What is the status of the PL-pseudoisotopy stability theorem?

Question

Suppose that $M$ is a compact PL-manifold (possibly with boundary) and let $C^{PL}(M)$ denote the (simplicial) group of PL isomorphisms of $M \times I$ relative to $M \times \{0\} \cup \partial M \times I$, i.e. $PL(M\times I,M \times \{0\} \cup \partial M \times I)$. There is a stabilization map $\sigma: C^{PL}(M) \to C^{PL}(M \times I)$. A hypothetical PL-pseudoisotopy stability would be:

There is a function $f(k)$ so that the map $\sigma: C^{PL}(M) \to C^{PL}(M \times I)$ is $k$-connected for $n = \dim(M) \geq f(k)$.

In other words, the homotopy groups of the PL-pseudoisotopy space stabilize as one multiplies with intervals.

The history of this theorem is as follows as far as I understand. Hatcher in 'Higher Simple Homotopy Theory' gives an outline of a possible proof of this with $f(k)$ approximately $3k$. In 1988, following Hatcher's outline for the PL-case, Igusa gave a very detailed proof for the smooth case. If $M$ is a compact smooth manifold, $C^{DIFF}(M)$ is the topological (or simplicial) group $DIFF(M\times I,M \times \{0\} \cup \partial M \times I)$ and $\sigma$ again the stabilization map, then the statement of the smooth pseudoisotopy stability theorem is as follows:

Let $n = \dim(M)$. The stabilization map $\sigma: C^{DIFF}(M) \to C^{DIFF}(M \times I)$ is $k$-connected for $n\geq\max(2k+7, 3k+4)$.

This implies, by an argument of Burghulea and Goodwillie, a PL-pseudoisotopy stability theorem for smoothable PL-manifolds M with the same range as Igusa's theorem. Also a general PL-pseudoisotopy stability theorem by smoothing theory implies the smooth pseudoisotopy stability theorem. However, as far as I can find the general PL case is still open (see e.g. Waldhausen-Jahren-Rognes, page 22). My questions are thus as follows:

• Is the general PL-pseudoisotopy stability theorem still open? If not, is there a reference for a detailed proof?
• If the answer to the previous question is yes, do people expect the PL-pseudoisotopy stability to be true? Is the range expected to the be same as Igusa's?
• If the answer to the previous question is yes, how is a hypothetical proof expected to go? Is it a matter of making Hatcher's outline precise to get something similar to Igusa's proof or are new ideas needed?

Answer 1

As you say, Hatcher once argued that the map $\sigma_M^{PL}:C^{PL}(M)\to C^{PL}(M\times I)$ is $k$-connected where $k$ is roughly $n/3$, but the proof was not all there.

And as you say Igusa later proved that the analogous map $\sigma_M^{DIFF}:C^{DIFF}(M)\to C^{DIFF}(M\times I)$ is $k$-connected where again $k$ is roughly $n/3$.

Since Igusa's argument was in its broad outlines modeled on Hatcher's, it is easy to imagine that someone might be able to go back and fix Hatcher's proof. But apparently nobody ever has.

Concerning the deduction of PL stability from smooth stability, as you say, it only works in the case of smoothable PL manifolds. There's nothing tricky about it. Here's the smoothing theory ideas we need:

Smoothing theory identifies the homotopy fiber of $Diff(M)\to PL(M)$ (diffeomorphisms fixed on the boundary to PL homeomorphisms fixed on the boundary) with the space of sections (fixed on the boundary) of a bundle over $M$ with fiber $PL_n/O_n$. Likewise it identifies the homotopy fiber of $C^{DIFF}(M)\to C^{PL}(M)$ with sections of a bundle whose fiber is the homotopy fiber of $PL_n/O_n\to PL_{n+1}/O_{n+1}$. And if we write $F^{PL}(M)$ for the homotopy fiber of $\sigma^{PL}$ and likewise for $DIFF$ then it identifies the homotopy fiber of $F^{DIFF}(M)\to F^{PL}(M)$ with sections of a bundle whose fiber is the homotopy fiber of a map $$fiber(PL_n/O_n\to PL_{n+1}/O_{n+1})\to \Omega\ fiber (PL_{n+1}/O_{n+1}\to PL_{n+2}/O_{n+2})$$

Now, by the Alexander trick the spaces $PL(D^n)$, $C^{PL}(D^n)$, and $F^{PL}(D^n)$ are contractible. Igusa's theorem tells us that $F^{DIFF}(D^n)$ is roughly $n/3$-connected. It follows that the space $$fiber(PL_n/O_n\to PL_{n+1}/O_{n+1})\to \Omega\ fiber (PL_{n+1}/O_{n+1}\to PL_{n+2}/O_{n+2})$$ becomes roughly $n/3$-connected after looping $n$ times. (Also this space is already known to be better than $n$-connected; in fact $fiber(PL_n/O_n\to PL_{n+1}/O_{n+1})$ is known to be $(n+1)$-connected). So this space is about $4n/3$-connected.

