Smooth (closed, connected, orientable) 3-dimensional manifolds are very special, in that for any 3-manifold $M$ there are two handlebodies, $V$ and $W$, of genus $g$ and an orientation reversing homeomorphism $f$ of their boundaries so that $M=V\ \cup\_f W$. Such a decomposition is called a Heegaard splitting.


I want to know: Does this kind of symmetric handlebody decomposition extend into higher dimensions? More specifically, given an $n=2k+1$ manifold $M$, can we construct a $V$ and $W$ by attaching handles $D^i \times \small{D^{n-i}}\ (i\leq k)$ to an $n$-disk, and find an orientation reversing homeomorphism $f:\partial V\rightarrow\partial W$ so that $M=V\ \cup\_f W$? Since $f$ is only continuous here, it might not provide a unique smooth structure for $M$; could we remedy this by requiring $f$ to be a diffeomorphism instead? After all, any exotic sphere $\Sigma\in\Theta\_n$ can be constructed by gluing two copies of $D^n$ together with an orientation reversing diffeomorphism of the boundary (except possibly $n=4$?).



Higher dimensional Heegaard splittings?
Every odd-dimensional manifold has an open book decomposition (T. Lawson, Topology 17, 189-192 (1978)) and is thus a twisted double. In the even dimensions there is an asymmetric
Witt group obstruction to the existence of an open book decomposition (F. Quinn, Topology 18,
55-73 (1979)), which is also the obstruction to being a twisted double. In the simply-connected 4k-dimensional case this is the original Winkelnkemper signature obstruction. There is an account of open books and twisted doubles in Chapters 29,30 of my book "High-dimensional knot theory" (Springer Monograph, 1998)


