1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homotopy equivalence

  1. Jul 17, 2012 #1
    Hi, I am stuck on two problems from Allen Hatcher's book, Algebraic Topology.

    The problem statement, all variables and given/known data
    4. A deformation retraction in the weak sense of a space X to a subspace A is a homotopy [itex]f_t[/itex]: X→X such that [itex]f_0[/itex]=[itex]1_X[/itex] (the identity map on X), [itex]f_1[/itex](X) [itex]\subset[/itex] A, and [itex]f_t (A) \subset A[/itex] for all t. Show that if X deformation retracts to A in this weak sense, then the inclusion [itex]i_A:[/itex]A→X is a homotopy equivalence.

    The attempt at a solution
    We want to show that there exists f:X→A such that [itex]f i_A[/itex] [itex]\approx[/itex] [itex]1_A[/itex] and [itex]i_A f \approx 1_X[/itex]. Define F:A[itex]\times[/itex]I→A by F(t,x) = [itex]f_t i_A[/itex]. We then have F(0,x) = [itex]f_0 i_A[/itex] = [itex]1_A[/itex] and F(1,x)=[itex]f_1 i_A[/itex].

    Now, for each t [itex]\in[/itex] [0,1] define [itex]X_t=f_t(X)[/itex]. Let [itex]i_{X_t} : X_t → X [/itex] be the natural inclusion. Now, define G:X[itex]\times[/itex]I→X to be G(t,x) = [itex]i_{X_t} f_t[/itex]. Then G(0,x) =[itex] i_X f_0 = 1_X[/itex] and G(1,x) = [itex]i_{X_1} f_1[/itex]. Since [itex]f_1(X)\subset A[/itex], it follows that [itex]i_{X_1} f_1 = i_A f_1[/itex].

    I don't know how to prove that F is continous. I am not sure if G is even continous, its the only thing i could come up with. Also I haven't used the fact that [itex]f_t (A) \subset A[/itex] for all t. What did I miss?

    The problem statement, all variables and given/known data
    9. Show that a retract of a contractible space is contractible.

    The attempt at a solution
    So we are given a retract from a space X to a subset A. We know that there exists a homotopy between [itex]1_X[/itex] and a constant map. I am not sure where to go from here. I just wanted to ask that if say X contracts to [itex]x_0[/itex], does A necessarily contain [itex]x_0[/itex]? Because if [itex]x_0\in A[/itex] then we can just restrict the homotopy to A and we would have shown that A is contractible.

    Any help would be appreciated.
     
  2. jcsd
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted



Similar Discussions: Homotopy equivalence
  1. Homotopy Problem (Replies: 6)

  2. Homotopy Classes (Replies: 2)

Loading...