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
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.
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries.
Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.
