# Are non-PL manifolds CW-complexes?

Can every topological (not necessarily smooth or PL) manifold be given the structure of a CW complex?

I'm pretty sure that the answer is yes. However, I have not managed to find a reference for this.

• @algori : I thought you had posted an (important sounding) comment? Why did you delete it? – A grad student Aug 27 '10 at 4:48
• It turns out that my first comment was a bit wrong. Here are the slides of A. Ranicki's talk in Orsay. www.maths.ed.ac.uk/~aar/slides/orsay.pdf It says on p. 5 there that a compact manifold of dimension other than 4 is a CW complex. There is a related conjecture that says that each closed manifold of dimension $\geq 5$ is homeomorphic to a polyhedron (there are 4-manifolds for which this is false). See arxiv.org/pdf/math/0212297. I'm not sure what if anything is known about the noncompact case. – algori Aug 27 '10 at 4:50
• Update: recent work of Davis, Fowler, and Lafont front.math.ucdavis.edu/1304.3730 shows that in every dimension ≥6 there exists a closed aspherical manifold that is not homeomorphic to a simplicial complex. – Lee Mosher May 1 '13 at 16:10
• Hatcher's Algebraic Topology p. 529 has a paragraph answering this question very clearly for compact manifolds (not including results in 2013 of course). However his references are to two long dense books, without page specification. – hsp Sep 3 '13 at 15:47