2012-11-13
So Rob Kirby and I have come up with the notion of a trisection of a $4$-manifold, as a completely natural generalization of a Heegaard splitting of a $3$-manifold. I want to think right now about generalizing this to arbitrary dimensions. There should be a good notion of a Heegaard $(n-1)$-splitting on an $n$-manifold, and maybe more generally a Heegaard $m$-splitting of an $n$-manifold. (This latter will probably only make sense for certain $k$'s and $n$'s, in particular I'm hoping it works for $m=n-1$ and $m=n$. This is because we also have observed that it makes sense to talk about trisections of $3$-manifolds, which actually correspond to nothing more than open book decompositions.)
So, a Heegaard $m$-splitting of an $n$-manifold $X$ should, first off, be a decomposition of $X$ into $m$ codimension-$0$ pieces, $X_1, \ldots, X_m$, each diffeomorphic to $\natural^{k_1} S^1 \times B^{n-1}$ , for some fixed $k_1$. Then the pairwise intersections $X_i \cap X_j$, for $i \neq j$, should be diffeomorphic to $\natural^{k_2} S^1 \times B^{n-2}$, for some fixed $k_2$. Triple intersections $X_i \cap X_j \cap X_k$, for $i \neq j \neq k$, should be diffeomorphic to $\natural^{k_3} S^1 \times B^{n-3}$, and so on. All the way down to the complete intersection $X_1 \cap X_2 \cap \ldots \cap X_m$, which should be the common boundary of all the $(m-1)$-fold intersections, i.e. it should be diffeomorphic to $\sharp^{k_{m-1}} S^1 \times S^{n-m}$.
I guess this is a perfectly well-defined concept. The immediate questions are:
- Does $\sharp^k S^1 \times S^{n-1}$ have a natural Heegard $(n-1)$-splitting?
- Does $S^n$ have a simplest nontrivial Heegaard $(n-1)$-splitting which we can use for a stabilization operation?
It's also fairly clear from this that the cases m=n-1 and m=n should be the most tractable, because there the central fiber $X_1 \cap \ldots \cap X_m$ is either a surface or a collection of circles. Let's focus on $m=n-1$.
OK, I think the answer to question (1) is easy but I'm not seeing how to write it right now. More later.