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Consider two smoothly homotopic maps $f_1,f_2:M \to S^1$ from a compact smooth $n$-manifold $M$ to the unit circle.

How do I show

$$f_0 \sim f_1 \Rightarrow f_0 + g \sim f_1 + g$$

for all $g:M \to S^1$

I am thinking since $f_0 \sim f_1$, than there exists a smooth map $F:[0,1]\times M \to S^1$ such that

$$F[0,p]=f_0(p) \text{ and } F[1,p]=f_1(p)$$

we can than add $g$ to both sides

$$F[0,p]+g=f_0(p)+g \text{ and } F[1,p]+g=f_1(p)+g$$

than I want to show $F+g$ is smooth? But I don't know is $g$ smooth or not.

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  • $\begingroup$ how to define $+$? ($f+g(x)= f(x) g(x)$?) $\endgroup$ – user99914 May 27 '15 at 9:26
  • $\begingroup$ Isn't $+$ is just addition in the usual sense? Since $S^1 \cong \mathbb{R}/ \mathbb{Z}$, so addition is the group operation here? $\endgroup$ – SamC May 27 '15 at 9:31
  • $\begingroup$ Okay, I was thinking of $\mathbb S^1 = \{e^{i\theta}: \theta\in \mathbb R\}$ so I write the multiplicative notation. Now the concept is clear. $\endgroup$ – user99914 May 27 '15 at 9:33
  • $\begingroup$ If $g$ is not smooth there is no way that your map is smooth. $\endgroup$ – user99914 May 27 '15 at 9:35
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    $\begingroup$ There's another way round if you are to use this: math.stackexchange.com/questions/176399/… $\endgroup$ – user99914 May 27 '15 at 9:41

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