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Homework Help: 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
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