Talk:Simplicial manifold

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)

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.

Clean-up begun

I have made some cosmetic improvements to the article, and explained or re-stated the generalities in the opening paragraph. An encyclopedia entry should include some plain-language description even of technical topics, and I've tried to do this. I don't feel there is enough material to warrant a topic split, but the subject is sufficiently notable to have and expand this entry. As to common usage, the need in Physics for these concepts disallows the strict definition in some regards. In Causal dynamical triangulation the researchers are actually using the technique to try to find the correct number of dimensions, at various levels of scale, so assigning a fixed number of dimensions beforehand would be artificial. Thus; the need for simplicial manifolds in CDT is for a more general framing of this concept.

And the prime authors of CDT just had a feature article in the July '08 Scientific American.

I echo the thoughts of Omar above that you folks are both right, and that Physics demands we use broader definitions. Therefore; both a coherent statement of the strict Mathematics definition, and a looser statement of the concept which allows for the most common usage, seem to be helpful or necessary for this entry. As the primary author of the WP article on CDT, which may have been this article's reason to be, I thank the other contributors for posting an entry for this subject.

JonathanD (talk) 15:23, 6 August 2008 (UTC)

Tried to give the article a firm base-structure

The subject of simplicial manifolds has always been a central part of mathematics and it will continue to bee so. I hope that this article can continue to grow within the two main sections "geometrical simplicial manifolds" and "abstract simplicial manifolds". The subject is large and it might need additional sections in the future. Gofors (talk) 06:15, 18 March 2010 (UTC)

Wrong Subject

A simplicial manifold is a contravariant functor from the category of finite ordered sets $\Delta$ to the category of manifolds with submersions as covers.

As such it is a "tower of manifolds" m_n each consisting of "n-simplices" ... But in a different manner than stated in the article.

The description as a manifold build from simplexes is something else. It is the geometric realization functor of an simplicial set in the category of manifolds...

The article should be completely rewritten, or better delete it all...

To get into the subject look for articles on higher Lie theory, Lie n-groupoids or Lie n-algebroids, since Lie n-groupoids are simplicial manifolds, that are "Kan" in some special way...

Hence Simplicial manifolds should be a subsection of Higher Lie Theory

(Mircomaster (talk) 18:33, 25 July 2011 (UTC))

Delete the page

This whole page should be simply deleted. The term simplicial manifold is not a standard term in manifold theory. And the exposition on the current page is just a mess. (These two points are of course related - the page has not received serious attention because the title contains a non-standard term.)

Jfdavis (talk) 17:40, 28 March 2012 (UTC)

Dubious statements

I'm tagging a bunch of stuff as dubious. Let me list them out:

A simplicial manifold is a simplicial complex for which the geometric realization is homeomorphic to a topological manifold.

Isn't this the same thing as a Triangulation (topology)? That article seems better than this one...

A manifold made from simplices can be locally flat, or can approximate a smooth curve, just as a large geodesic dome...

Whaaaat? flatness requires a metric or at least differentiable structure and that is absent in the definition. So this sentence cannot possibly be right.

This notion of simplicial manifold is important in Regge calculus and Causal dynamical triangulations as a way to discretize spacetime by triangulating it. A simplicial manifold with a metric is called a piecewise linear space.

Well, OK, true, if you've endowed it with lots of extra structure needed to study quantum gravity. But neither here nor there, for this article, So I'm deleting this. 67.198.37.16 (talk) 21:47, 6 May 2016 (UTC)

Never mind. I got bold, and just deleted and re-wrote, trying to capture everything the page was trying to say, while also addressing all of the comments made throughout this talk page, above. Its now a stub that no longer says anything that is obviously wrong or inane. 67.198.37.16 (talk) 22:25, 6 May 2016 (UTC)

Article cleanup proposal

Here's a way to split up the article: we can have this article discuss simplicial manifolds in the sense of simplicial objects in the category of smooth manifolds, and have redirects to Piecewise-Linear manifold for the other definition. This redirect could be at the top, and in a definition section where the alternative uses could be discussed, but have them point to piecewise-linear manifolds, since Regge's theory of simplicial manifolds is essentially constructing a traingualization via simplices of a manifold, which is awfully close to PL-manifolds, an a subclass of objects.

Also, here's some useful references for populating this page with material

• https://arxiv.org/abs/0905.4044 - Presenting higher stacks as simplicial schemes
• The references on https://ncatlab.org/nlab/show/Kan+simplicial+manifold#NSS12
• Should also reference Duskin hypergroupoids
• Also, taking a covering of a space, and taking the cech nerve gives examples of simplicial manifolds
• It would be interesting to add a Lie groupoid and show how it can be embedded as a simplicial manifold

In addition, material on cosimplicial manifolds should also be included, since they form the basis for various theories of derived manifolds. See

• http://web.archive.org/web/20210702222329/https://www.maths.ed.ac.uk/~v1ranick/papers/spivakd.pdf
• https://people.maths.ox.ac.uk/~joyce/YorkHandout2.pdf

Kaptain-k-theory (talk) 22:39, 2 July 2021 (UTC)