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This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?

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Are you hoping to avoid sweating over barycentric subdivision of singular chains, or just to disguise the sweat? [Presumably any proof has to show somehow that chains are local.] –  Tim Perutz May 1 '10 at 13:51
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As pointed out in the "derivation" section of the wikipedia page en.wikipedia.org/wiki/Mayer%E2%80%93Vietoris_sequence, Mayer-Vietoris may be derived from the Eilenberg-Steenrod axioms and the long exact sequence. Since it is independent of the dimension axiom, it applies to extraordinary homology theories, such as K-theory. –  Ian Agol May 1 '10 at 14:51
    
In my limited experience, I don't think I've seen how to do this, but I've always wanted to think of the MV spectral sequence where the open cover is a resolution/(co)fibrant replacement of the space to which you then apply the cohomology functor. I think I ran across this notion somewhere, once upon a time, but I could be misremembering. Can this be made sense of? –  Aaron Bergman May 1 '10 at 19:20
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I'm in a slightly cranky mood, so I feel like objecting to the title of your question. There is a big difference between viewing something in a mathematically mature way and viewing something in a fancy abstract framework! I'm a reasonably mathematically mature guy, and I think of MV as a generalization of the inclusion/exclusion principle for counting elements of a set (exercise : write out the MV ex seq for X a finite set w/ the discrete topology and U,V subsets with X=U \cup V. Interpret it as the inclusion/exclusion principle). As Agol said, though, the proof just uses the E-S axioms. –  Andy Putman May 2 '10 at 18:43

4 Answers 4

up vote 42 down vote accepted

The Mayer-Vietoris sequence is an upshot of the relationship between sheaf cohomology and presheaf cohomology (a.k.a. Cech cohomology).

Let $X$ be a topological space (or any topos), $\mathcal U$ a covering of $X$. Let $\mathop{\rm Sh}X$ be the category of sheaves on $X$ and $\mathop{\rm PreSh}X$ the category of presheaves. The embedding $\mathop{\rm Sh}X \subseteq \mathop{\rm PreSh}X$ is left-exact; its derived functors send a sheaf $F$ into the presheaves $U \mapsto \mathrm H^i(U, F)$. For any presheaf $P$, one can define Cech cohomology $\mathrm {\check H}^i(\mathcal U, P)$ of $P$ by the usual formulas (this is often done only for sheaves, but scrutinizing the definition, one sees that the sheaf condition is never used). One shows that the $\mathrm {\check H}^i(\mathcal U, -)$ are the derived funtors of $\mathrm {\check H}^0(\mathcal U, -)$; and of course for a sheaf $F$, $\mathrm {\check H}^0(\mathcal U, F)$ coincides with $\mathrm H^0(\mathcal U, F)$. The Grothendieck spectral sequence of this composition, in the case of a covering with two elements, gives the Mayer--Vietoris sequence.

There is also a spectral sequence for finite closed covers, which is obtained as in anonymous's answer.

I guess that this can also be interpreted as Tilman does, in a different language (I am not a topologist).

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Amazing! This forum lets you tap into the accumulated insights of such talented people. I'm hooked. Thanks Angelo, and thanks Tilman - I'm not familiar with "homotopy colimits" but I'll look some things up on it on the web. –  James D. Taylor May 1 '10 at 15:45
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Thanks. I don't know about talented; it's just that I have been around for a fairly long time. –  Angelo May 1 '10 at 16:48

Here's an answer somewhat different from those already given.

Associated to any homotopy pullback square, there's a long exact sequence of homotopy groups often called the Mayer-Vietoris sequence. It comes from weaving together the long exact sequences for, say, the two vertical maps in the square, which have homotopy equivalent homotopy fibers. (This weaving is a standard homological algebra exercise, and appears somewhere in Hatcher's book...)

Now, to build the Mayer-Vietoris sequence in cohomology for a CW complex X written as a union of subcomplexes $X = A\cup B$, just note that the homotopy pushout square formed by $A\cap B$, A, B, and X becomes a homotopy pullback square after applying Map(-, K(G, n)), where G is the coefficient group you're using. The Mayer-Vietoris homotopy sequence is now precisely the M-V sequence in cohomology.

(Annoyingly, for a fixed value of n this only gives you some of the sequence.)

It would be interesting to see a variant of this for homology, maybe using the infinite symmetric product? I suppose the place to look would be the book by Aguilar-Gitler-Prieto, where homology is introduced entirely in terms of symmetric products. The relevant bit seems to be missing from the Google preview.

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This answer is related to Tilman's: Let $U$ and $V$ be the open sets covering $X$. For $S$ an open subset of $X$, let $\mathbb{Z}_S$ be the pushforward to $X$ of the sheaf of locally constant integer valued functions on $S$. Then we have a short exact sequence of sheaves

$$0 \to \mathbb{Z}_X \to \mathbb{Z}_U \oplus \mathbb{Z}_V \to \mathbb{Z}_{U \cap V} \to 0$$

and the corresponding long exact sequence is the Mayer-Vietores sequence in cohomology.


This answer can be generalized easily to any open cover of $X$: you have a long exact sequence of sheaves:

$$0 \to \mathbb{Z}_X \to \bigoplus \mathbb{Z}_{U_i} \to \bigoplus \mathbb{Z}_{U_i \cap U_j} \to \cdots$$

which gives a spectral sequence

$$\bigoplus H^p(U_{i_1} \cap U_{i_2} \cap \cdots U_{i_q}) \to H^{p+q}(X).$$

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This works for finite closed covers, not for open covers. –  Angelo May 1 '10 at 13:52
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I believe that this does work, except that I think you want to use the extension by zero $j! \mathbb{Z}$ rather than the pushforward, and the exact sequence of sheaves goes in the opposite direction - you get the spectral sequence by applying Hom from this to the desired sheaf. –  Tyler Lawson May 1 '10 at 16:08
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The cohomology of $j! \mathbb{Z}$ is not the cohomology of the open subset. If $X$ is compact, it is the cohomology with compact support; in this way you can get Mayer-Vietoris for Borel-Moore homology. –  Angelo May 1 '10 at 16:33
    
Sorry, are you saying $Ext(j_! \mathbb{Z},F)$ is not the cohomology of F restricted to the open subset? –  Tyler Lawson May 1 '10 at 17:27
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I am sorry, I misunderstood what you were saying. You are absolutely right, you can also get Mayer-Vietoris this way. I had never thought of this. –  Angelo May 1 '10 at 21:26

Maybe you're looking for the Mayer-Vietoris spectral sequence, the homology spectral sequence for a homotopy colimit? The MV-sequence is a two-line spectral sequence, thus an exact sequence.

The general form is $$ E^2_{p,q} = colim_p H_q(X_\bullet) \Rightarrow H_{p+q}(hocolim X_\bullet) $$ You can think of this as a composite functor spectral sequence.

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This probably works only for numerable coverings (i.e. ones with a subordinate partition of unity) since, as far as I know, only those define homotopy colimit diagrams. By the way: Does this work for arbitrary homology theories? –  Lennart Meier Jan 20 at 14:28

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