Low Dimensional Topology

October 21, 2008

Orbit spaces of Morse functions

Filed under: Misc. — Jesse Johnson @ 10:22 am

A paper on the arXiv by Sergiy Maksymenko [1] caught my eye the other day.  (It was posted about a year ago, but I saw it when it was updated.)  The paper considers the following problem:  Let f  be a Morse function on a surface S and let F be the orbit of f in $C^\infty(S)$ under all diffeomorphisms of S.  (The orbit is a subset of the space of all smooth functions on S.)  You can put a topology called the Whitney $C^\infty$ topology on $C^\infty(S)$, which induces a topology on the orbit set.  Maksymenko shows that this space is homotopy equivalent to a finite cell complex and that the fundamental group of this subcomplex is a subgroup of some braid group over that surface.

There are two reasons this caught my eye.  First, I like this idea of considering the space of topological objects that are isotopic to a fixed subset or structure on a larger manifold.  For example, Hatcher [2] has looked at the space of knots isotopic to a fixed knot in the 3-sphere.  The homotopy type of such a space is determined by a combination of the homotopy type of the diffeomorphism group of the original manifold and by the symmetries of the subset or structure.

The second reason is that it reminds me of Hatcher and Thurston’s paper on mapping class groups of surfaces [3], in which they define the pants complex of a surface.  Rather than finding a cell complex that encodes a single orbit inside $C^\infty(S)$, they find a cell complex that encodes the overall structure of $C^\infty(S)$, with each vertex representing a collection of orbits (all of which induce the same pants decomposition) and with edges and higher dimensional cell encoding how the orbits fit together to form the entire space.  The idea of approximating the topology of an infinite dimensional space with a cell complex comes up less explicitly in other areas as well, such as thin position and even the stabilization problem for Heegaard splitings.

Before I end this post, I just wanted to suggest one more thing:  One could ask this same question for Morse functions in any smooth manifold.  For an irreducible 3-manifold, I would guess that the space is always contractible, but that’s just a guess, not a conjecture.  For a reducible 3-manifold and a Morse function in which a level surface is a reducing sphere, it seems like there’s room for a non-trivial loop in the orbit space, though maybe not.  But this seems like an interesting problem.

[3] Hatcher, A.; Thurston, W. A presentation for the mapping class group of a closed orientable surface. Topology 19 (1980), no. 3, 221–237.

1. For a reducible 3-manifold and a Morse function in which a level surface is a reducing sphere, it seems like there’s room for a non-trivial loop in the orbit space, though maybe not.

Could you elaborate a little further? I’m picturing $latex S^2\times S^1$. But now I don’t know why the same intuition wouldn’t apply to any 3-manifold that fibres over the circle.

Comment by Henry Wilton — October 21, 2008 @ 12:08 pm

2. My latex didn’t work. Why not?

Comment by Henry Wilton — October 21, 2008 @ 12:08 pm

3. Actually, after thinking about it, maybe my intuition about reducing spheres was wrong. You want a diffeomorphism that takes the Morse function to itself and is isotopic to the identity, but not by an isotopy that fixes the Morse function. If you have a neighborhood of high genus level sets then the connected component of the diffeomorphism group of that surface is contractible. Thus if the diffeomorphism is isotopic to the identity on the top and bottom surfaces then it is isotopic to the identity on the whole neighborhood of surfaces.

The diffeomorphism group of the torus, on the other hand, has a large fundamental group. So on a neighborhood of level tori, a diffeomorphism may be equal to the identity on the top and bottom tori, but may not be isotopic to the identity in between. For example, if you have a solid torus whose boundary is a level set and whose interior contains a single index-one and a single index-zero critical point then you can drag the two critical points around the core of the torus (not fixing the Morse function) and back to themselves. This will give you a non-trivial element of the fundamental group of the orbit space. You could probably do something similar if the torus bounds a Seifert fibered space on one side.

The fundamental group of the diffeomorphism group of a sphere is Z_2 (is this right?) so you might be able to do something like that by filling a ball with some critical points, then dragging them around. But you want a sphere bounding a ball so that there is an isotopy of the submanifold bounded by the sphere that induces a non-trivial path in the diffeomorphism group of the sphere. If it bounds anything non-trivial then this probably won’t happen (will it?)

So I was wrong about the reducing spheres, but I now think it’s an even more interesting problem than I did when I wrote that this morning.

Comment by Jesse Johnson — October 21, 2008 @ 7:49 pm

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