If a space X has with contractible universal cover then any map $S^n> \rightarrow X$ can be extended to $D^{n+1} \rightarrow X$
$D^{n+1}$ is the ball with dimension $n+1$
How should I approach this proof? Ihave drawn a diagram with the spaces, the maps and also the induced maps for the fundamental groups. I see that it should be true, but do not know how to start it.
Let $\hat X$ be the universal cover of $X$ and $p:\hat X\rightarrow X$ the covering map, there exists a family of maps $h_t:\hat X\rightarrow \hat X$ such that $h_0=Id_X, h_1$ is constant. Let $f:S^n\rightarrow X$, we can lift it to a map $\hat f:S^n\rightarrow \hat X$ since $p$ is a Serre fibration $g:D^{n+1}\rightarrow \hat X$ defined by $\hat g(x)=h_{\|x\|}(\hat f({x\over\|x\|}))$, if $x\neq 0, \hat g(x)=x_0$ where $x_0$ is the image of $h_1$. and $g=p\circ\hat g$.