2015-09-23
OK, yesterday I defined a "genus $g$ Heegaard splitting of a $1$-manifold", which is sort of absurd, but I'm trying to organize my low-dimensional definitions carefully enough to say something nontrivial as we go up in dimension. There is the brief question of whether we should think about Heegaard splittings of $0$-manifolds. Let's try:
A genus $g$ Heegaard splitting of a $0$-manifold $M$ is a decomposition $M=M_1 \cup M_2$ such that (1) each $M_i \cong \natural^g S^0 \times B^0$ and (2) $M_1 \cap M_2 \cong \#^g S^0 \times S^{-1}$.
Condition (2) means that $M_1 \cap M_2 = \emptyset$. For condition (1), recall that $\natural$ means "glue by identifying balls in the boundary", and since the boundary here is empty, $\natural$ should mean disjoint union. Also, the ball is the identity element for boundary connected sum, so $\natural^g X$ is $B^n$, where $n$ is the dimension of $X$. So $\natural^g S^0 \times B^0$ would seem to mean one point if $g=0$, and otherwise $2g$ points. This doesn't sound good, e.g. $B^0$ doesn't have a Heegaard splitting. Well, maybe this only works for an even number of points ($B^0$ is somehow not "closed"). But it also looks like a collection of $6$ points doesn't have a Heegaard splitting. Maybe the problem is the definition of $\natural$ for $0$-manifolds, but this is enough. Maybe the real problem is just that we hit $(-1)$-dimensional stuff in our definition.
What I really want to think about (which may run into the same problem I now realize) is:
A $(g,k)$ trisection of a $1$-manifold $M$ is a decomposition $M=M_1 \cup M_2 \cup M_3$ where (1) each $M_i \cong \natural^k S^0 \times B^1$, (2) each $M_i \cap M_j \cong \natural^g S^0 \times B^0$ and (3) $M_1 \cap M_2 \cap M_3 \cong \#^g S^0 \times S^{-1} = \emptyset$.
Remember that $\natural^k S^0 \times B^1$ is $(k+1)$ arcs and $\natural^g S^0 \times B^0$ is one point if $g=0$ otherwise $2g$ points. E.g. this seems fine ($k=0$, $g=0$):And so does this ($k=1$, $g=1$):
But one more step and it falls apart, I think. $k=3$ will give pairwise intersections being $3$ points. Unless we change the way we alternate? No that won't fix it either.
Conclusion: there is no reasonable definition of a trisection in dimension less than $2$ because the triple intersection has to be codimension $2$ and shouldn't be empty, just as there is no reasonable definition of a Heegaard splitting in dimension $0$ because the pairwise intersection needs to be codimension $1$ and nonempty. So, tomorrow, we'll look at Heegaard splittings of surfaces.