[幾何] homotopy

[rtyxn (Steve)]

Let S^n be the n-sphere and let f:S^n→S^n be a mapping such that

f(x)≠-f(-x) for each x\in S^n. Then a mapping F:S^n ×[0,1]→S^n is well

defined by

          tf(x)+(1-t)f(-x)
F(x,t)=----------------------
        ∥tf(x)+(1-t)f(-x)∥  .


We can see from the above that f(x) is homotopic to f(-x).

Now, I want to ask you a question. Is f(-x) homotopic to F(x,1/2)?

The answer is yes, but why is that? Is it possible to apply the definition

to determine whether f(-x) and F(x,1/2) are homotopic? I mean, can we

construct a homotopy between f(-x) and F(x,1/2)?

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→ Desperato   : t改成2t不行嗎
Did you mean

           sf(x)+(1/2-s)f(-x)
F(x,2s)=------------------------  ??
         ∥sf(x)+(1/2-s)f(-x)∥

Sorry, I don't see any difference. What I'm going to do is to find a homotopy

between f(-x) and the mapping

  [f(x)+f(-x)]/2
------------------
∥[f(x)+f(-x)]/2∥ .

→ silvermare  : If f(x)=-f(-x), then F(x,1/2) is not well-define
Sorry, I missed a minus sign. After correcting the sign error, we know the

denominator is never zero for any x and any t. The mapping is well-defined.

→ Vulpix      : 應該是他要避開的情況寫錯了吧，f(-x)=f(x)不會矛盾
Thx.
※ 編輯: rtyxn (140.112.233.124), 04/16/2019 07:40:19
推 Vulpix      : The homotopy is simply F(x,t/2).
!!!!!!!!!!!!!!!!!!!!!!!!!  Thank you!
※ 編輯: rtyxn (140.112.233.124), 04/16/2019 14:16:35
推 silvermare  : In fact, F(x,s) ~ F(x,r), for any s,r in [0,1]