How to prove the inclusion is a homotopy equivalence?

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In summary, the conversation discusses the concept of a deformation retraction in the weak sense of a space X to a subspace A, and how this can be used to prove that the inclusion of A into X is a homotopy equivalence. The conversation also touches on the issue of composing the homotopy f_t with the inclusion i, as their domains and codomains are different sets.
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qinglong.1397
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How to prove the inclusion is a homotopy equivalence?

Homework Statement



A deformation retraction in the weak sense of a space X to a subspace A is a homotopy [tex]f_t: X\rightarrow X[/tex] such that [tex]f_0=Id_x, f_1(X)\subset A,[/tex] and [tex] f_t(A)\subset A[/tex] for all t. Show that if X deformation retracts to A in this weak sense, then the inclusion [tex]A\hookrightarrow X[/tex] is a homotopy equivalence.

Homework Equations





The Attempt at a Solution



I have one problem. Let [tex]r: X\rightarrow A[/tex] be the retraction. I can prove that [tex]ri\simeq Id_A[/tex]. Then I have to prove that [tex]ir\simeq Id_X[/tex]. Then there is a problem. To achieve this goal, I have to use the homotopy [tex]f_t[/tex]. This homotopy is defined on the domain X. It seems that I have to compose [tex]f_t[/tex] with inclusion, but the codomain of [tex]f_t[/tex] is different from the domain of [tex]i[/tex]. So I do not know how to do. Please give some hints. Thanks a lot!:shy:
 
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No one knows the answer? What a pity! Hope someone can help me!
 
  • #3


qinglong.1397 said:
It seems that I have to compose [tex]f_t[/tex] with inclusion, but the codomain of [tex]f_t[/tex] is different from the domain of [tex]i[/tex].

Isn't it more important that [tex]i[/tex] be defined on the image of [tex]f_t[/tex]?
 
  • #4


fzero said:
Isn't it more important that [tex]i[/tex] be defined on the image of [tex]f_t[/tex]?

Can [tex]i[/tex] be defined on the image of [tex]f_t[/tex]?

I do not think so. That is because the domain of [tex]i[/tex] and the codomain of [tex]f_t[/tex] are different sets. For example, if [tex]t=0[/tex], [tex]f_0=Id_X[/tex] whose codomain is [tex]X[/tex], but the domain of [tex]i[/tex] is [tex]A[/tex]. So the composition [tex]if_0[/tex] makes none sense.
 
  • #5


Is there someone who can help me?
 

Related to How to prove the inclusion is a homotopy equivalence?

1. How do you define a homotopy equivalence?

A homotopy equivalence is a mathematical concept that describes a relationship between two topological spaces. It means that there exists a continuous function between the two spaces that is both one-to-one and onto, and its inverse is also continuous. This allows for a continuous transformation between the two spaces, and thus they are considered equivalent in terms of their topological properties.

2. What is the process for proving inclusion is a homotopy equivalence?

To prove that inclusion is a homotopy equivalence, you need to show that there exists a continuous function between the two spaces that satisfies the definition of a homotopy equivalence. This involves constructing the function and its inverse, and then showing that they are both continuous. Additionally, you must demonstrate that the function and its inverse are one-to-one and onto.

3. What is the significance of proving that inclusion is a homotopy equivalence?

Proving that inclusion is a homotopy equivalence is important because it allows for a better understanding of the topological structure of a space. It also helps to identify if two spaces are equivalent and if there exists a continuous transformation between them. This can have implications in various fields of mathematics, such as algebraic topology and differential geometry.

4. What are the common techniques used to prove inclusion is a homotopy equivalence?

There are various techniques that can be used to prove inclusion is a homotopy equivalence. Some common ones include constructing a deformation retract, using the fundamental group, and using the concept of homotopy equivalences to show that the two spaces have the same homotopy groups. Another approach is to use the concept of a homotopy fiber, which involves constructing a fiber bundle and showing that it is homotopy equivalent to the original space.

5. Are there any special tools or properties that are helpful in proving inclusion is a homotopy equivalence?

Yes, there are certain tools and properties that are helpful in proving inclusion is a homotopy equivalence. These include the use of deformation retracts, the fundamental group, and the concept of homotopy equivalences. Additionally, the properties of homotopy groups, fibrations, and homotopy fibers can also be useful in proving this type of equivalence.

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