Tags: #subjects/homotopy #subjects/algtop/obstruction-theory #subjects/algtop/characteristic-classes
The rough idea of obstruction theory is simple. Suppose we want to construct some kind of function on a CW complex \(X\). We do this by induction: if the function is defined on the k-skeleton \(X_k\), we try to extend it over the \((k + 1){\hbox{-}}\)skeleton \(X^{k+1}\). The obstruction to extending over a \((k + 1){\hbox{-}}\)cell is an element of \(\pi_k\) of something. These obstructions fit together to give a cellular cochain on \(X\) with coefficients in this \(π_k\). In fact this cochain is a cocycle, so it defines an “obstruction class” in \(H_{k+1}(X; π_k(something))\). If this cohomology class is zero, i.e. if there is a cellular \(k{\hbox{-}}\)cochain \(η\) with \(0 = δη\), then \(η\) prescribes a way to modify our map over the \(k{\hbox{-}}\)skeleton so that it can be extended over the \((k + 1){\hbox{-}}\)skeleton