Writing down an explicit homotopy

In summary, a homotopy is a continuous deformation and the task at hand involves constructing a piece-wise function that satisfies certain conditions. This can be achieved by understanding loop multiplication and proving its homotopy invariance. Sources such as Wikipedia can provide more information on this topic. This exercise is a step towards showing that loop multiplication forms a group, and one can compare different groups using group homomorphisms. The technicalities to keep in mind include reparameterization and the restriction of loops to have a fixed base point.
  • #1
Mikaelochi
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TL;DR Summary
I've never written down an explicit homotopy before which is part of the reason I'm so lost on this problem.
HW9Q2.png

I understand that a homotopy is a continuous deformation. The only thing I really remember is something in my notes like this: F(s, 0) = a(s) for 0≤s≤1 and F(s, 1) = a * Id(s). Basically I have to construct some sort of piece-wise function such that I go something like two loops half of the time and stay in place at one point the other half of the time. Anyway, any help is greatly appreciated.
 
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  • #2
Mikaelochi said:
Summary:: I've never written down an explicit homotopy before which is part of the reason I'm so lost on this problem.

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I understand that a homotopy is a continuous deformation. The only thing I really remember is something in my notes like this: F(s, 0) = a(s) for 0≤s≤1 and F(s, 1) = a * Id(s). Basically I have to construct some sort of piece-wise function such that I go something like two loops half of the time and stay in place at one point the other half of the time. Anyway, any help is greatly appreciated.
I suspect that if you review the definition of multiplication of loops a homotopy will become obvious.

To be entirely rigorous you might like to prove that your homotopy is continuous.
 
  • #3
lavinia said:
I suspect that if you review the definition of multiplication of loops a homotopy will become obvious.

To be entirely rigorous you might like to prove that your homotopy is continuous.
Do you know a good source on that?
 
  • #5
@Mikaelochi

It seems that these exercises are designed to get one used to the idea that homotopy classes of loops form a group. For this one needs a loop multiplication that is homotopy invariant. That is: a.b is homotopic to c.d whenever a is homotopic to c and b is homotopic to d. One thinks of a homotopy as a one parameter family of loops that continuously deforms one loop into the other.

Your problem was one of the steps in showing that multiplication forms a group. It shows that multiplication on the right by the constant loop is a right identity.

You also need to show that the constant loop is a left identity and then prove the existence of inverses and the associative law of multiplication. These are all good exercises.

At each base point one gets a different group and one might ask how these groups are related. In mathematics one can compare groups using group homomorphisms. The groups are considered to be in some sense the same if a homomorphism from one to the other has an inverse. This is true for different base points and your exercise on the two base points on the circle is asking you to define an isomorphism.

To get started note that if α is a path connecting the base points p and q and λ is a loop at q then then α followed by λ followed by α in reverse is a loop at p.

BTW: Convince yourself that loop multiplication without homotopies is not a group.

Technicalities:

- By convention loops are defined on the unit interval. So following one loop by another does not give a loop unless it is reparameterized.

- Loops are restricted in your case to have a fixed base point. So a homotopy through loops must keep the base point fixed.
 
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Related to Writing down an explicit homotopy

1. What is an explicit homotopy?

An explicit homotopy is a mathematical concept used in topology to describe the continuous deformation of one map into another. It is a way of showing that two maps are homotopic, or continuously equivalent, by explicitly defining a path between them.

2. How is an explicit homotopy written down?

An explicit homotopy is typically written down as a formula or equation that describes the path of deformation between two maps. This formula should be continuous and show how the maps change from one to the other.

3. What is the purpose of writing down an explicit homotopy?

The purpose of writing down an explicit homotopy is to prove that two maps are homotopic. This is important in topology because it allows us to understand the properties of spaces by studying their continuous deformations.

4. How does writing down an explicit homotopy help in understanding topological spaces?

By writing down an explicit homotopy, we can see how two maps are continuously equivalent and how they differ from each other. This helps us to understand the topological properties of spaces and how they are related to each other.

5. Are there any applications of writing down an explicit homotopy?

Yes, there are many applications of writing down an explicit homotopy in mathematics and other fields. For example, it is used in differential geometry to study the curvature of surfaces, and in physics to describe the behavior of particles in a system.

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