Talk:Simplicial manifold

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[edit] Confusion

I'm a bit confused about the definition given here and the relation to piecewise linear manifolds.

  • I've also seen simplicial manifolds defined as simplicial complexes that are also topological manifolds. Is this equivalent to the definition given here?
  • Are simplicial manifolds piecewise linear, or does one need to require that the star of every point is piecewise linearly equivalent to an n-simplex (rather than just homeomorphic)?
  • I remember reading something about the double suspension of a Poincare sphere being a manifold that was homeomorphic to a simplicial complex but was not a PL manifold. What is the relation to the concept defined here?
  • Is the definition here the correct one?

If anyone who knows this stuff could work on this page and the PL manifold page it would be appreciated. -- Fropuff 03:20, 16 November 2006 (UTC)

Answers:

  • No. The example you give (double suspension of a homology 3-sphere that is not the 3-sphere) will give a triangulation of the 5-sphere where the link of some vertices will not even be homeomorphic to the 4-sphere, which is required by the definition in the article.
  • Simplicial manifold (as described here) only need have stars of vertices homeomorphic to a ball. So if you take the triangulation of S^6 given by suspending the nonPL triangulation of S^5, then you are in trouble. The link of a suspension point will only be homeomorphic, but not PL equivalent to the standard triangulation of the 5-sphere.
  • The relation is that this example shows there is something funny about this definition.
  • I'm not sure. We have here something that is not the same as a manifold that is a simplicial complex, but not the same as a PL manifold. --C S (Talk) 18:05, 17 November 2006 (UTC)

Thanks for the response C S. That clarifies things. I am still a little suspicious of the definition given here. If anyone has a good reference for this page I'd like to see it. -- Fropuff 19:38, 17 November 2006 (UTC)

I took a look at Causal_dynamical_triangulation which is apparently a reason for the existence of this stub. From what I could make of it (the phrasing was a little vague), the simplicial manifolds mentioned there seem have triangulations coming from the triangulation of a +1 dim higher PL triangulation restricted to the boundary of the PL manifold. The definition of "piecewise linear space" here is also odd. I'm beginning to think this is just some physicists lingo for PL manifold (where homeomorphism is corrected to mean PL equivalent). It may be standard or not. It may even be that different physicists use different definitions of simplicial manifold without realizing they are different. If they are only interested in 3-dimensional "slices", then of course there is no difference in that case. --C S (Talk) 20:34, 17 November 2006 (UTC)
I suspect you are right. After a little searching it does seem that the phrase occurs much more frequently in the physics literature. I've come across at least 3 inequivalent definitions. It's probably just a case of physicists being careless with their definitions. The more rigorous treatments seem to define a simplicial manifold as a simplicial complex that is also a PL manifold. -- Fropuff 04:01, 18 November 2006 (UTC)
You both are right. In first Regge's article (T. Regge "General relativity without coordinates". Nuovo Cim. 19: 558-571, 1961) the situation is strongly unclear, but I would say that he is using a simplicial manifold. Subsequent articles (like R Friedberg and TD Lee - Nuclear Physics B, 1984) make evidently use of PL structures. The fact is that when you are doing physics you cannot restrict your working environment with narrow definitions. Omar.zanusso (talk) 17:44, 31 May 2008 (UTC)

[edit] simplicial objects

Simplicial manifold very commonly means a simplicial object in the caterogy of manifolds. I added this second meaning of the term to this entry, but perhaps they should be split into two different entries.