# Show Regular Homotopy is an Equivalence Relation

• MxwllsPersuasns
In summary, the problem is to show that regular homotopy of regular curves is an equivalence relation, meaning it satisfies reflexivity, symmetry, and transitivity. A homotopy is a function that deforms one curve into another while preserving the endpoints. To show reflexivity, we need to show that a curve is regularly homotopic to itself. For symmetry, we need to show that if one curve is regularly homotopic to another, then the second curve is also regularly homotopic to the first. Finally, for transitivity, we need to show that if two curves are regularly homotopic to a third curve, then they are also regularly homotopic to each other. This can be done by using a smoothing function to
MxwllsPersuasns

## Homework Statement

Show that regular homotopy of regular curves γ : I → Rn is an equivalence relation, that is:
i) γ ∼ γ (where the symbol ∼ stands for “regularly homotopic”);
ii) γ ∼ γ˜ implies ˜γ ∼ γ;
iii) γ ∼ γ˜ and ˜γ ∼ γˆ implies γ ∼ γˆ (here you have to use a smoothing function).

## Homework Equations

Not entirely sure

## The Attempt at a Solution

So my issue here is that I was sick with the flu for a week and so I missed the week we (apparently) went over homotopy and now I'm quite lost on what to even do for this problem. My understanding of homotopy and how to show two curves are homotopic is extremely limited but this is what (I think) I know.

A homotopy is some kind of function that takes two endpoints and then deforms a curve (which is attached to those two endpoints at either end) into another curve (It changes the actual curve right? And it's not something that just changes the parameterization of a single curve?). The function has two parameters (at least it seems) where the first denotes the length along the interval (which is being mapped onto the curve) and the 2nd denotes incremental steps in the deformation of one curve to the other. That's about all I know..

So it looks like I basically need to show that a regular homotopy of regular curves satisfies the three conditions that define an equivalence relation: i) reflexivity, ii) symmetry and iii) transitivity. I'm unsure of what the structure of a homotopy looks like (is there some general structure like an ellipse is γ(t) = <acos(t), bsin(t)>) or what exactly to manipulate to show that i), ii) and iii) are true.

If anyone can offer me any advice or guidance at all I would appreciate it more than you could imagine.

Nevermind all, I pondered it for a bit and with the help of wikipedia I think I got it.

## 1. What is regular homotopy?

Regular homotopy is a concept in topology that describes a continuous deformation of one shape into another. More specifically, it refers to a continuous transformation of one closed curve into another in a topological space, while keeping the endpoints fixed.

## 2. How is regular homotopy related to equivalence relations?

Regular homotopy is an equivalence relation, meaning that it satisfies the properties of reflexivity, symmetry, and transitivity. This means that every shape is homotopic to itself (reflexivity), any two shapes that are homotopic to each other are also homotopic in the reverse direction (symmetry), and if shape A is homotopic to shape B and shape B is homotopic to shape C, then shape A is also homotopic to shape C (transitivity).

## 3. What is the significance of showing that regular homotopy is an equivalence relation?

Showing that regular homotopy is an equivalence relation is important because it allows us to classify shapes into equivalence classes. This means that we can group shapes together based on their homotopy, which can help us better understand and analyze their topological properties.

## 4. How is regular homotopy different from other types of homotopy?

Regular homotopy is a stricter concept compared to other types of homotopy, such as homotopy through paths or homotopy through maps. It requires that the endpoints of the curves remain fixed throughout the transformation, whereas other types of homotopy may allow for the endpoints to move.

## 5. Can regular homotopy be extended to higher dimensions?

Yes, regular homotopy can be extended to higher dimensions. In two dimensions, it is known as regular isotopy, and in three dimensions, it is known as regular cobordism. These concepts follow the same principles as regular homotopy but apply to higher-dimensional shapes and spaces.

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