$\mathbf {The \ Problem \ is}:$ Let $X$ be a compact simplicial complex and $Y$ be based, connected space with $f:X\to Y.$ If $X$ is simply connected and $f_k:=f\mid_{X^k}$. Show that $f_1$ is nullhomotopic where $X^k$ is the $k-$th sub complex of $X.$
Also show that if a specific nullhomotopy has been selected for $f_k$ then there is an obstruction to extending it to a nullhomotopy for $f_{k+1}$ and show that it is a combinatorial cochain $z$ of dimension $k+1$ on $X$ with coefficients in $π_{k+1}(Y,*).$
$\mathbf {My \ approach}:$ As, $S^1$ is a finite simplicial complex then any map from $S^1$ to $X$ is homotopic to a simplicial map $s$ (by Simplicial Approximation theorem) and $s$ is nullhomotopic. Now, again $Y$ has a CW and hence a simplicial approximation $Y^s$ but then $f$ becomes homotopic to a simplicial map .
I can't approach any further . Geometrically, I can think that $X^1$ is convex combination of some subset of $X^0$ and the $1-$simplices in $X^1$ can be contracted to a point, but I am very sloppy here .
Do I need to invoke geometric realization of $X$ here ?
And I am clueless about 2nd problem . Can anyone suggest a good reference for Obstruction theory for finite simplicial complexes?
Thanks in advance for any help .