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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.

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I realize this has already been answered, but Hatcher's Algebraic Topology also has a nice section on local coefficients, relating the covering space and bundle-of-groups points of view –  Greg Friedman Mar 15 '13 at 18:27
    
Thank you Greg. I looked more closely at Hatcher's section 3.H on Local Coefficients, and it does contain very helpful material. –  Martin Frankland Apr 12 '13 at 23:34

1 Answer 1

up vote 5 down vote accepted

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).

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Thank you very much! Chapter VI does have the material I was looking for. In particular, the sequence I wrote in Question 1 seems to be correct. –  Martin Frankland Mar 12 '13 at 5:16

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