to show that birth-death moves are not necessary?

In the case of a single critical point of index n, we get disk bundles over $S^n$. In this case, it seems that the second map is in fact an isomorphism; see my previous question Smooth classification of vector bundles. Can this approach be generalized to show that the second map is always an isomorphism?


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