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If I have two concatenated paths $f_{1} \cdot g_{1}$ and $f_{2} \cdot g_{2}$ where $f_{1} \cong f_{2}$ and $g_{1} \cong g_{2}$, is it fair to say

$$f_{1} \cdot g_{1} \cong f_{2} \cdot g_{2}$$

by the homotopy $(f \cdot g)_{t} = (1-t)(f_{1} \cdot g_{1})+t(f_{2} \cdot g_{2})$?

ABC
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  • What does + mean? – Randall Feb 25 '19 at 04:05
  • @Randall its addition? I'm wanting to do a line segment homotopy from $f_{1} \cdot g_{1}$ to $f_{2} \cdot g_{2}$. Sorry I'm a bit confused by your question. – ABC Feb 25 '19 at 04:14
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    General topological spaces don’t have an addition. – Randall Feb 25 '19 at 04:16
  • @Randall Ah, I understand your question now. So all we can say about a homotopy $h$ is $h(x,0) = f_{1} \cdot g_{1}$ and $h(x,1) = f_{2} \cdot g_{2}$ if we are dealing in some general topological space $X$. – ABC Feb 25 '19 at 04:27
  • What is $\cdot$ here? Composition? – qualcuno Feb 25 '19 at 04:44
  • @GuidoA Yes its the product path. First $f$ then $g$. – ABC Feb 25 '19 at 04:47
  • I am then confused: if you can compose the paths, they should all have a (co) domain contained in $[0,1]$. The latter is contractible, so any path $[0,1] \to [0,1]$ is homotopic to the constant path $c_0$. In particular we get $f_1g_1 \simeq f_2g_2$. Or do you mean that only the $g_i$ are paths? In any case, the codomain of the $f_i$ should have a vector space structure or something similar to make sense of the homotopy that you propose. Would you mind elaborating? – qualcuno Feb 25 '19 at 04:56
  • @GuidoA. I am supposed to find a formula for a path homotopy between the paths $f_{1}g_{1}$ and $f_{2}g_{2}$ on a topological space $X$ where each $f_{i},g_{i}$ are themselves paths and $f_{1} \cong f_{2}$ and $g_{1} \cong g_{2}$ – ABC Feb 25 '19 at 06:29

1 Answers1

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Hint: consider for each $t \in I$ the (path) composition $H_t := F_t*G_t$ where $F: f_1 \simeq f_2$ and $G: g_1 \simeq g_2$. Prove that this defines a homotopy of paths: to justify the 'gluing' of the functions, use that both $F$ and $G$ fix the extrema of their corresponding paths.

qualcuno
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