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I'm reading a textbook on differential geometry and the author proceeds to define a weaker version of homotopy.

Two continuous curves $c_0, c_1: [a,b] \to U \subseteq \mathbb{R}^n$ with the same endpoints are homotopic if there exists a continuous map $H: [a,b] \times [0,1] \to U$ such that $H(s,0) = c_0(s), H(s,1)=c_1(s)$ and $H(a,t) = c_0(a), H(b,t) = c_0(b).$ If the last condition is dropped, we say $H$ is a free homotopy.

A proof of line integral invariance under homotopy proceeds and at the end the author says

If $c_0, c_1$ are closed curves ($c_0(a) = c_0(b), c_1(a) = c_1(b)$) that are freely homotopic, then $H(a,t), H(b,t)$ are equal curves.

This is equivalent to every curve in the free homotopy being closed, but I don't see why this has to be true. In fact, why can't we slightly open a loop and then shrink it to a point? There has to be some continuity problem with what I'm imagining since in the case of a simple loop around the origin, this contradicts the fact that $\mathbb{R}^2 \setminus \{0\}$ is not simply connected. In fact, I didn't even think about this issue until reading through the book caused me to imagine it.

The continuity issue must be very subtle since the partial derivatives (with respect to $s, t$) exist and are continuous in the example of slightly opening a loop. I suspect that letting $(s,t)$ approach $(a, 0)$ from different directions will seal the deal. With that in mind, how do I go back and prove $c_t(s) = H(s,t)$ is a closed curve for all $t$?

Update: The book is Differential Forms and Applications by Do Carmo. In fact, the author defines simple-connectedness as every closed curve being freely homotopic to a point and says $\mathbb{R}^2 - \{0\}$ is not simply connected later, so my example better not work. But now I've changed my mind and am starting to think it works again.

Let $C_t$ be the unit circle with the top half dilated to have radius $1+t$ and center at $(t,0).$ Consider a free homotopy from $C_0$ to $C_1$ back to $C_0.$ We must already have a violation of continuity at $t=0$ else after getting to $C_{\epsilon},$ we can just shrink to a point. The exact equation for $H : [0, 2\pi] \times [0,1] \to \mathbb{R}^2$ is $H(s,t) = (t + (1+t)\cos s, (1+t)\sin s)$ when $s \le \pi$ and this is clearly a continuous function of $s, t.$ Thus, $H$ is continuous on $A = [0, \pi] \times [0,1], B = [\pi, 2\pi] \times [0,1], A \cap B = \{\pi\} \times [0,1],$ so is continuous on the whole domain $A \cup B.$

Display name
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  • Your reasoning looks good to me.Perhaps if you identify the book, someone who knows it will be able to explain. – Rob Arthan Mar 07 '21 at 14:34
  • @RobArthan Updated. – Display name Mar 07 '21 at 14:45
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    I am afraid I don't have access to the book.The condition $H(a, t) = H(b, t)$ should surely be part of the definition of free homotopy between closed curves - i.e., at each level of the homotopy you should have a closed curve - so the homotopy is really a homotopy of maps on a circle (see https://math.stackexchange.com/questions/2458349/base-point-homotopy-vs-free-homotopy-example). – Rob Arthan Mar 07 '21 at 15:08
  • @RobArthan Here's the book, it's page 22: https://kupdf.net/download/do-carmo-differential-forms-and-applications_59f25837e2b6f5f325f7bc1c_pdf You can see whether the author implicity assumes that condition in the following pages. It might be somewhere where the element of angle is used, but I haven't fully checked every detail regarding $\omega_0.$ – Display name Mar 07 '21 at 15:12
  • Thanks for the link. The book is just wrong about this. Paul Frost's answer confirms the suspicions I raised in my prvious comment. – Rob Arthan Mar 07 '21 at 22:13

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do Carmo's definition of a free homotopy between curves does not really make sense because then any two curves $c_0, c_1: [a,b] \to U$ with the same endpoints are freely homotopic - and that would be fairly uninteresting. To see that take $$H(s,t) = \begin{cases} c_0(a +(b-a)s(1-2t)) & t \le 1/2 \\ c_1(a +(b-a)s(2t-1)) & t \ge 1/2 \end{cases}$$ Note that for $t = 1/2$ both lines give $H(s,1/2) = c_0(a) = c_1(a)$. Also observe that $H(a,t) = c_0(a) = c_1(a)$ for all $t$, but in general we do not have $H(b,t)$ constant.

I think what do Carmo really wants to define is free homotopy for closed curves. If $c_0,c_1$ are closed curves, not nessarily based at the same point of $U$, then a free homotopy of closed curves is a homotopy such $H_0 = c_0, H_1 = c_1$ and all $H_t$ closed curves. See for example Characterizing simply connected spaces .

Paul Frost
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