Poincaré conjecture proof's precision relative the derivative number

Question

First of all, this is a question from amateur in geometric topology.

Since most probably I won't be able to follow currently accepted proofs (they are lengthy and field-specific), I have to ask this question that bothers me quite a lot. In layman's terms, the proof equates (in classification sense) an outer, continuous, surface of a 3-dimensional voluminous object A to the same of object B.

However, as we know, a slope in a specific point of the surface can be defined down to any derivative (1st, 2nd, 3rd, etc.), up to infinite number of derivatives.

So, the question is: how "deep" the existing proofs go?

Answer 1

First of all, it is impossible to formulate the statement of the (3-dimensional) Poincare Conjecture (abbreviated as PC3) in layman terms: Any attempt to do so, will pile a lie, upon a lie, upon a lie.

Let me give a mathematical answer to what, I think, you are asking: In the proof of the Poincare Conjecture Perelman establishes the existence of an infinitely differentiable diffeomorphism between a (smooth) homotopy 3-dimensional sphere and the ordinary 3-dimensional sphere.

I doubt you will understand much of what I was saying. Here is a "study list" of what you will have to read and (understand) in order to understand what I wrote above and the statement of the PC3:

Pick any textbook on topology and read about:

1. Definition of a topology in terms of open subsets.

2. Definition of a continuous map between topological spaces.

3. Definition of a homeomorphism.

4. Definition of a compact topological space.

5. Definition of a Hausdorff topological space.

6. Definition of a topological manifold.

7. Definition of a simply-connected topological space.

Now, you are ready for the topological statement of the PC3, but not of the paragraph that I wrote. Here is an alternative (a short-cut):

1. Read a vector-calculus textbook, covering notions of differentiability in Euclidean spaces of arbitrary dimensions (yes, even though you are only interested in the 3-dimensional conjecture). For instance, Chapter 9 of Rudin's book "Foundations of mathematical analysis" will be fine.

2. Read section 1 in the 1st chapter of

Guillemin, Victor; Pollack, Alan, Differential topology, Providence, RI: AMS Chelsea Publishing (ISBN 978-0-8218-5193-7/hbk). xviii, 222 p. (2010). ZBL1420.57001.

covering the notions of differentiable manifolds, smooth maps and diffeomorphisms.

3. You still have to learn about open and compact subsets of Euclidean spaces (of arbitrary dimension). One option is to read chapter 2 of Rudin's "Foundations of mathematical analysis."

4. You also have to learn about simple-connectivity of subsets of Euclidean spaces. This material you will have to pick up elsewhere.

Now, you are ready for the statement of PC3 and of the paragraph that I wrote.

If you are very motivated, it might take you a few months..