When is a manifold a cw complex

It only takes a minute to sign up. Kirby showed that a compact 4-manifold has a handlebody structure if and only if it is smoothable: 1 and 2. When is a compact topological 4-manifold a CW complex? Without Loss Of Generality, every compact manifold has such a cover. So fix one finite good cover, and refine it to a finite good cover by closed sets. This doesn't literally produce a CW complex, in that the attaching of cells isn't in cellular order; however it is a finite process, amenable to iterative correction, and the Reader can supply the details as needed.

Sign up to join this community. The best answers are voted up and rise to the top. Ask Question. Asked 10 years, 2 months ago. Active 6 years, 3 months ago. Viewed 2k times. Improve this question. Community Bot 1 2 2 silver badges 3 3 bronze badges. Andrew Ranicki Andrew Ranicki 3, 1 1 gold badge 33 33 silver badges 25 25 bronze badges. Small note, but your question was effectively asked in another form, here: mathoverflow. Apologies - I hadn't seen the earlier posting mathoverflow.

Add a comment. Active Oldest Votes. See also the following three numbered conclusions. Or, in simpler terms, CW is a lot more flexible than, say, PL. For compact 4-manifolds the problem of existence of a CW-complex as well as a handlebody structure is open, see Question 1.

Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. CW complexes and manifolds Ask Question. Asked 7 years, 11 months ago. Active 4 years, 2 months ago. Viewed 8k times. Paulo Henrique 1, 12 12 silver badges 21 21 bronze badges. Add a comment.

Active Oldest Votes. I'm not sure how the classification proceeds for open manifolds. Community Bot 1. Neal Neal 30k 2 2 gold badges 61 61 silver badges bronze badges. Moishe Kohan Moishe Kohan Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown.