## Pseudoisotopy in low dimensions

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Recall that the space $P(M)$ of (smooth) pseudoisotopies of the compact manifold $M$ is defined as the space of all diffeomorphisms $M\times I\to M\times I$ that fix every point in $(M\times 0)\cup (\partial M \times I)$. I have worked on these things in high-dimensional cases, but I realize that there are big gaps in my knowledge in low-dimensions. I am wondering how much of what I don't know is known.

If the dimension of $M$ is less than $2$ then $P(M)$ is contractible. According to the Smale Conjecture $P(D^2)$ (or equivalently $P(S^2)$) is contractible. I have the impression that Hatcher also proved that $P(M)$ is contractible when $M$ is a two-dimensional annulus. Is it perhaps known, or believed, that this holds for all compact $2$-manifolds? Or maybe it's not true. (I could have started by just asking Allen Hatcher or Ryan Budney, but I thought others might be interested in any answers.)

And what is known in the next dimension?

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## 1 Answer

When $M$ is a compact $2$-manifold, with or without boundary, $P(M)$ is known. When $M$ is a 3-manifold there's bits and pieces known, especially once you get to more fine detail like pseudo-isotopy embedding spaces. But at the level of $P(M)$ I don't think there's a complete description for a single $3$-manifold. For $4$-manifolds the situation is worse, but again there are some things known for pseudo-isotopy embedding spaces.

As you mention, $P(S^2)$ and $P(D^2)$ are contractible by the Smale Conjecture. I think $P(M)$ is contractible for an arbitrary $2$-manifold. The argument goes like this: You look at locally trivial fiber bundle $P(M) \to Diff(M)$. The fiber over the identity map is $Diff(M \times I)$, the group of diffeomorphisms fixing the entire boundary $\partial M \times I \cup M \times \partial I$. The other fibers are empty because diffeomorphisms are prescribed up to isotopy by their action on $\pi_1 M$. Except in a few cases, $Diff(M)$ has contractible components, so showing $P(M)$ is contractible is equivalent to showing $Diff(M \times I)$ is contractible. But $M \times I$ has incompressible annuli in it, corresponding to $C \times I$ where $C \subset M$ is a closed curve in $M$ that does not bound a disc in $M$. Hatcher's work on spaces of incompressible surfaces kicks in and tells you the space of these incompressible annuli is contractible. So you can reduce studying $Diff(M \times I)$ to things like $Diff(A \times I)$ where $A$ is an annulus or a pair of pants, but then you have vertical incompressible discs you can use and reduce to the point that $A$ is itself a disc.

The above argument works for any surface other than $\mathbb RP^2$, a torus or a klein bottle. But similar arguments cover these cases.

Here's the reference for the results of Hatcher's I'm using, it also defines the terminology like "incompressible annulus, disc" and so on. http://www.math.cornell.edu/~hatcher/Papers/emb.pdf

Allen had a student (Kiralis) who wrote some papers on pseudo-isotopy diffeomorphisms of 3-manifolds. That might be a place to look.

But the kind of things that I remember are mostly along the lines of pseudo-isotopy embeddings of knots and links in $D^3$ and $S^3$. I'm about to hop on an airplane. I'll edit this response sometime in the next few days and add some of these observations if they haven't already been made by someone else.

edit 1: Here are two unrelated observations.

(a) Let $N$ be a co-dimension zero solid torus in $M=\mathbb R^3$ or $M=S^3$. There's the pseudo-isotopy embedding fibration $P(N,M) \to Emb(N,M)$. If $N$ is an unknotted solid torus, the question of what the map the image of the map $\pi_0 P(N,M) \to \pi_0 Emb(N,M)$ is, this is a long-standing hard problem in knot theory. Another way to say it is `which knots in $S^3$ bound a disc in $D^4$?'. These knots are called slice knots. Ralph Fox has the Slice Ribbon Conjecture, which might be described as a hopeful combinatorial answer to the question. There are many useful tools for determining whether or not a given knot is slice,starting with the Alexander module and more recently tools from Heegaard Floer theory.

Two examples:

• Look at the class of knots that are a connect-sum of torus knots. Using the Alexander module Litherland proved such a knot bounds a disc in $D^4$ if and only if in the prime decomposition, the number of times a prime summand appears (like say a right-handed trefoil) is equal to the number of times its mirror-image appears (the left handed trefoil, in my example). Related previous MO thread

• Paolo Lisca has used Heegaard Floer theory to determine when a connect sum of 2-bridge knots in $S^3$ bounds a disc in $D^4$ REF, but in this case the answer is more elaborate.

(b) Just like how diffeomorphism groups $Diff(D^n)$ and spaces of long knots have actions of cubes operads, pseudo-isotopy embedding spaces and diffeomorphism groups also have such actions. I'm pretty sure there's a pseudo-isotopy splicing operad as well, but I haven't written out the details.

edit 2: Rick Litherland generalized a result of Zeeman (Deforming twist-spun knots TAMS 250 (1979) 311--331) showing that "Deform-spun knots" have complements that frequently fibre over $S^1$. This process called "Deform spinning" is just the boundary map in the pseudo-isotopy fibre sequence for spaces of knots. One of the nice things about this is Litherland gives a prescription for what the fibre is. In the case where you're looking at the pseudo-isotopy sequence for long knots in $\mathbb R^3$, it generates long embeddings of $\mathbb R^2$ in $\mathbb R^4$. So the fibre is a 3-manifold with boundary a sphere. This process produced some embeddings of 3-manifolds in the 4-sphere that nobody had known about at the time, like the once-punctured Poincare Dodecahedral Space (which without a puncture does not embed in $\mathbb R^4$, at least, not smoothly, it does admit a tame topological embedding). I got interested in this case largely because it represents sort of an extreme end of the terrain of your dissertation.

edit 3: I forgot to mention, I kept on pushing trying to understand why your dissertation broke down in co-dimension two. In some sense my paper "An obstruction to a knot being deform-spun via Alexander polynomials" is an answer. In a way it's not, since co-dimension two deform-spinning is more of a free loop space construction than a based loop space construction. But I found the exercise informative.

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