| 1. | Spaces that are homotopy equivalent to a point are called contractible.
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| 2. | Any fiber bundle over a contractible CW-complex is trivial.
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| 3. | Since is a fibration with contractible fibre, sections of exist.
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| 4. | Non-contractible in general have non-trivial de Rham cohomology.
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| 5. | Furthermore, every cone is contractible to the vertex point by the homotopy
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| 6. | For example, an open ball is a contractible manifold.
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| 7. | All manifolds homeomorphic to the ball are contractible, too.
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| 8. | In particular, the universal covering of such a space is contractible.
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| 9. | Every collapsible complex is contractible, but the converse is not true.
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| 10. | Since the half-plane is contractible, all bundle structures are trivial.
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