**Background.** I'm trying to compute some homology groups using a Mayer-Vietoris argument, but I really need local coefficients.


**Question 1.** What does the Mayer-Vietoris sequence look like when using local coefficients?


Consider an open cover $X = U \cup V$ with inclusion maps
$$\begin{array}{ccccc}
& & U \cap V & &\\
& i \swarrow & & \searrow j &\\
U & & & & V\\
& k \searrow & & \swarrow l &\\
& & X & &\\
\end{array}$$
and a coefficient module $M$ on $X$. (Assume all four spaces are path-connected if needed.) I believe that the Mayer-Vietoris sequence takes the form
$$\ldots \to H\_n(U \cap V; (ki)^\*M) \to H\_n(U;k^\* M) \oplus H\_n(V;l^\*M) \to H\_n(X;M) \to$$
$$\to H\_{n-1}(U \cap V; (ki)^\*M) \to \ldots$$
where $k^\*M$ denotes the restriction of $M$ to $U$ along the inclusion $k \colon U \to X$. Is that correct?


**Question 2.** Are there good references for homology with local coefficients, and in particular the Mayer-Vietoris sequence in that context?


Sections 5.3 and 5.4 of *[Lecture Notes in Algebraic Topology](http://www.indiana.edu/~jfdavis/teaching/m623/book.pdf)* by J. Davis and P. Kirk are a good start, especially Theorem 5.13 and the remark afterwards.


**Question 3.** Are there good references that treat local coefficients as functors from the fundamental groupoid $\Pi\_1(X) \to Ab$ and describe homology with local coefficients in that context?


I wouldn't mind reducing the problem to the case of path-connected spaces, but I feel like the argument would be cleaner without such reductions or choices of basepoints.



Mayer-Vietoris sequence in homology with local coefficients
Whitehead's ["Elements of Homotopy Theory"](http://rads.stackoverflow.com/amzn/click/0387903364), in particular chapter VI, seems to have everything you ask for. (Note that the Mayer-Vietoris sequence is a formal consequence of excision and the long exact sequence of a pair; see section 2.3 of Hatcher's book).


