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 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.
Best Answer
Whitehead's "Elements of Homotopy Theory", 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).