Trying to understand Homeotopy Equivalence

Question sheets have become popular for the Math Counts Meetup. For me, they are nice to have because my discussions can get very, very off track and it keeps things more focused. The questions can be used to explain a topic in a logically inductive sort of way. They can help guide the participants from a place they might be at and helps guide them along to answer, not by showing them the answer but letting people work on the ideas in a guided manner. This presents its own trouble though. What questions do we start with, what question do we use?

phillypoincare
Background is modified from Wiki Commons, Sphere is modified from a design by Nick Saporito. You should really check his videos out.

I don’t understand the Poincaré Conjecture  or Grigori Perelman’s proof. The simple reason why is that I have never taken a topology class, ever. What is on my side is I love research and I love trying to understand a concept. To begin with, I identified in my last post some questions to start with.

  1. What does ‘Compact connected and simply connected’ mean?
  2. What is a 3-manifold?
  3. What is a 3-sphere?
  4. What is a Hamilton Program?
  5. What is Ricci Flow?

This is what I have to work with, and expanding the questions out into something more substantial strikes me as the most obvious path. Let us start with “What does ‘Compact connected and simply connected’ mean?”

The first thing to define is Homotopy Equivalence. Here’s a warm-up question you may have seen:

8809 = 6
7111 = 0
2172 = 0
6666 = 4
1111 = 0
3213 = 0
7662 = 2
9313 = 1
0000 = 4
2222 = 0
3333 = 0
5555 = 0
8913 = 3
8096 = 5
7777 = 0
9999 = 4
7756 = 1
6855 = 3
9881 = 5
5531 = 0

2581 = ????

The concept is that children can solve this intuitively, but engineers have trouble with. It says something about the purity of a child’s mind, I guess. I’m using it as an example of Homeotopy Equivalence. This is because the Poincare conjecture connects topology with geometry, and it’s important that we understand that in 2 dimensions any space with one hole can be transformed into any other space with one hole, but it can’t become space with no holes without closing up a hole, and it can’t become space with many holes without puncturing the surface. We can squish, turn upside down, and stretch, but we cannot tear or crease.

At the moment, I’m stuck with finding problems to explain this concept due to most of the math being complicated. So first, a 0 and a 9 are both 1 because they both only have one hole, 8 is a 2 because it has two holes and  so on with the rest of the numbers. A tail can squished to a point so a 9 can squished into a 0. I’m starting here for “What is simply connected space” because other papers tend to start with the concept of homeomorphic surfaces. This seems like a good stopping point for today, and over the next week I’ll hopefully work towards more questions to  help understanding this topic.

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