Timeline for How does the Framed Function Theorem simplify Cerf Theory?
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32 events
| when toggle format | what | by | license | comment | |
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| Mar 27, 2012 at 17:09 | comment | added | Nikolai Mnev | John, for Pontryagin - this is going and using higher torsions stuff of cause. | |
| May 5, 2011 at 1:01 | vote | accept | Daniel Moskovich | ||
| May 4, 2011 at 16:28 | comment | added | Sergey Melikhov | That'd be great! melikhov (at) mi.ras.ru | |
| May 4, 2011 at 14:24 | comment | added | John Klein | I have a message written to me from John Rognes which compares what Mnev did to what J-R-W do. If you wish to I can send it to you. | |
| May 4, 2011 at 12:21 | comment | added | Sergey Melikhov | I also think that the Hatcher-Steinberger result that Wh$^{\rm PL}$ classifies PL fibrations, and especially Mnev's result that BH$^{\rm PL}$ as defined above classifies PL bundles are proved way harder than they could be. The problem (which Mnev emphasizes in his section 1.2) arises because they want to triangulate the base by a simplicial complex, and I think it will disappear completely if we triangulate the base by a poset. Apparently this does not simplify all of Mnev's theory, only one its consequence. | |
| May 4, 2011 at 12:10 | comment | added | Sergey Melikhov | John, I've heard several talks on this subject (combinatorial formulas for characteristic classes from Mnev's work) by Georgy Sharygin in 2010 (you can find his email on the arxiv). Somewhat related work is being done by Alexander Gaifullin (see his papers on the arxiv). I appreciate your answers and I realize that the parametrized PL $h$-cobordism theorem as stated above, and certainly the high connectivity of the PL stabilization map, may not have optimal proofs in the literature, so maybe they do not yet fully qualify as discrete generalized Cerf theory. I hope to work on this some day. | |
| May 4, 2011 at 11:26 | comment | added | John Klein | Sergey, I realize now why it took me so long to understand what you are writing. Your definition of the Whitehead space is not the same as Waldhausen's and to equate the two, one needs to prove a theorem. | |
| May 4, 2011 at 11:14 | comment | added | John Klein | Sergey, I just looked at Mnev's paper (which I wasn't aware of before). That's really cool! A naive thought occurs to me: I wonder if it can be used to give a combinatorial description of Pontryagin classes. | |
| May 4, 2011 at 4:00 | comment | added | Sergey Melikhov | Objects are finite posets, morphisms are order-preserving maps whose point-inverses have contractible geometric reazilations (let me call these CE-maps). Is there anything wrong with that? For Wh$^{PL}(P)$, where P is a poset, objects are finite posets $Q$ containing $P$ as a subposet and such that the inclusion $|P|\to |Q|$ is a homotopy equivalence (sorry this I forgot to say!) and morphisms are CE-maps fixed on $P$. | |
| May 4, 2011 at 3:53 | comment | added | John Klein | ....and the "other stuff" I mentioned above is needed to relate manifolds to higher algebraic K-theory. As far as geometry goes, you're correct. By the way, what I called "categorical definition" is taken by me to mean using Waldhausen's machinery. | |
| May 4, 2011 at 3:49 | comment | added | John Klein | Okay, that's fine. I understand what you are trying to say. The Jahren-Rognes-Waldhausen paper contains a version of Hatcher' parametrized h-cobordism theorem + other stuff, the latter which relates your definition of the Whitehead space to the categorical definition (as you know doubt know, Hatcher's original statement was never proved; the J-R-W paper proves a modified version). The "other stuff" is where the triangulations and/or thickenings are to be implemented. By the way, in your definition of the PL Whitehead space, can you please be more explicit as to the morphisms you are using? | |
| May 4, 2011 at 3:45 | comment | added | Sergey Melikhov | My definition of Wh$^{\rm PL}(M)$ is implicit in Steinberger's paper that Jahren-Rognes-Waldhausen cite. Mnev mentions it explicitly. I suspect that my version of the parametrized PL $h$-cobordism theorem above would be easiest to prove in the same language of posets. | |
