Let M be an arbitrary smooth manifold.
Then, is there an open covering $\{U_i\}$ of $M$ such that each $U_i$ is constractible anytime?
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If M is simply-connected, then does it hold? – math student Feb 14 '20 at 14:38
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Yes! This comes almost from the definition.
When we define the manifold structure on $M$, we choose an open cover $\{U_i\}$ and diffeomorphisms from the $U_i$ to an open set in $\mathbb{R}^n$ (with extra conditions).
One thing we could do is require that the $U_i$ map diffeomorphically to the open unit ball. This means that each $U_i$ is contractible, and they form a cover by assumption.
Santana Afton
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Thank you for answering my question. How can we constract a diffeomorphically map f_i:U_i →B_i ?, where B_i is the open unit ball. – math student Feb 14 '20 at 13:30