Confusion with the basics of Topology (Poincare conjecture)

[shiv23mj]

Hi there I am trying to get into topology
I am looking at the poincare conjecture
if a line cannot be included
as it has two fixed endpoints
by the same token
isn't a circle a line with two points? that has just be joined together
so by the same token the circle is not allowed?
Can i get a clarification

 

[fresh_42]

Can i get a clarification

As soon as I figure out what your questions are.

Hi there I am trying to get into topology

Fine. What do you read and why?

I am looking at the poincare conjecture

So you want to get into topology by one of the most complicated theorems topology has to offer? Ambitious plan. Good luck!

if a line cannot be included
as it has two fixed endpoints

A line has a boundary, yes, and the Poincaré conjecture is a statement for objects without a boundary.

by the same token
isn't a circle a line with two points?

Which two points? They aren't anymore after you glued them together. Forgotten. Lost. Gone.

that has just be joined together

And lost its boundary when glued together.

so by the same token the circle is not allowed?

The circle is allowed in the one-dimensional case. However, the statement sounds a bit stupid in this case because it becomes almost trivial: Every one-dimensional closed manifold of the homotopy type of a circle is homeomorphic to the circle. It means: any closed line without any crossings can (topologically) be seen as a circle.

The Poincaré conjecture (now theorem) is about specific topological objects (compact, simply connected, three-dimensional manifolds without boundary) and a criterion when two of them are topologically equivalent, namely to a 3-sphere, the surface of a four-dimensional ball.

The key is to understand what topological equivalent means. It basically means a continuous deformation that can be reversed. Not allowed are cuts, e.g. creating holes. That is why topologists consider a mug and a donut to be the same thing:

Spoiler: Animated picture. Put into spoiler tags to avoid the annoying movements.

 

[mathwonk]

The Poincare conjecture says, that every compact, connected 3 - manifold (without boundary) which has trivial fundamental group, is homeomorphic to the 3-sphere.

Thus you must learn the concepts of homeomorphism, manifold, compactness, connectedness, and fundamental group, to even understand the statement.

I recommend reading the book Algebraic topology, an introduction, by William Massey, at least the first two chapters. You will find there most of the proof of the 2 dimensional analogue of the Poincare conjecture, already very instructive and interesting.

here's a used copy for under $10!
https://www.abebooks.com/servlet/Se..._f_hp&tn=algebraic topology&an=William Massey