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Show Regular Homotopy is an Equivalence Relation

  1. Feb 21, 2017 #1
    1. The problem statement, all variables and given/known data
    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).

    2. Relevant equations
    Not entirely sure

    3. 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.
     
  2. jcsd
  3. Feb 21, 2017 #2
    Nevermind all, I pondered it for a bit and with the help of wikipedia I think I got it.
     
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