Postnikov k-invariant

Assume AC, let $(X,x_0)$ be a connected based space, and let $n\ge 2$. Put $A={\pi }_n(X,x_0)$. Present the Postnikov-stage map by a based fibration

$K(A,n)⟶P_{n}X\stackrel{q_n}{\to }{P}_{n-1}X$

with the identification of the fiber over the specified basepoint with $K(A,n)$ fixed. Its primary obstruction to a section, in the sign convention of the local cellular obstruction cochain, is the $(n+1)$st Postnikov k-invariant

$k_{n+1}(X):=o_{n+1}(q_n)\in H^{n+1}(P_{n-1}X;\mathcal{A})\mathrm{.}$

Here $\mathcal{A}$ is the local system whose monodromy is the ${\pi }_1(X,x_0)$-action on $A$. If $X$ is simple, this action is trivial and the chosen identification makes $\mathcal{A}$ the constant system $A$, so

$k_{n+1}(X)\in H^{n+1}(P_{n-1}X;A)\mathrm{.}$

Representability then identifies this class with a based homotopy class

${\kappa }_{n+1}:P_{n-1}X⟶K(A,n+1)\mathrm{.}$

A fiber-homotopy-equivalent stage, together with the stated base and fiber-group identifications, transports the obstruction class to the same k-invariant. If those identifications are changed, the corresponding automorphism of $A$ acts on the class. For a nontrivial ${\pi }_1$-action, the local-coefficient class above is still the definition, but no untwisted cohomology formula or ordinary map to $K(A,n+1)$ is asserted here.