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I need to reference the following result. Do you know a good source?

The following conditions on an $n$-dimensional simplicial complex $S$ are equivalent: a) $S$ is an $n$ manifold; b) The link of every vertex of $S$ is homeomorphic to the $(n - 1)$-sphere; c) The link of every $k$-simplex is homeomorphic to the $(n - k - 1)$-sphere.

edited May 10 2010 at 14:54

asked May 10 2010 at 9:01

Kestutis Cesnavicius
1,198●5●12

	I think this title is a little misleading as the term "simplicial manifold" is usually reserved for a simplicial object in the category of manifolds. – David Carchedi May 10 2010 at 11:49
	Changed the title. – Kestutis Cesnavicius May 10 2010 at 15:01
	To get a simple correct statement in the category implied by the tags, (a) should refer to S as a combinatorial n-manifold; and (b) and (c) should refer to the links in question as being PL homeomorphic to the appropriate sphere. – Allan Edmonds May 10 2010 at 17:49

Andy Putman · Accepted Answer · 2010-05-10 12:41:40Z UTC

The usual term for objects like this is "combinatorial manifolds".

However, the result is not quite true as you have stated. Definitely b is true if and only if c is true, and b implies a. However, a does not imply b. There definitely exist simplicial complexes which do not satisfy b or c but which are topological manifolds. For example, the famous double suspension theorem of Cannon (weaker versions were proved by Edwards) says that if $X$ is a homology $n$-sphere, then the space $Y$ obtained by suspending $X$ twice is homeomorphic to the $(n+2)$-sphere. If neither $X$ nor the suspension of $X$ is an actual sphere (for instance, this holds if $X$ is the Poincare homology sphere), the vertices of $Y$ corresponding to the suspension points will then not satisfy b.

Roland van der Veen · Answer 2 · 2010-05-10 09:52:01Z UTC

0

I'm pretty sure you can find this kind of results in Rourke and Sanderson's "Introduction to Piecewise linear topology".

http://www.amazon.com/Introduction-Piecewise-Linear-Ergebnisse-Mathematik-Grenzgebiete/dp/0387111026

edited May 10 2010 at 9:52

answered May 10 2010 at 9:26

Roland van der Veen
531●2●7

It has parts of the claim as exercises on p. 22 and p. 24. – Kestutis Cesnavicius May 10 2010 at 10:03

Characterization of combinatorial manifolds in terms of links

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