| 31. | For example, the punctured Poincar?sphere is an acyclic, 3-dimensional manifold which is not contractible.
|
| 32. | So, why shouldn't it be possible to do calculus on a locally contractible first countable space?
|
| 33. | In particular, if " C " has an initial or final object, its nerve is contractible.
|
| 34. | The fibre in this case is EG, which is a contractible space by the definition of a classifying space.
|
| 35. | According to Brouwer fixed point theorem every Kinoshita who found an example of a compact contractible space without the FPP.
|
| 36. | But not every homology 3-sphere is diffeomorphic to the boundary of a contractible compact smooth 4-manifold.
|
| 37. | The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity.
|
| 38. | The real line is also contractible, and as such all of its homotopy groups and reduced homology groups are zero.
|
| 39. | Exercises are performed by pushing inwards on the contractible cylinders or by pulling the cables outwards, away from the cylinders.
|
| 40. | In other words, while the composition law is not uniquely defined, it is unique up to a contractible choice.
|
