Topology and Geometry

Introduction to Topology and Geometry

This is a course that Henry and Ben are studying during 2021.

Lecture 1

Starting from Lecture 1 of this course, we have realized that the Mobius strip is a very powerful mathematical idea. --Benkoo (talk) 03:35, 18 July 2021 (UTC)

The Mobius strip is a strip twisted one or more times. One twist is equal to $180^{o}$. Before the strip becomes a Mobius strip, it can be divided into two sides. We will name them red and blue. Then, you twist the strip to turn it into a Mobius strip. If the Mobius strip has an odd twist the blue side will be connected to the red side. If you have an even twist, the blue side will be connected to the blue, and the red will be connected to red. If you start to cut the middle of the blue part and the red part of the Mobius strip you will get two different outcomes:

1. The Mobius strip has an odd twist so you will get a bigger Mobius strip

2. The Mobius strip has an even twist then you will get two Mobius strips. (that are the same length and same number of twists as the Mobius strip before it was cut)

In Topology and Geometry There are three points to remember.

1. There is so much more to mathematics than numbers and formulas. (For example, there is pictorial thinking)

2. Always draw pictures whenever you work on mathematics.

3. There is so much more to pictures than photos of objects.

In Topology and Geometry, you should learn to see and draw things that can't be seen physically. For example take the Mobius strip. When you are doing the experiment of cutting the Mobius strip you will still know what will happen but if you draw it out it will be easier to understand what is happening.

Lecture 2

This lecture is about

1. Solving problem by deformation
2. Understanding by turning it to a higher dimension
3. Introduction to Basic Building Blocks of Topology and Geometry
   1. n-ball $B^{n}$
   2. (n-1)-sphere $S^{n},n-1$ (Don't know why I can't write the )
   3. what is the different between circle and disk

Lecture 3

This Lecture is about

1. The Operation of I:product
   1. m-cube $I^{m}$
2. m-torus $T^{m}$
3. The multiplication of shape in Topology and Geometry
4. Quotient in topology
   1. all kinds of quotient example
   2. using cut to understanding quotient

Lecture 4

This Lecture is about

1. Quotient in topology
   1. using cut to understanding quotient
2. Introduction to $\Sigma g$ and $Ng$
3. Homeomorphism

Lecture 5

This Lecture is about

1. The transformation between $\Sigma g$ and $Ng$
2. Overflow
3. The L dimension object vs K dimension object in M dimension

Lecture 6

This Lecture is about

1. Isotopic
2. The relationship between isotopic and homeomorphic
3. outside the shape inside the shape
4. The positive and negative intersection
   1. There is no tangent vector intersection in Topology

Lecture 7

This Lecture is about

1. The positive and negative intersection
   1. There is no tangent [vector] intersection in Topology
2. Intersect transversely

Lecture 8

This Lecture is about

1. Jordan curve theorem
   1. If you have a closed curve which does not intersect itself it will divide the plan into two parts.
2. Fixed Point Theorem

Lecture 15

This Lecture is about

Also, we should make proper reference, and it will show at the Reference section.

Some interesting websites that referred to this lecture series.