# Loops as maps from $S^1\to X$, Hatcher 1.15 [duplicate]

I am working on Hatcher's problem 1.1.5.

Show that for a space $$X$$, the following three conditions are equivalent.

$$\textit{a)}$$ Every map $$S^1\to X$$ is homotopic to a constant map.

$$\textit{b})$$ Every map $$S^1\to X$$ extends to a map $$D^2\to X$$

$$\textit {c})$$ $$\pi_1(X,x_0)=0$$.

I have shown that $$a$$ implies $$b$$.

Now I want to prove that $$b$$ implies $$c$$. Therefore, I let $$[f]\in\pi_1(X,x_0)$$. Then, $$f:I\to X$$ with $$f(0)=f(1)=x_0$$. I want to show that $$f$$ is homotopic to the constant path. Since $$S^1\cong I/_{0\sim 1}$$, $$f$$ induces a map $$f':S^1\to X$$ with $$f'(1)=x_0$$. By $$b$$, this map can be extended to a map $$\tilde{f'}:D^2\to X$$. Since $$D^2$$ is convex, $$D^2$$ is contractible, there exists a homotopy $$\tilde{H}:I\times D^2\to X$$, $$\tilde{H}_0=const_{x_0}$$, $$\tilde{H}_1=Id_{D^2}$$. I don't know how I can go on from this, There is also a point where I have to transfer back to $$f$$ without $$'$$.

For $$c$$ implies $$a$$: Let $$[f]\in\pi_1(X,x_0)$$. Then $$f:I\to X$$ is homotopic to the constant path $$const_{x_0}$$. Also here, I don't know how to go on.

## marked as duplicate by Lee Mosher, Xander Henderson, José Carlos Santos, positrón0802, CesareoFeb 18 at 19:43

Hint: If $$f':S^1 \to X$$ extends to a map $$\tilde f: D^2 \to X$$, this means $$f'$$ can be written as a composition $$(S^1,1) \xrightarrow i (D^2,1) \xrightarrow {\tilde f} (X,x_0)$$ where $$i$$ is the inclusion. Can you find a homotopy $$H_t$$ rel $$1$$ from $$i$$ to some constant map $$c$$? What if you compose it with $$\tilde f$$?
For (c) $$\implies$$ (a), given a map $$g:S^1\to X$$, can you find a homotopy from $$g$$ to a loop at $$x_0$$?
• What do you mean with, "$H_t$ rel 1 from i to some constant map c? And with $(S^1,1)$ the circle whereby 1 is mapped to $x_0$? – user408856 Feb 14 at 7:11
• @James: I mean a map $H: S^1\times I \to D^2$ such that $H(x,0)=i(x)$ and $H(x,1)=1$ and $H(1,t)=1$ for all $t\in I, x\in S^1$. So the homotopy starts with the embedding $i$ and ends with a constant map, and is fixed at the point $1$ during the time interval. – Stefan Hamcke Feb 14 at 13:34
• Since $D^2$ is convex, the homotopy $H(x,t)=(1-t)\cdot i(x)+t\cdot(1,0)$ will do? – user408856 Feb 16 at 9:40