For a commutative square of spaces (of manifolds, or of simplicial sets:)) $S=(A\to B\sqcup C\to D)$: $$S=\left(\begin{array}{ccc} A & \to & B \newline \downarrow & & \downarrow \newline C & \to & D\end{array}\right)$$ I am trying to understand when the relative homotopy of $(B,A)$ is isomorphic up to a certain degree to the one of $(C,D)$. According to my poor understanding of algebraic topology, this condition is closely related with $S$ being a homotopy pullback square. Yet I am most interested in reformulating this condition cohomologically (or, maybe, homologically?). Certain examples and http://ncatlab.org/nlab/show/fiber+sequence#LongSequCoh seem to suggest that the cohomology of $D$ should be something like $H^*(B)\otimes_{H^*(A)}H^\ast(C)$ (in lower cohomological degrees). Yet I do not understand how to define the latter object. Is there a nice way to do this (in particular, I would not like to consider some weird spaces in the case when $A,B,C,D$ are manifolds)?
Moreover, I would like to have something like a Hurewicz theorem i.e. I would like this cohomological condition to be equivalent to the original homotopy one. Is there a way to do this if all the spaces are simply connected? Is there a nice way to deal with $\pi_1$ in this setting if $A,B,C,D$ are not simply connected (possibly, by passing to the universal covers)?
I would be deeply grateful for any hints, and for any references here (especially, for the ones that are not too abstract, and that 'do not deal with weird spaces'). The last remark: I would like to relate this matter with etale homotopy types (and so, with certain completions of homotopy types and homotopy groups).:)
