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How to prove the inclusion is a homotopy equivalence?

  • #1
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:
 

Answers and Replies

  • #2


No one knows the answer? What a pity! Hope someone can help me!!!
 
  • #3
fzero
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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


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?
 

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