Understanding Manifolds: Exploring Dimensions and Shapes in Mathematics

[Kidphysics]

Homework Statement

Taken from Wiki:
a manifold is a mathematical space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold. Thus, a line and a circle are one-dimensional manifolds, a plane and sphere (the surface of a ball) are two-dimensional manifolds, and so on into high-dimensional space.

Homework Equations

A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball.

The Attempt at a Solution

The dimensions of a manifold are=n-1 dimensions of shapes and objects of reality?

 

[hunt_mat]

What you're referring to in actual fact are the boundaries of trhese shapes. For example the sphere is the boundary of a solid ball. The reason for this is because the solid ball is contractable to a point. When you take the boundary you reduce the dimension by one.

 

[Kidphysics]

a solid ball is contractible to a point? I do not understand that since a sphere is supposed to exist in 3 dimensions and and a point has only one.

Is there a way to think of manifolds and boundaries in terms of spatial dimensions?

The circle is the boundary of a sphere making it a 2 dimensional manifold?

 

[Kevin_Axion]

I think it means that the sphere is simply connected therefore a loop on the surface can be compressed to a point unlike a torus. This was shown in Perelman's Ricci Flow with Surgery proof of the Poincare Conjecture to be true for a 3-dimensional closed surface.
"Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere." - Wikipedia
Solution:
http://en.wikipedia.org/wiki/Solution_of_the_Poincaré_conjecture

 

[hunt_mat]

The circle id the boundary of a solid disc. Topology is weird, get used to it.

 

[Kidphysics]

I think it means that the sphere is simply connected therefore a loop on the surface can be compressed to a point unlike a torus.

Thank you Kevin, I have seen pictures of this and I know exactly what you're referring to.

The circle id the boundary of a solid disc. Topology is weird, get used to it.

Lol yes it is very abstract, can anyone point me in a good direction for elementary reading on the subject?

 

[hunt_mat]

Topology: A Geometric Approach (Oxford Graduate Texts in Mathematics) by Terry Lawson