---- > [!theorem] Theorem. ([[Mayer-Vietoris theorem]] for [[singular homology]]) > > Let $X=A \cup B$ be a [[topological space]] [[cover|covered]] by two opens. We have [[inclusion map|inclusions]] > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEACAHW4GM6aTgCEQAX1LpMufIRQBGclVqMWbduMkgM2PASJl5y+s1aIQoiVN2yiio9RNrzXXnyZDhnALycAGuLKMFAA5vBEoABmAE4QALZIZCA4EEiKyXRYDGwAFhAQANYgjqpmILz4OHQA+hpWIDHxidQpSABMLZnZ5nmFxSqmbBUQVdWi1Ax0AEYwDAAK0npyINFYITk4mlGxCYgdyamIAMydWbn5RSWD5gBWY-2TM-OLtuar65v1jbvprcen3RAvUuA2cIDudQoYiAA > \begin{tikzcd} > A \cap B \arrow[r, "\iota_A", hook] \arrow[d, "\iota_B"', hook] & A \arrow[d, "j_A", hook] \\ > B \arrow[r, "j_B"', hook] & A \cup B = X > \end{tikzcd} > \end{document} > ``` > > **1.** Then there are **connecting [[linear map|homomorphisms]]** $\partial_{MV}:H_{n}(X) \to H_{n-1}(X)$ such that the following sequence, called the **Mayer-Vietoris sequence**, is [[exact sequence|exact]]: > > ![[CleanShot 2025-04-01 at [email protected]]] > > **2.** This Mayer-Vietoris sequence is *natural*, in the sense that it is compatible with [[continuous|morphisms]]: if $f:X=A \cup B \to Y=U \cup V$ is such that $f(A) \subset U$ and $f(B) \subset V$, then the diagram > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAAkB9YMAagEYAvgAoAGgEoQg0uky58hFGX5VajFmy48BIgJqTps7HgJF+5VfWatEHTmGEBBAAQAdVwGM6aZwCEDMiAYxgpmpCrUVhq2XA4Aqm6e3s4AagFG8qYoAEwWkeo2dg6O4okQaMxwzrHC-lKBwZmKyLkRataa9sJxpe7lldVdafUZJs0AzHnt0UVi6UFyY0STbVGFNfpSqjBQAObwRKAAZgBOEAC2SGQgOBBI5tOFR5wAVCDUDHQARjAMAAqLUK2E5YXYACxwIxApwuV2otyQuUebHcaDoJzwjG4AFkUoIoTDLogkQjEAAWfIdWzCdz4HB0biOQTiV5lCpMKo01x0hn+V4Es5Eh6kybI2yo9GYhg4vHvECfH7-QFZEAg8GQwzQwVIUWkgCslJmXJ53DizNZfXZnNpEHpnDS-I+31+AJCKrVEIFsPJ8LuiAAbIbCsIAFacEqsgC0zlDnD5b01hKQBpufoA7EG2LGelGY2GHW8nYrXU02B6NYEkwHfUgM2LofzE9riTXEKK1lmjs4AD6MxJeHy+c0JyvNimp5OZ6lHXsRl5sgbCLu9+NbQRAA > \scriptsize > \begin{tikzcd} > H_{n+1}(X) \arrow[d, "f_*"'] \arrow[r, "\partial_{MV}"] & H_n(A \cap B) \arrow[r, "(\iota_{A})_* \oplus (\iota_B)_*"] \arrow[d, "(f |_{A \cap B})_*"] & H_n(A) \oplus H_n(B) \arrow[r, "(j_A)_* - (j_B)_*"] \arrow[d, "(f|_A)_* \oplus (f |_B)_*"] & H_n(X) \arrow[d, "f_*"] \\ > H_{n+1}(Y) \arrow[r, "\partial_{MV}"'] & H_n(U \cap V) \arrow[r, "(\iota_{U})_* \oplus (\iota_V)_*"'] & H_n(U) \oplus H_n(V) \arrow[r, "(j_U)_* - (j_V)_*"'] & H_n(Y) > \end{tikzcd} > \end{document} > ``` > > commutes for all $n$. (That is, we get a [[chain map]] between the Mayer-Vietoris sequences for each space's covering in the obvious way). > > **3.** How is $\partial_{MV}$ defined? For certain elements of $H_{n}(X)$, we can easily specify what $\partial_{MV}$ does to them. If $[a+b] \in H_{n}(X)$ is such that $a \in C_{n}(A)$ and $b \in C_{n}(B)$, then[^1] $\partial_{MV}([a+b])=[d_{n}(a)]=[-d_{n}(b)] \in H_{n-1}(A \cap B).$ > (The meat of the proof will be to show that every element of $H_{n}(X)$ can be made to look like that.) > [!proof]- Proof. ([[Mayer-Vietoris theorem]]) > Summary: > 1. With notation as in [[small simplices theorem]], write $C_{\bullet}(A+B)=C_{\bullet}^{\mathcal{U}}(A \cup B)=C_{\bullet}^{\mathcal{U}}(X)$ > 2. The map $C_{\bullet}(A) \oplus C_{\bullet}(B) \xrightarrow{j_{A}-j_{B}} C_{\bullet}(A+B)$ is surjective. Argue that its [[kernel]] is $C_{\bullet}(A \cap B)$ > 3. We now have SES of chain complexes. (Corollary of) the snake lemma (corollary) gives the sequence with