How to prove that every continuous function $f:M\to X$ from the Möbius strip $M$ into a simply connected space $X$ is null-homotopic? Thanks in advance.
[Math] Question about null-homotopic function
algebraic-topologyhomotopy-theory
algebraic-topologyhomotopy-theory
How to prove that every continuous function $f:M\to X$ from the Möbius strip $M$ into a simply connected space $X$ is null-homotopic? Thanks in advance.
Best Answer
Since $M \simeq S^1$, take WLOG $f:S^1 \to X$. By definition a loop in $X$ is a path $g:I \to X$ such that $g(0)=g(1)$. Notice $S^1=I/\sim, 0\sim1$. So any map $f:S^1 \to X$ is just a loop and induces a loop in $\pi_1(X)$.
$\pi_1(X)=0$ so the loop has to be homotopic to a constant (trivial) loop and hence by definition null-homotopic.