(This is a follow-up question to this one).
As it is nicely outlined in an answer to this question, homotopy groups behave well with respect to (Serre)-fibrations and (co)homology groups behave well with respect to cofibrations. The Serre spectral sequence calculates what happens with (Serre)-fibrations and (co)homology. My question is: Is there an analogue to the Serre spectral sequence for cofibrations and homotopy groups?
There is, sort of, and the idea was developed in the paper "Induced fibrations and cofibrations" by Tudor Ganea in a 1967 paper appearing in the transactions.
The idea is roughly this: given a cofibration $$ A \to X \to C , $$ let us set $X_1 = \text{hofiber}(X \to C)$, where the latter means the homotopy fiber. Then there is a preferred lift $A \to X_1$. Define $C_1$ to be the mapping cone of this map, so we have a cofiber sequence $A \to X_1 \to C_1$. We now iterate the forgoing to get a sequence of spaces $$ \cdots \to X_n \to \cdots \to X_1 \to X_0 = X $$ together with compatible maps $A \to X_n \to C_n$. Under mild hypotheses, the map $$ A \to {\text{holim}}_n \phantom{,} X_n $$ is a weak homotopy equivalence (for example, it will be the case when $A$ and $C$ are $1$-connected).
The homotopy groups tower above gives rise to a spectral sequence where the $E_1$-term is given by $\pi_p(X_q,A)$ (I am being sloppy about the indexing conventions here). In a certain stable range (roughly, $p < 2(qa + c)$ if I recall, where $A$ is $a$-connected and $C$ is $c$-connected), these latter groups are identified with $$ \pi_p(A^{[q]}\wedge \Omega C) $$ where $A^{[q]}$ is the $q$-fold smash product of $A$ with itself.
A problem, which I'm not sure how to solve, is how does one identify the $k$-invariants of the tower to obtain a description of the $E_2$-term in this stable range?
Added: On second reflection, the above is not really an analogue of the Serre spectral sequence. The Serre spectral sequence for $F \to E \to B$ is given by filtering $B$ as a CW complex and pulling back the fibration along skeleta. Dually, the thing to do would be to approximate the cofibration $A\to X \to C$ by a Postnikov tower $A^i$, and the take cobase change of $A \to X$ along $A\to A^{(i)}$ to get a tower $X^{(i)}$ with compatible maps from $X$ into it. Under mild hypotheses, this ought to converge to $X$ when taking the homotopy inverse limit. Then the thing to do would be to consider the homotopy spectral sequence of the tower $X^{(i)}$. However, I'm skeptical that there's a description of the $E_2$-term in terms of $A$ and $C$.
