From: asimov@nas.nasa.gov (Daniel A. Asimov)
Newsgroups: sci.math.research
Subject: Non-PL Triangulation of Manifolds -- What's the Latest???
Date: 10 Jul 1996 20:25:19 GMT
[Note: All manifolds are assumed to be Hausdorff and paracompact, and to
have a countable base.]
Around 1969, R. Kirby & L. Siebenmann first showed that there exist topological
manifolds that admit no PL structure.
It was since shown that there could exist a manifold triangulated as a
simplicial complex but with a non-PL triangulation. (E.g., the double
suspension of a triangulated nontrivial homology 3-sphere gives a non-PL
triangulation of S^5. Of course, S^5 admits other triangulations that are PL.)
QUESTION: Do there exist topological manifolds that admit no triangulation
(PL or not) whatsoever??? In any case, what is known about the range of
dimensions for which this may be possible?
References to the literature would be appreciated.
Dr. Daniel Asimov
Senior Research Scientist
Mail Stop T27A-1
NASA Ames Research Center
Moffett Field, CA 94035-1000
asimov@nas.nasa.gov
(415) 604-4799 w
(415) 604-3957 fax
==============================================================================
From: hempel@math.rice.edu (John Hempel)
Newsgroups: sci.math.research
Subject: Re: Non-PL Triangulation of Manifolds -- What's the Latest???
Date: 12 Jul 1996 18:00:28 GMT
In article <4s13jf$bh9@cnn.nas.nasa.gov>,
Daniel A. Asimov wrote:
.........
>QUESTION: Do there exist topological manifolds that admit no triangulation
>(PL or not) whatsoever??? In any case, what is known about the range of
>dimensions for which this may be possible?
>
>References to the literature would be appreciated.
E_8 is such a manifold.
cf Akbulut and McCarthy, "Casson's Invariant for Oriented Homology
3-spheres," Princeton U. Press 1990.
==============================================================================
From: Allan Edmonds
Newsgroups: sci.math.research
Subject: Re: Non-PL Triangulation of Manifolds -- What's the Latest???
Date: Fri, 12 Jul 1996 15:21:12 +0000
John Hempel wrote:
>
> In article <4s13jf$bh9@cnn.nas.nasa.gov>,
> Daniel A. Asimov wrote:
> .........
> >QUESTION: Do there exist topological manifolds that admit no triangulation
> >(PL or not) whatsoever??? In any case, what is known about the range of
> >dimensions for which this may be possible?
> >
> >References to the literature would be appreciated.
>
> E_8 is such a manifold.
>
> cf Akbulut and McCarthy, "Casson's Invariant for Oriented Homology
> 3-spheres," Princeton U. Press 1990.
See Galewski and Stern, Classification of simplicial triangulations of topological
manifolds, Annals of Math. 111(1980), 1-34, for the higher dimensional situation.
It is an open question in higher dimensions (say >= 5 in the case of empty boundary)
whether *every* topological n-manifold can be triangulated as a simplicial complex. The
theorem of Galewski and Stern reduces the question to the existence of a PL homology
3-sphere H of mu-invariant 1 such that the connected sum H#H bounds a contractible PL
4-manifold.
==============================================================================
From: ruberman@binah.cc.brandeis.edu
Newsgroups: sci.math.research
Subject: Re: Non-PL Triangulation of Manifolds -- What's the Latest???
Date: 15 Jul 1996 14:42:53 GMT
In article <4s13jf$bh9@cnn.nas.nasa.gov>, asimov@nas.nasa.gov (Daniel A.
Asimov) writes:
>[Note: All manifolds are assumed to be Hausdorff and paracompact, and to
>have a countable base.]
>
>Around 1969, R. Kirby & L. Siebenmann first showed that there exist topological
>manifolds that admit no PL structure.
>
>It was since shown that there could exist a manifold triangulated as a
>simplicial complex but with a non-PL triangulation. (E.g., the double
>suspension of a triangulated nontrivial homology 3-sphere gives a non-PL
>triangulation of S^5. Of course, S^5 admits other triangulations that are PL.)
>
>QUESTION: Do there exist topological manifolds that admit no triangulation
>(PL or not) whatsoever??? In any case, what is known about the range of
>dimensions for which this may be possible?
>
>References to the literature would be appreciated.
The existence of non-triangulable (as simplicial complexes) manifolds follows
from work of Freedman (J. Diff. Geom. 17, (1982)) combined with closely related
works of Casson (cf Akbulut-McCarthy: Casson's invariant for oriented homology
3-spheres, Princeton Math. Notes 36 (1990)) or Taubes (J. Diff Geom. 25
(1986)).
Freedman constructs many non-smoothable manifolds (eg the so-called E8
manifold, which is spin with signature -8 and so would contradict Rohlin's
theorem if smoothable.) If E8 were triangulable, an elementary argument shows
that it is smooth away from its vertices, and that the link of some vertex must
be a homotopy 3-sphere with non-zero Rohlin invariant. But Casson's invariant
shows that there is no such homotopy sphere. Taubes' work on 4-manifolds with
periodic ends could alternatively be applied--in this setting it implies that
there is no open 4-manifold with the homology of E8- point, with an end of the
form (homotopy-sphere) x [0,infinity).
The existence of non-triangulable manifolds in higher dimensions is still
unknown, although it is related to problems about homology 3-spheres through
work of Galewski-Stern and T. Matumoto in the mid-70's.
Daniel Ruberman