Confusion with the basics of Topology (Poincare conjecture)

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

shiv23mj said:

Can i get a clarification

As soon as I figure out what your questions are.

shiv23mj said:

Hi there I am trying to get into topology

Fine. What do you read and why?

shiv23mj said:

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!

shiv23mj said:

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.

shiv23mj said:

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.

shiv23mj said:

that has just be joined together

And lost its boundary when glued together.

shiv23mj said:

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:

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