In

W. W. Boone, W. Haken, and V. Poenaru, On Recursively Unsolvable Problems in Topology and Their Classification, Contributions to Mathematical Logic (H. Arnold Schmidt, K. Schütte, and H. J. Thiele, eds.), North-Holland, Amsterdam, 1968.

a combinatorial manifold is defined as a simplicial complex with the property that the star of every vertex is combinatorially equivalent to the standard $n$-simplex. (two simplicial complexes are combinatorially equivalent if they possess linear subdivisions, the associated abstract simplicial complexes of which are isomorphic) This is equivalent to the condition that the link of every vertex be a combinatorial $(n-1)$-sphere (=boundary of the standard $n$-simplex) if the underlying manifold has no boudnary.

However, in

A. Ranicki (ed.), The Hauptvermutung Book, K-Monographs in Mathematics, vol. 1, Kluwer Academic Publishers, Dordrecht, Boston, London, 2010.

on page 4, this condition is used to define the term

``combinatorial manifold (or PL manifold)''.

I find this very weird;
a PL manifold should be defined as a topological manifold with a maximal atlas of homeomorphisms with PL coordinate changes (and I know a lot of authors who use this definition).

The obvious question now is:
Is a simplicial complex, the vertices of which have $S^{n-1}$ as link,
the same as a topological manifold with a maximal atlas of homeomorphisms piecewise linear coordinate changes?
Of course, Ranickis nomenclature implies that it does.

Obviously, the condition on the links can be used to construct such an atlas. However, the converse puzzles me, as it seems to be equivalent to the question if every manifold with a maximal PL atlas admits a triangulation.

If anyone could point me to an article where this problem is addressed, I would be thrilled.

Best regards, Malte

`non-linear'' (I guess this is what you mean by`

make the transition maps linear''). Anyway, you're a very sweet man, thanks a lot! $\endgroup$ – Malte Jan 14 '12 at 20:01