Can anyone tell me if there is a Mayer-Vietoris sequence for an arbitrary homotopy pushout (hence homotopy pullback) of spectra and an arbitrary (co)homology theory. If this comes from some easy way of writing down a pushout/pullback as a fiber sequence, it'd be really cool to see that spelled out (as if I were a baby). Also, it would also be really neat to know if there are general conditions on a model category or on the (co)homology theory to make this true. 


Thanks in advance. :-) 



Mayer-Vietoris Sequence for Arbitrary Bicartesian Square of Spectra
Mayer-Vietoris sequences can be obtained from excision isomorphisms. 


Anything worthy of the name "homology theory" will give a long exact sequence 
$$\dots \to h\_n(A)\to h\_n(X)\to h\_n(A\to X)\to h\_{n-1}(A)\to \dots$$
for each morphism $A\to X$. And for a square, a.k.a. map of morphisms $(C\to B)\to (A\to X)$, it will give you a map from the long exact sequence of $C\to B$ to that of $A\to X$. 


If the square happens to induce an isomorphism $h\_n(C\to B)\to h\_n(A\to X)$ for every $n$ (as it will if it is a homotopy pushout square) then the desired map $h\_n(X)\to h\_{n-1}(C)$ is given by composing $h\_n(X)\to h\_n(A\to X)\cong h\_n(C\to B)\to h\_{n-1}(C)$, and a little diagram chase gives you the exactness you want. 


Depending on how one axiomatizes the notion of homology theory, the fact that homotopy pushout squares gives isomorphisms in relative homology is either an axiom or a consequence of the axioms. 


There is no need to work with spectra or other stable objects here.


Note that if you had something like a homology theory except that it gave isomorphisms $h\_n(C\to B)\to h\_n(A\to X)$ for homotopy pullbacks instead of for homotopy pushouts then you would get a "Mayer-Vietoris sequence" for pullback squares. That's how it is for homotopy groups and based spaces, except that things get funny down around $\pi\_1$ and relative $\pi\_2$. 


