Mathematically mature way to think about Mayer–Vietoris - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T22:30:40Z http://mathoverflow.net/feeds/question/23175 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris Mathematically mature way to think about Mayer–Vietoris James D. Taylor 2010-05-01T12:55:25Z 2010-05-02T17:53:45Z <p>This question is short but to the point: what is the "right" abstract framework where Mayer-Vietoris is just a trivial consequence?</p> http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris/23178#23178 Answer by Tilman for Mathematically mature way to think about Mayer–Vietoris Tilman 2010-05-01T13:19:01Z 2010-05-01T13:19:01Z <p>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.</p> <p>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.</p> http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris/23179#23179 Answer by anonymous for Mathematically mature way to think about Mayer–Vietoris anonymous 2010-05-01T13:50:24Z 2010-05-01T13:50:24Z <p>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</p> <p><code>$$0 \to \mathbb{Z}_X \to \mathbb{Z}_U \oplus \mathbb{Z}_V \to \mathbb{Z}_{U \cap V} \to 0$$</code></p> <p>and the corresponding long exact sequence is the Mayer-Vietores sequence in cohomology.</p> <hr> <p>This answer can be generalized easily to any open cover of $X$: you have a long exact sequence of sheaves:</p> <p><code>$$0 \to \mathbb{Z}_X \to \bigoplus \mathbb{Z}_{U_i} \to \bigoplus \mathbb{Z}_{U_i \cap U_j} \to \cdots$$</code></p> <p>which gives a spectral sequence </p> <p><code>$$\bigoplus H^p(U_{i_1} \cap U_{i_2} \cap \cdots U_{i_q}) \to H^{p+q}(X).$$</code></p> http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris/23180#23180 Answer by Angelo for Mathematically mature way to think about Mayer–Vietoris Angelo 2010-05-01T14:11:05Z 2010-05-01T16:40:04Z <p>The Mayer-Vietoris sequence is an upshot of the relationship between sheaf cohomology and presheaf cohomology (a.k.a. Cech cohomology).</p> <p>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.</p> <p>There is also a spectral sequence for finite closed covers, which is obtained as in anonymous's answer.</p> <p>I guess that this can also be interpreted as Tilman does, in a different language (I am not a topologist).</p> http://mathoverflow.net/questions/23175/mathematically-mature-way-to-think-about-mayervietoris/23238#23238 Answer by Dan Ramras for Mathematically mature way to think about Mayer–Vietoris Dan Ramras 2010-05-02T07:15:59Z 2010-05-02T17:53:45Z <p>Here's an answer somewhat different from those already given. </p> <p>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...)</p> <p>Now, to build the Mayer-Vietoris sequence in cohomology for a CW complex X written as a union of subcomplexes <code>$X = A\cup B$</code>, just note that the homotopy pushout square formed by <code>$A\cap B$</code>, 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.</p> <p>(Annoyingly, for a fixed value of n this only gives you some of the sequence.)</p> <p>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.</p>