It follows that that space of sections over $M$ is about $n/3$-connected, so the fiber of $F^{DIFF}(M)\to F^{PL}(M)$ is about $n/3$-connected; so $F^{PL}(M)$ is about $n/3$-connected, just like $F^{DIFF}(M)$. In other words, the $PL$ stability result for $M$ follows from the $DIFF$ stability result for $M$ and for $D^n$ as long as the $n$-manifold $M$ is smoothable. You lose just one degree of connectivity; $k$ becomes $k-1$.

For the deduction of DIFF stability from PL stability (if we knew PL stability), something more is needed, because you don't have the Alexander trick; you don't have some $n$-manifold for which $C^{DIFF}(M)$ is highly connected. This is where Burghelea and I had an idea. Use a relative connectivity argument. Suppose $M$ is obtained by attaching a handle $H$ to $M'$, where $H=D^p\times D^q$, $p+q=n$, and $M'\cap H=D^p\times S^{q-1}$. It was known, using Morlet's disjunction lemma, that the space of concordance embeddings of $H$ in $M$, $CE(H,M)$, has a $(2n-2p-4)$-connected map to the $p$th loopspace of $CE(\ast,M)$ where $\ast$ is a point, if $n-p\ge 3$, and in the case when $M$ is a disk it is also true that $CE(\ast,M)$ is $(2n-5)$-connected. So the suspension map $\sigma^{DIFF}:CE(H,M)\to CE(H\times I,M\times I)$ is a map between roughly $(2n-2p)$-connected spaces, therefore a roughly $(2n-2p)$-connected map.

That paragraph was all DIFF. There is a fibration sequence $$ C(M')\to C(M)\to CE(H,M)$$ (DIFF or PL). If we have PL stability, so that $C^{PL}(M')\to C^{PL}(M'\times I)$ and $C^{PL}(M)\to C^{PL}(M\times I)$ are roughly $n/3$-connected maps, then $CE^{PL}(H,M)\to CE^{PL}(H\times I,M\times I)$ is also about that good.

Now use smoothing theory to conclude that that homotopy fiber of $$fiber(PL_n/O_n\to PL_{n+1}/O_{n+1})\to \Omega\ fiber (PL_{n+1}/O_{n+1}\to PL_{n+2}/O_{n+2})$$ must, after looping $p$ times (sections over $H=D^p\times D^q$ fixed on $S^{p-1}\times D^q$), be highly connected, the number being the smaller of, roughly, $2n-2p$ and $n/3$. Choosing $p$ so as to maximize $min(p+2n-2p,p+n/3)$, we find that the space is roughly $7n/6$-connected. We conclude that the comparison map $F^{DIFF}(M)\to F^{PL}(M)$ is about $n/6$-connected. Since we assumed $F^{PL}(M)$ to be $n/3$-connected, we learn that $F^{DIFF}(M)$ is about $n/6$-connected. We lost about half of the connectivity, but we got something.

Later it became clear that one could do better: Using a refinement of Morlet's disjunction lemma (proved in my thesis), I know that the Hatcher suspension for smooth concordance embeddings $\sigma:CE(H,M)\to CE(H\times I,M\times I)$ is almost $(2n-p)$-connected, much better than the connectivity of the two spaces involved, as long as $n-p\ge 3$. (I believe that my former student Guowu Meng had a proof of this, never published, using his 1992 thesis. I know a somewhat different proof.) Using this you can get from PL stability to DIFF stability without that loss of half of the connectivity.

Returning to the original question, this suggests a different approach to going from DIFF stability to PL stability. If $M$ is not smoothable, it can still be obtained from a smoothable manifold by attaching handles of index at least $3$, in fact of index at least $8$. If $M$ is $H\cup M'$ and the desired result holds for $M'$ then to get it for $M$ it would be enough to have it for $$\sigma:CE^{PL}(H,M)\to CE^{PL}(H\times I,M\times I).$$ It would be enough if the result about stability of DIFF concordance embeddings was valid also for PL. And I am sure that if that result of my 1982 thesis, a "multirelative" version of the disjunction lemma, could be replicated in the PL category then the result about suspension could be so replicated, too.

Now, that result was proved by completely different methods from those of Igusa. The one is sort of parametrized Morse theory. It's all about trying to match up the projection $M\times I\to I$ with the projection $N\times I\to I$ when you have a family of diffeomorphisms $M\times I\cong N\times I$. The other is about trying to match up the projection $H\times I\to H$ with the projection $M\times I\to M$ when you have a family of embeddings $H\times I\to M\times I$.

One funny thing: in my long-ago student days, when I was trying to prove the multirelative smooth disjunction lemma, at one point I considered trying to work in the PL category instead. I had discovered that the method I was developing was a smooth analogue of a parametrized version of a PL technique called sunny collapsing. I even read something about parametrized PL sunny collapsing, but I couldn't understand it, so I went back to the smooth category and faced up to some singular sets and finished the project.

In short, one potential strategy for getting PL stability would involve adapting Igusa's parametrized Morse theory to the PL category (fixing the details of Hatcher's proof). Another completely different strategy would involve adapting my parametrized sunny collapsing to the PL category (fixing the details of parametrized PL sunny collapsing).