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:

  1. Does $\sharp^k S^1 \times S^{n-1}$ have a natural Heegard $(n-1)$-splitting?
  2. 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.


Waldhausen's theorem about Heegaard splittings of $S^3$ states that, for each $g \geq 0$, there exists a unique genus $g$ Heegaard splitting of $S^3$, obtained by $g$ stabilizations of the standard genus $0$ splitting. In terms of Morse theory this means that, if two Morse functions $f_0, f_1 : S^3 \to \mathbb{R}$ both have a single minimum and a single maximum and $g$ critical points each of index $1$ and $2$, then there exists $f_0$ and $f_1$ are homotopic via a generic homotopy $f_t$ with no births or deaths.

Put this way, the first question is: Is there a proof in the same spirit as this statement, i.e. a Cerf-theoretic proof about eliminating cusps in the Cerf graphic of a generic $1$-parameter family?

The second question is: Well, this Morse-theoretic statement suggests an obvious generalization to higher dimensions: If $f_0, f_1 : S^n \to \mathbb{R}$ are two Morse functions with the same numbers of critical points of each index, can they be connected by a homotopy with no births or deaths? We may need to constrain this further, such as: both have exactly one min and one max, or, even, they have one min and one max and NO critical points of any other index other than the middle dimensions. This version is meaningless, I guess, if $n$ is even, because then that leaves ONLY critical points in index $n/2$ (and $0$ and $n$), but with none in index $n/2 \pm 1$, there's nothing to cancel, and so actually the only option is NO crit points in index $n/2$ either.


I was inspired by a talk by Clayton Shonkwiler in the UGA topology seminar, on integral definitions of Milnor's triple linking number. What I have been thinking about is much simpler, just about the ordinary (double) linking number. There is an integral definition of this linking number (Gauss's integral) which is explained as the degree of the following map from $S^1 \times S^1 \to S^2$:

Given a $2$-component link $L$, realized by two disjoint embeddings $\phi_1, \phi_2 : S^1 \to \mathbb{R}^3$, we map $(\theta_1, \theta_2) \in S^1 \times S^1$ to $\phi_2(\theta_2) - \phi_1(\theta_1) / \| \phi_2(\theta_2) - \phi_1(\theta_1) \|$. Call this map $\Phi_L : T^2 \to S^2$. The degree of $\Phi_L$ is the only invariant needed to determine the homotopy class of $\Phi_L$. The linking number is a complete invariant of two-component links up to link homotopy (link homotopy means the individual components can cross themselves, but distinct components should remain distinct). I believe that, using some other easier definition of the linking number, there exists an easy direct proof of this fact that linking number determines link homotopy type.

So I believe that the logical conclusion of the above facts is that, for any $\Phi : T^2 \to S^2$, $\Phi$ is homotopic to $\Phi_L$ for some $L$ and, for any homotopy $\Phi_t$ from $\Phi_0 = \Phi_{L_0}$ to $\Phi_1 = \Phi_{L_1}$, there is a link homotopy $L_t$ such that the homotopy $\Phi_t$ is homotopic to $\Phi_{L_t}$. This is what I want to see directly and that is what I have been thinking about a bit.

The point is that an arbitrary $\Phi : T^2 \to S^2$ is not of the form $\Phi_L$ for a link $L$. The question is how to recover a candidate $L$ given $\Phi$.

If I can understand this then maybe I can understand the triple linking situation nicely and geometrically.



Here is a fact: There is a measure of complexity of orientable surfaces, called genus, such that, if a surface $\Sigma$ if sufficiently complex (genus greater than $1$), then $\pi_1(\mathop{Diff}_0(\Sigma))=0$, where $\mathop{Diff}_0$ is the identity component of the self-diffeomorphism group. Is there any such measure of complexity in higher dimensions? Do "sufficiently complex" $3$-manifolds have $\pi_1(\mathop{Diff}_0) = 0$?

The contex for this question is the problem of recovering a smooth $n$-manifold from combinatorial data associated to a Morse $2$-function (generic smooth map to a $2$-manifold). If $\pi_1(\mathop{Diff}_0(F)) \neq 0$ for some regular fiber $F$, then in some sense we have no hope of writing down something combinatorial in the base that will determine the total space, because we can always remove a neighborhood $B^2 \times F$ of $F$ and glue it back in using a nontrivial loop of self-diffeomorphisms of $F$.

This issue in turn arose from (1) a paper Rob Kirby and I are writing and (2) discussions with Bruce Bartlett about his work with Chris Douglas, Chris Schommer-Pries and Jamie Vicary on the $1$-$2$-$3$ cobordism $2$-category. In the latter case, they understand this category by thinking about Morse $2$-functions on $3$-dimensional cobordisms between $2$-dimensional cobordisms; anyway, the regular fibers are disjoint unions of circles, and $\pi_1(\mathop{Diff}_0(S^1)) = \mathbb{Z}$, so the problem arises. But it seems it can be kept track of using knowledge about Dehn twists and the fact that the mapping class group of a surface is generated by Dehn twists.

Here is another question arising from my discussions with Bruce Bartlett: Is there a simple Morse/Cerf theory proof that the mapping class group of a surface is generated by Dehn twists (forget relations or finite presentations)? I would allow as input the facts that the mapping class group of a disk is trivial, that of a cylinder is generated by the Dehn twist about its core circle, and that of a pair of pants in which we allow the two cuffs to switch places is generated by Dehn twists and braid generators (half Dehn twists). My idea is then you pick your favorite Morse function $f$ on your surface and first argue (using the input facts above) that $f$-level-preserving automorphisms of the surface are generated by Dehn twists. Then you consider an arbitrary automorphism $\phi$ of the surface and consider a generic homotopy from $f$ to $f \circ \phi$. Then the decomposition of this homotopy into elementary moves (critical value crossings and births and deaths) should somehow fill in the rest of the picture.