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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 $n=4$?).

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High-dimensional manifolds have handle decompositions, this is one of the standard generalizations of handlebody decompositions. Have you looked at Milnor's h-cobordism notes, or Kosinski's textbook? The manifolds $V$ and $W$ are usually described as the unions of the handles of dimension less than half that of $M$, and the corresponding dual handles respectively. –  Ryan Budney Nov 16 '11 at 6:37
That said, what would you want to do for a manifold like $\mathbb CP^2$? What I describe really only works for odd-dimensional manifolds. –  Ryan Budney Nov 16 '11 at 6:41
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up vote 8 down vote accepted

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)

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This idea of a "twisted double" is exactly what I was thinking of. I didn't want to get my hopes up by asking if you could split $M$ using two copies of the SAME manifold $W$, but I guess in the odd-dimensional case you always can. :) I think the appropriate course of action now would be to check out some of your books! –  William Nov 18 '11 at 0:11
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Any closed connected n-manifold admits a Morse function f with one critical point of index zero and one critical point of index n (see e.g. Matsumoto's "Introduction to Morse Theory", Theorem 3.35). If n is odd, you can slide all handles of index $< \frac{n}{2}$ to below $f^{-1}(\frac{1}{2})$ and all handles of index $> \frac{n}{2}$ to above $f^{-1}(\frac{1}{2})$, and "split along" the connected $n-1$-manifold $f^{-1}(\frac{1}{2})$.
Parenthetically, $4$--manifolds do have their own concepts of Heegaard diagrams. Namely, the union N of the 0 handle with all the 1-handles, together with frames closed curves on the boundary $\partial N$ along which we are to attach 2-handles. See e.g. Section 5.3 in Matsumoto's book for details.

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