If MnM^n is a compact topological manifold (not necessarily with additional structure), is there an upper bound on the number of charts needed to cover MM ? Does this bound depend on the dimension of MM ?

Thanks in advance…

Cheers

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a related thread

– t.b.

Dec 20 ’11 at 6:42

I have seen a related thread about my question. But it seems the manifolds they talk about are triangulated manifolds and i would like to know an answer that does not depend on triangulability of manifolds. And i would like to have a good reference. Thanks in advance !!

– onebengaltiger

Dec 20 ’11 at 19:05

I have examined the related thread a bit more, and it seems i could find some answers by looking at Ostrand’s theorem and Kirby-Siebenmann handle decompositions for TOP manifolds. Thanks for the tip about the related thread !!!

– onebengaltiger

Dec 22 ’11 at 5:11

Someone’s trying to edit the question to remove the word “compact”… wtf? how is that a reasonable edit??

– 6005

2 days ago

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1 Answer

1

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This is an old question, but maybe somebody still cares.

First, let us define what a “chart” means in the context of topological manifolds MnM^n:

Definition. A chart on MM is an open subset U⊂RnU\subset R^n and a topological embedding f:U→Mnf: U\to M^n. We say that MM is covered by charts (Ui,fi),i∈J(U_i, f_i), i\in J, if

M=⋃i∈Jfi(Ui).

M=\bigcup_{i\in J} f_i(U_i).

Note that in this definition I do not require the open sets UU to be connected, which is important. I also require MM to be Hausdorff and 2nd countable.

Then:

Theorem. Every topological nn-dimensional manifold MM (compact or not) admits a cover by n+1n+1 charts.

Proof. Note that MM has topological dimension nn and is a normal topological space (since MM embeds in some RNR^N and, hence, is metrizable). Let W{\mathcal W} be an open cover of MM by subsets homeomorphic to open balls in RnR^n. Thus, by Ostrand’s theorem on colored dimension, there exists a set {Vi:i=0,…,n}\{{\mathcal V}_i: i=0,…,n\} such that:

Elements of each Vi{\mathcal V}_i are certain pairwise disjoint open subsets of MM.

The union

V=n⋃i=0Vi

{\mathcal V}=\bigcup_{i=0}^{n} {\mathcal V}_i

is an open cover of MM.

Each element of V{\mathcal V} is contained in an element of W{\mathcal W}.

By 2nd countability property of MM, each Vi{\mathcal V}_i is at most countable and each of its elements is homeomorphic to an open subset of RnR^n (part 3). Therefore, for each ii, the (disjoint!) union

Ti=⋃V∈ViV

T_i=\bigcup_{V\in {\mathcal V}_i} V

admits a topological embedding gi:Ti→Ui=gi(Ti)⊂Rng_i: T_i\to U_i=g_i(T_i)\subset R^n. Its inverse fif_i is a chart on MM. Since V{\mathcal V} is a cover of MM, it follows that we obtained a cover of MM by n+1n+1 charts (Ui,fi),i=0,…,n(U_i, f_i), i=0,…,n. qed