some $H_{*}^{\mathcal{U}}$s mixed in. [[small simplices theorem]] finishes. > > Let $X=A \cup B$, with $A,B$ open in $X$. Using notation as in the [[small simplices theorem]] (which will be crucial in our argument), we let $\mathcal{U}:=\{ A,B \}$, and write $C_{\bullet}(A+B):=C_{\bullet}^{\mathcal{U}}(X)$. The general element of $C_{*}^{\mathcal{U}}(X)$ is sum of [[singular simplex|singular simplices]] living in $A$ and $B$ (hence the notation $C_{*}(A+B)$). We have a natural [[chain map]] $C_{\bullet}(A) \oplus C_{\bullet}(B) \xrightarrow{j_{A} - j_{B}} C_{\bullet}(A + B)$ > that is [[surjection|surjective]].[^2] The [[kernel of a module homomorphism|kernel]] consists of $(x,y)$ such that $j_{A}(x)-j_{B}(y)=0$, i.e., $j_{A}(x)=j_{B}(y)$. But $j$ doesn't really *do* anything — it just forgets that the [[singular simplex|simplicies]] lie in $A$ or $B$. So this means $y=x$ is a chain in $A \cap B$. It follows that $\operatorname{ker }(j_{A}-j_{B}) = C_{\bullet}(A \cap B)$. We deduce that we have a [[short exact sequence]] of [[chain complex of modules|chain complexes]] $0 \to C_{\bullet}(A \cap B) \xrightarrow{\iota_{A} \oplus \iota_{B}} C_{\bullet}(A) \oplus C_{\bullet}(B) \xrightarrow{j_{A}-j_{B}}C_{\bullet}(A+B) \to 0.$ > Then the [[long exact sequence on homology induced by short exact sequence of chain complexes]] (a corollary of [[the snake lemma]]) gives a long [[exact sequence]] > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAB12BjKCHBAL6l0mXPkIoAjOSq1GLNgAkA+mAAUAQWWcARkwYMYOAJQghI7HgJEATDOr1mrRCBXqAQtvZ6DR0+ZAMS3EiAGZ7OSclVTUAYS8fQxMzYUDRKwlkaRtZRwUXFWAwAFpJAU0E-ST-VKCxaxQ7HId5Z1dlItLyz10qvxSLeszw5sj89s6yuMrfZIC6jKIAFlJRvLbOHj5BWvSQxtJJXNa2TjQ6ACc8RmUAKjNZGCgAc3giUAAzC4gAWyQANmoOAgSAA7AEvr8kGQQMCkNIxht2OcrlgbvcId8-ogEXDEHZEWwsHcBiBIdiCXjwoSXAArEmYqGIal4gAc1B0MDAUCQoWIjOx7NhIMQKxAnO5SGKfIFSDFeIArC0oi5iRjUuSkErhQDleN6RiKAIgA> > \begin{tikzcd} > \cdots \arrow[r, "\partial"] & H_n(A \cap B) \arrow[r, "(\iota_A)_* \oplus (\iota_B)_*"] & H_n(A) \oplus H_n(B) \arrow[r, "(j_A)_* - (j_B)_*"] & H_n^{\mathcal{U}}(X) \arrow[ld, bend left] & \\ > & & \partial \arrow[ld, bend right] & & \\ > & H_{n-1}(A \cap B) \arrow[r, "(\iota_A)_* \oplus (\iota_B)_*"] & H_{n-1}(H_n(A) \oplus H_n(B)) \arrow[r, "(j_A)_* - (j_B)_*"] & H_{n-1}^{\mathcal{U}}(X) \arrow[r] & \cdots > \end{tikzcd} > \end{document} > ``` > > By the [[small simplices theorem]], we can replace $H_n^{\mathcal{U}}(X)$ with $H_{n}(X)$, and this is the result. > > ---- #### [^1]: To see that this even makes sense — that $\partial_{MV}([a+b])$ as defined actually lives in $H_{n-1}(A\cap B)$ — note that certainly $d_{n}(a+b)=0$, since $a+b$ is a cycle. So $d_{n}(a)=-d_{n}(b)$. Now, $d_{n}(a) \in C_{n-1}(A)$ and $-d_{n}(b) \in C_{n-1}(B)$. Hence $d_{n}(a)=-d_{n}(b) \in C_{n-1}(A) \cap C_{n-1}(B)=C_{n-1}(A \cap B)$. Since $d^{2}=0$, $d_{n}(a)=-d_{n}(b) \in C_{n-1}(A \cap B)$ is a cycle, and so $[d_{n}(a)]=[-d_{n}(b)]$ indeed makes sense to talk about as an element of $H_{n-1}(A \cap B)$. [^2]: It is surjective because every element of $C_{\bullet}(A+B)$ arises as a sum $a+b$ of an element $a$ in $C_{\bullet}(A)$ and an element $b$ in $C_{\bullet}(B)$; equivalently, as a difference $a-(-b)$. ----- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` - [ ] MV sequence for a pair (relative MV)![[CleanShot 2025-04-14 at [email protected]]] - [ ] cohomology