| May 4, 2011 at 3:38 | comment | added | Sergey Melikhov | ... we're concerned with any real geometric application, we won't be able to use a theorem that we cannot even state in an effective language. So the only formulation of the parametrized PL h-cobordism theorem that I understand is the one that I gave above. And it does work: if we have an $n$-parameter pseudo-isotopy, $n$ finite, it is easy to triangulate it by an $n$-parameter pseudo-isotopy of posets in the above sense, extending the given triangulation of $M$ (this is similar to Hatcher's "iterated mapping cylinder decomposition"). | |
| May 4, 2011 at 3:37 | comment | added | John Klein | Concerning your definition of $\text{Wh}^{\text{PL}(M)$---that's not the definition Waldhausen gives. The usual definition the cofiber of the assembly map $(M_+) \wedge A(\ast) \to A(M)$ (Waldhausen gives a specific model for the domain of the assembly map which is the $K$-theory of the category of simple maps (When $M = \ast$, a simple map is a map of based finite simplicial sets which as contractible point inverses). My recollection is that its highly non-trivial to show that the cofiber of the assembly map is equivalent to the definition you are giving. | |
| May 4, 2011 at 3:07 | comment | added | Sergey Melikhov | John, I'm not convinced that "triangulation is not a priori part of the structure". A PL manifold (or polyhedron) is a space with a compatible family of triangulations. Such objects don't even form a set, so Jahren-Rognes-Waldhausen always fix an embedding in $\Bbb R^\infty$. Is it any better than fixing a triangulation? Even so, this is a highly non-constructive definition of a polyhedron, and the only way I know to make it combinatorial is via Alexander's theorem on stellar subdivisions. But we don't have such a parametrized theorem. The trouble with non-constructiveness is that whenever ... | |
| May 4, 2011 at 0:59 | comment | added | Sergey Melikhov | Also, the issue of non-canonical triangulations is not going to be very hard in the one-parameter case - in that case Hatcher's original (rather short) arguments should apply already. | |
| May 4, 2011 at 0:36 | comment | added | Sergey Melikhov | Thanks, John, I'll think about the relative triangulation issue. As for the absolute one, indeed I was avoiding it, but it seems to have been treated by Mnev, arxiv.org/abs/0708.4039 | |
| May 4, 2011 at 0:23 | comment | added | Sergey Melikhov | ("Original category"=from the first comment, "smaller category"=from the second comment; sorry for being slow, I have to recall these things.) In the above, $M$ has no boundary; if it does, the homeomorphism must send $\partial M\times I$ to itself. Finally, the map $dirlim_n BH^{\rm PL}(M\times I^n)\to Wh^{\rm PL}(M)$ is the identity on objects, and may adjust a morphism by composing it with a projection. Does it help? | |
| May 4, 2011 at 0:11 | comment | added | John Klein | Note: The triangulation is not a priori part of the structure. Also Note: for an $h$-cobordism $W$ from $M$ to $N$, we are looking for a way to produce relative triangulation of $W$ (that is relative to $M$). There is no canonical way to accomplish that. | |
| May 4, 2011 at 0:02 | comment | added | John Klein | There is something non-trivial that you might be glossing over in the PL case: you need to choose an explicit triangulation to get from an h-cobordism to a simplicial complex (or finite poset). In any case, as I've said Jahren-Rognes-Waldhausen go the other way. Concerning Kirby's theorem, I simply have know idea since I've never thought about that. My "answer" to Daniel's question had to do with the framed function theorem as a replacement for Cerf theory. The original application of Cerf theory was to pseudoisotopies. | |
| May 3, 2011 at 23:59 | comment | added | Sergey Melikhov | Let me anyway say that $Wh^{\rm PL}(M)$, where $M$ is a specific poset with geometric realization a PL manifold, is the classifying space of the category whose objects are that of the original category containing $M$ as a subposet, and morphisms are that of the original category fixed on $M$. Now $BH^{\rm PL}(M)$ will be the classifying space of the category whose objects are those of the smaller category that contain $M$ as a subposet and are PL homeomorphic to $M\times [0,1]$ extending the identification $M=M\times\{0\}$, and morphisms are those of the smaller category fixed on $M$. | |
