Poincare Conjecture: Explaining Lower Dimension Equivalent

[Phyzwizz]

I've been doing a project on Henri Poincare and I am attempting to explain his conjecture to my Calculus class so I am using the common lower dimensional equivalent to do so. If a rubber band is wrapped around an object and becomes smaller and smaller until it is a point than that object is homeomorphic to a sphere and contrast this with the case of a torus.

My problem is that I am confused as to whether the Poincare conjecture involves wrapping a one dimensional line around a four dimensional figure or if it is a two dimensional plane around a four dimensional figure. I am only briefly mentioning this so the answer shouldn't be too complex.

 

[lavinia]

you want a 1 dimensional line. The 3 sphere though is a 3 dimensional manifold - meaning that locally it looks exactly like a small region in three dimensional space.

Not every rubber band will shrink to a point by itself. It may need a little nudge. This can be seen on the regular sphere. Stretch a rubber band on it that forms a great circle. It will sit tight until without moving until you push it slightly. Then it will shrink to a point.

 

[Phyzwizz]

Thank you that was very helpful!