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The generalized PL Poincare hypothesis states that in dimension $n$ there is a unique PL manifold that has the homotopy-type of $S^n$. It's known to be true in all dimensions except perhaps $n=4$.

For $n \geq 5$ the standard argument uses rather explicit handle manipulations to come up with the proof (proving the h-cobordism theorem in the process), which is somewhat spiritually-similar to (although more complicated than) the proofs in dimension $n=2$.

The proof in dimension $n=3$ is very different than the above two cases.

But I wonder, perhaps there is a proof that avoids the h-cobordism theorem, perhaps there is a more direct proof? Has there been much discussion of this in the literature?

One thought would be to find an appropriately simplified proof of the Farrell fibering theorem (when a manifold fibers over $S^1$), one that perhaps allows you to reduce $S^n$ recognition into a homotopy-unknot recognition problem.

I imagine back in the 60's and 70's there was some discussion of these topics but I wouldn't know where to look.

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  • $\begingroup$ I am taking a class with Farrell. He often say "this (seemingly trivial statement) is equivalent to Poincare conjecture" when I claim it was obvious... $\endgroup$
    – Bombyx mori
    Commented Dec 18, 2013 at 7:41

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The Poincare conjecture predates the h-cobordism theorem, and the original proof of it in Smale's paper does not prove the h-cobordism theorem (though the handle manipulations in it are what inspired the h-cobordism theorem, so this is maybe not a good answer).

A very different way of proving the Poincare conjecture is Stallings's proof using "engulfing". A nice modern source for this is Chapter 9 of Ferry's notes here.

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  • $\begingroup$ Hmm, I suppose I disagree with you. The h-cobordism theorem was just the natural setting for Smale's proof of the Poincare conjecture. It wasn't much more than a recasting of the argument into its "natural" form. And Stallings proof I lump in with the handle-by-handle type arguments such as the h-cobordism theorem. $\endgroup$
    – Ryan Budney
    Commented Dec 18, 2013 at 0:09
  • $\begingroup$ @RyanBudney : I'd be impressed if you managed to give an engulfing-style proof of the h-cobordism theorem; the flavor of the handle manipulations in it are pretty different from those that show up in Smale's work. $\endgroup$
    – Andy Putman
    Commented Dec 18, 2013 at 0:21
  • $\begingroup$ (which is not to say that I wouldn't be interested in a completely different proof of the PC; however, I kind of doubt that one exists) $\endgroup$
    – Andy Putman
    Commented Dec 18, 2013 at 0:24
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    $\begingroup$ I'm starting to think maybe there's one does exist. I don't have a coherent argument, but when I put my next paper up on the arXiv you'll see some of my reasons why I have these hopes. $\endgroup$
    – Ryan Budney
    Commented Dec 18, 2013 at 0:34
  • $\begingroup$ @RyanBudney : I look forward to seeing it! I always enjoy your papers. $\endgroup$
    – Andy Putman
    Commented Dec 18, 2013 at 0:39

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