| May 3, 2011 at 23:43 | comment | added | Sergey Melikhov | Now there is a special type of a morphism $f:X\to Y$ which I call a subdivision and Mnev calls an assembly; the definition is that the geometric realization of $f^{−1}(Y_{\le p})$ is PL homeomorphic to the cone over the geometric realization of $f^{−1}(Y_{<p})$ keeping the latter fixed, for each $p\in Y$. We then have a smaller category, whose objects are finite posets with geometric realization a PL manifold, and morphisms just the subdivisions (to be cont'd) | |
| May 3, 2011 at 23:23 | comment | added | Sergey Melikhov | John, thanks for details. I have no idea about the passage from (0) to (1), but is it needed for Kirby's theorem? Then, I don't see any difficulty with writing down the map from the space of stabilized PL $h$-cobordisms of a manifold $M$ to Wh$^{\rm PL}(M)$ (though showing it's a homotopy equivalence certainly needs work). Perhaps we're speaking about different spaces. By Wh$^{\rm PL}$ I mean the classifying space of the category whose objects are finite posets and whose morphisms are order-preserving maps with contractible point-inverses. (Wh$^{\rm PL}(M)$ is then clear.) | |
| May 3, 2011 at 23:02 | comment | added | Sergey Melikhov | (Oops, a bit too many diffs in one line.) | |
| May 3, 2011 at 23:01 | comment | added | John Klein | Sergey, what was actually published by Jahren-Rognes-Waldhausen was also very difficult: The idea of that proof is to relate: (0) $A(\ast)$ (which is defined using based simplicial sets) to classifying spaces of (1) finite polyhedra and then to (2) categories of manifolds. The passage from (0) to (1) takes up the bulk of their paper, and the passage from (1) to (2) goes via thickening theory (which is in essence the reverse direction to Morse theory. | |
| May 3, 2011 at 22:57 | comment | added | John Klein | Sergey, I don't think that's the issue. What I was alluding to was a different program for showing Waldhausen's theorem on $A(X)$ which relates $A(X)$ to geometry (in the simplest case: the space of stabilized $h$-cobordisms of a disk is supposed to map to $\text{Wh}^{\text{diff}}(\ast)$ as a homotopy equivalence. The real difficulty is to write down the appropriate map. The expansion space approach gives an alternate description of $\text{Wh}^{\text{diff}}(\ast)$ as a moduli space of cell complexes. The difficulty a lot to do with the geometry of the birth/death singularities. | |
| May 3, 2011 at 22:52 | comment | added | Sergey Melikhov | John, is the difficulty in the difficult step related to the difference between Wh$^{\rm Diff}$ and Wh$^{\rm PL}$? I suspect that for purposes such as justification of Kirby's calculus doing everything piecewise-linearly would simplify matters by omitting a huge chunk of Cerf theory that amounts to proving that a PL 4-manifold has a unique Diff structure. So, would not the PL version of the generalized Cerf theory, in the form of the parametrized PL $h$-cobordism theorem, give a shorter route to such applications? | |
| May 3, 2011 at 21:47 | history | edited | John Klein | CC BY-SA 3.0 |
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| May 3, 2011 at 21:43 | comment | added | John Klein | In effect yes. It supercedes Cerf Theory. Cerf theory is the one-parameter special case. Also, in some sense Waldhausen theory is a stabilized version of Cerf theory (where the manifolds are stabilized with respect to dimension by taking the product with disks). | |
| May 3, 2011 at 21:40 | history | edited | John Klein | CC BY-SA 3.0 |
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| May 3, 2011 at 19:06 | comment | added | Daniel Moskovich | Wow! So if I read correctly, your answer is "yes"? When all these ideas appear, the Framed Function Theory will effectively replace Cerf Theory? At least, the "difficult step" will recover Kirby's theorem? | |
| May 3, 2011 at 18:56 | history | answered | John Klein | CC BY-SA 3.0 |