I am trying to solve the following problem but so far I cannot do it.
Let $X$ be a connected CW-space such that its homotopy group is 0 except for the fundamental group.
Let $M$ be a closed manifold and $g: M \to X$ be a map such that $g(m_0)=x_0$ , where $m_0$ and $x_0$ are base points of $M$ and $X$ respectively. Suppose we have a map $F: M \times [0,1] \to X$ such that $F(m,0)=g(m)$ and $F(m,1)=g(m)$ for all $m \in M$ and $F(m_0 \times [0,1])=x_0$.
I want to show that $F$ is homotopic to $\bar g$ relative to $M\times\{0,1\}$, where $\bar g: M\times [0,1] \to X$ is a map obtained by composing the projection to $M$ and the map $g$.
I am not sure how to use the condition that the higher homotopy of $X$ is zero.
(Or am I missing some conditions so that this statement is true?)
I appreciate any help.