Mayer-Vietoris sequence

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In algebraic topology and related branches of mathematics, the Mayer-Vietoris sequence^[1] (named after Walther Mayer and Leopold Vietoris) is an exact sequence that often helps one to compute homology groups. It is somewhat analogous to the Seifert-van Kampen theorem for homotopy groups.

Homology groups can often be computed directly using the tools of linear algebra (in simplicial homology). However, even so, eventually such computations become cumbersome, and it is useful to have tools that allow one to compute homology groups from others that one already knows (this approach, of course, is used everywhere in mathematics). The Mayer-Vietoris sequence is one of the most useful tools for this.

For X a topological space with two open subsets U and V whose union is X, we call (X,U,V) a triad. (It's sometimes possible to form triads out of non-open subsets, but it doesn't automatically work.) The Mayer-Vietoris sequence of the triad (X,U,V) is a long exact sequence which relates the (singular) homology groups of the space X to those of U, V, and their intersection A. The sequence runs:

$\cdots \to H_{n+1}(X) \to^{\!\!\!\!\!\!\partial}\, H_{n}(A) \to^{\!\!\!\!\!\!\varphi}\, H_{n}(U) \oplus H_{n}(V) \to^{\!\!\!\!\!\!\psi}\, H_{n}(X) \to^{\!\!\!\!\!\!\partial}\, H_{n-1} (A) \to \cdots \!$

Here, H_n(Y) is the n-dimensional homology group of some space Y. The maps between each homology group of the same dimension n are induced by the inclusions of A into U and V, and of U and V into X. More precisely, the mapping (φ) into the direct sum is a product map, and the map (ψ) out of the direct sum is a difference. The maps (∂) that lower the dimension ("step down") are boundary maps that come from the snake lemma.

The Mayer-Vietoris long exact sequence can also be written with cohomology groups:

$\cdots \longrightarrow H^{n}(X) \longrightarrow\, H^{n}(U) \oplus H^{n}(V) \longrightarrow \, H^{n}(A) \longrightarrow \, H^{n+1}(X) \longrightarrow \cdots \!$

One of the most immediate applications of the Mayer-Vietoris sequence is to prove that the nth reduced homology group of the sphere S^k is trivial unless n = k, in which case H_k(S^k) is isomorphic to the group of integers (Z). Such a complete classification of the homology groups for spheres starkly contrasts with what is known for homotopy groups of spheres; there is similar result when n < k, but not not much is known when n > k.