If an all maps f from an arbitrary manifold Y to manifold X are homotopic, then X is contractible

Question

Feeling silly that I'm not getting this proof.

I have the proof that if a manifold X is contractible then all maps from an arbitrary manifold $Y$ into $X$ are homotopic, but no its converse. any hints would be appreciated.

thanks!

Edit: My confusion was the word "arbitrary", I falsely interpreted it as "some fixed" manifold Y given the manifold X. English is my second language so thanks for bearing with me and my silly question.

Thank you

Answer 1

The devil is in the details and in this case it is in the word arbitrary. To conclude that $X$ is contractible, we need to know that for any manifold $Y$ it holds that any two maps $f, g\colon Y\to X$ are homotopic. Indeed, if this is the case, we can simply take $Y=X$ and conclude that any map $f\colon X\to X$ is homotopic to the identity.

The formulation of your question had confused me as well. Note, however, that if $Y=\{a\}$ is a one-point manifold then all maps $f,g\colon Y\to X$ are automatically homotopic. This shows that one needs to require something more.