Relative form A
relative form of the Mayer–Vietoris sequence also exists. If Y \subset X and is the union of the interiors of C \subset A and D \subset B , then the exact sequence is: \cdots\to H_{n}(A\cap B,C\cap D)\,\xrightarrow{(i_*,j_*)}\,H_{n}(A,C)\oplus H_{n}(B,D)\,\xrightarrow{k_* - l_*}\,H_{n}(X,Y)\, \xrightarrow{\partial_*} \,H_{n-1}(A\cap B,C\cap D)\to\cdots
Naturality The homology groups are
natural in the sense that, if f:X_1 \to X_2 is a
continuous map, then there is a canonical
pushforward map of homology groups f_*: H_k(X_1) \to H_k(X_2) such that the composition of pushforwards is the pushforward of a composition: that is, (g\circ h)_* = g_*\circ h_*. The Mayer–Vietoris sequence is also natural in the sense that if \begin{matrix} X_1 = A_1 \cup B_1 \\ X_2 = A_2 \cup B_2 \end{matrix} \qquad \text{and} \qquad \begin{matrix} f(A_1) \subset A_2 \\f(B_1) \subset B_2\end{matrix}, then the connecting morphism of the Mayer–Vietoris sequence, \partial_*, commutes with f_*. That is, the following diagram
commutes (the horizontal maps are the usual ones): \begin{matrix} \cdots & H_{n+1}(X_1) & \longrightarrow & H_n(A_1\cap B_1) & \longrightarrow & H_n(A_1)\oplus H_n(B_1) & \longrightarrow & H_n(X_1) & \longrightarrow &H_{n-1}(A_1\cap B_1) & \longrightarrow & \cdots\\ & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow & & f_* \Bigg\downarrow\\ \cdots & H_{n+1}(X_2) & \longrightarrow & H_n(A_2\cap B_2) & \longrightarrow & H_n(A_2)\oplus H_n(B_2) & \longrightarrow & H_n(X_2) & \longrightarrow &H_{n-1}(A_2\cap B_2) & \longrightarrow & \cdots\\ \end{matrix}
Cohomological versions The Mayer–Vietoris long exact sequence for
singular cohomology groups with coefficient
group G is
dual to the homological version. It is the following: \cdots\to H^{n}(X;G)\to H^{n}(A;G)\oplus H^{n}(B;G)\to H^{n}(A\cap B;G)\to H^{n+1}(X;G)\to\cdots where the dimension-preserving maps are restriction maps induced from inclusions, and the (co-)boundary maps are defined in a similar fashion to the homological version. There is also a relative formulation. As an important special case when G is the group of
real numbers, and the underlying topological space has the additional structure of a
smooth manifold, the Mayer–Vietoris sequence for
de Rham cohomology is \cdots\to H^{n}(X)\,\xrightarrow{\rho}\,H^{n}(U)\oplus H^{n}(V)\,\xrightarrow{\Delta}\,H^{n}(U\cap V)\, \xrightarrow{d^*}\, H^{n+1}(X) \to \cdots where \{U,V\} is an
open cover of X , \rho denotes the restriction map, and \Delta is the difference. The map d^* is defined similarly as the map \partial_* from above. It can be briefly described as follows. For a cohomology class represented by a closed p-form \omega on U\cap V, choose a
partition of unity \rho_U+\rho_V=1 subordinate to the open cover \{U,V\}, with \rho_U supported in U and \rho_V supported in V. Then \rho_V\omega extends by zero to a p-form on U, and \rho_U\omega extends by zero to a p-form on V. The (p+1)-forms -\,d\rho_V\wedge\omega on U and d\rho_U\wedge\omega on V agree on U\cap V, because d\rho_U=-\,d\rho_V there. Hence they glue to a global (p+1)-form on X=U\cup V, which is closed because it is locally exact (locally either as -d(\rho_V\omega) or d(\rho_U\omega)). The connecting homomorphism d^* sends [\omega] to the de Rham cohomology class of this global form. For de Rham cohomology with compact supports, there exists a "flipped" version of the above sequence: \cdots\to H_{c}^{n}(U\cap V)\,\xrightarrow{\delta}\,H_{c}^{n}(U)\oplus H_{c}^{n}(V)\,\xrightarrow{\Sigma}\,H_{c}^{n}(X)\, \xrightarrow{d^*}\, H_{c}^{n+1}(U\cap V) \to \cdots where U,V,X are as above, \delta is the signed inclusion map \delta : \omega \mapsto (i^U_*\omega,-i^V_*\omega) where i^U extends a form with compact support to a form on U by zero, and \Sigma is the sum. The connecting coboundary d^* arises by writing a compactly-supported p-form \omega on the union X=U\cup V as the sum \omega=\omega_U+\omega_V of a compactly supported p-form \omega_U on U and one on V. Because d\omega_U+d\omega_V=0 on the overlap, the form \xi that is d\omega_U on U and -d\omega_V on V is a global closed (p+1)-form, and d^*[\omega] is defined as [\xi].
Derivation Consider the
long exact sequence associated to the
short exact sequences of chain groups (constituent groups of
chain complexes) 0 \to C_n(A\cap B)\,\xrightarrow{\alpha}\,C_n(A) \oplus C_n(B)\,\xrightarrow{\beta}\,C_n(A+B) \to 0, where \alpha(x) = (x,-x) , \beta(x,y) = x + y , and C_n (A+B) is the chain group consisting of sums of chains in A and chains in B . It is a fact that the singular n -simplices of X whose images are contained in either A or B generate all of the homology group H_n (X) . In other words, H_n (A+B) is isomorphic to H_n (X) . This gives the Mayer–Vietoris sequence for singular homology. The same computation applied to the short exact sequences of vector spaces of
differential forms 0\to\Omega^{n}(X)\to\Omega^{n}(U)\oplus\Omega^{n}(V)\to\Omega^{n}(U\cap V)\to 0 yields the Mayer–Vietoris sequence for de Rham cohomology. From a formal point of view, the Mayer–Vietoris sequence can be derived from the
Eilenberg–Steenrod axioms for
homology theories using the
long exact sequence in homology.
Other homology theories The derivation of the Mayer–Vietoris sequence from the
Eilenberg–Steenrod axioms does not use the
dimension axiom. Consequently, the sequence extends from ordinary theories to
extraordinary or generalized (co)homology theories, such as
topological K-theory and
cobordism. In modern
stable homotopy theory, this is explained by
Brown representability: a reduced generalized cohomology theory is represented by a
spectrum E. Schematically, one may write \widetilde E^n(X)\cong[\Sigma^\infty X,\Sigma^nE], where \Sigma^\infty X is the
suspension spectrum of X and \Sigma^nE is the n-fold suspension (shift) of E. In this language, the Mayer–Vietoris sequence reflects the fact that the same excisive gluing pattern of spaces underlies each theory; what varies from one theory to another is the representing spectrum.
Sheaf cohomology From the point of view of
sheaf cohomology, the Mayer–Vietoris sequence is related to
Čech cohomology. Specifically, it arises from the
degeneration of the
spectral sequence that relates Čech cohomology to sheaf cohomology (sometimes called the Mayer–Vietoris spectral sequence) in the case where the open cover used to compute the Čech cohomology consists of two open sets. This spectral sequence exists in arbitrary
topoi. ==See also==