Are smooth mappings of homotopic loops homotpic?

Question

I am a physics student. Consider two manifolds $M$ and $N$ and a smooth mapping $f: M \rightarrow N$. If there are two loops $C_1$ and $C_2$ in $M$ which are homotopic (can be continuously deformed into each other), then are the loops $f(C_1)$ and $f(C_2)$ in $N$ also homotopic to each other?

If not, in what other circumstances will the loops $f(C_1)$ and $f(C_2)$ be homotopic? Please give me a textbook/article reference too because I have to cite it to support the claim.

Answer 1

Yes, the two loops in $N$ are always homotopic so long as the map $f$ is continuous.

Proof: Assume $f:M\to N$ is continuous. Suppose the closed loops $C_1: [0,1]\to M$ and $C_2: [0,1]\to M $ are homotopic. Then there exists a continuous function $h_M: [0,1]\times[0,1] \to M$ such that $h_M(0,t) = C_1(t)$ and $h_M(1,t) = C_2(t)$. We can then construct a homotopy $h_N: [0,1]\times[0,1] \to N$ by composing $h_M$ and $f$:
$$ h_N(s,t) = f(h_M(s,t)). $$ By construction, $h_N(0,t) = f(C_1(t))$ and $h_N(1,t) = f(C_2(t))$; and $h_N$ is the composition of continuous maps and is therefore itself continuous. So $h_N$ provides a map that continuously deforms $f(C_1(t))$ to $f(C_2(t))$, and thus the two curves are homotopic in $N$.

Note that:

• This did not require any differentiability or smoothness assumptions on $f$ or on the curves, only continuity.
• Nothing in the above requires that $C_1$ or $C_2$ be loops; they could equally well be paths with endpoints and the proof would still carry through as above.