Understanding Contractible Curves

[sadegh4137]

I read some text to find it's definition
Is it possible to tell me it's definition?


I read below statements about local and global geometry and I didn't understand it. is it possible tell me it.
"If M ( a manifold) has a trivial topology, a single neighborhood can be extended globally, and geometry is indeed trivial; but if M contains non-contractible curves, such as extension may not be possible."

 

[tiny-tim]

hi sadegh4137!

(if C₁ is the unit circle, ie [0,1] with 0 and 1 the same point)

a closed curve f:C₁ -> M on a manifold M is contractible if it can be contracted to a point,

ie if there's a continuous function g:[0,1] -> C₁^M such that each g(t) is continuous, g(0) is a single point, and g(1) is f

eg the surface of a torus is not contractible, since a circle that "loops" the hole cannot be shrunk to a point!

 

[WWGD]

hi sadegh4137!

(if C₁ is the unit circle, ie [0,1] with 0 and 1 the same point)

a closed curve f:C₁ -> M on a manifold M is contractible if it can be contracted to a point,

ie if there's a continuous function g:[0,1] -> C₁^M such that each g(t) is continuous, g(0) is a single point, and g(1) is f

eg the surface of a torus is not contractible, since a circle that "loops" the hole cannot be shrunk to a point!

Sorry to nitpick, Tiny Tim, but I think it is important to note that the contraction must be done

within the space ( I thinks this follows from your definition of g , but I think it is important to say it any way, since I think it brings room for confusion ), in case the space is embedded somewhere else. As an example, if we have

$S^1$ embedded in $\mathbb R^2$ , then note that $S^1$ --and any curve in it--

can be contracted to a point if we can work in $\mathbb R^2$ , but not so if, while doing the deformation , we must stay within $S^1$ .