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.

Let X be a topological space with two subsets U and V whose interiors cover X. The union of the interior of U and the interior of V is usually not disjoint. Then one can show that the singular simplices of X which images are contained in either U or V generate the same homology groups Hn(X) as all the singular simplices of X together.

Using this result on homology generation, one can define the Mayer–Vietoris sequence of the triad (X,U,V), 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) \xrightarrow{\partial_*}\, H_{n}(A) \xrightarrow{(i_*,j_*)}\, H_{n}(U) \oplus H_{n}(V) \xrightarrow{k_* - l_*}\, H_{n}(X) \xrightarrow{\partial_*}\, H_{n-1} (A) \to \cdots \!


Here, i : A \to U, j : A\to V, k : U \to X and l : V\to X are inclusion maps. The maps \partial_* that lower the dimension ("step down") are defined as follows. Let x\in H_n(X). Using the previous result on homology generation, x is the homology class of an n-cycle made out of n-simplices which images are in either U or V : \displaystyle x = u + v and \partial x = 0. Hence \partial u = -\partial v so the images of both those (n-1)-cycles are contained in U\cap V = A. We then define \partial_* x = \partial u \in H_{n-1}(A) and argue that this definition does not depend on representants for homology classes.

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 Sk is trivial unless n = k, in which case Hk(Sk) is isomorphic to the group of integers (Z). Such a complete classification of the homology groups for spheres starkly contrasts what is known for homotopy groups of spheres; there are similar results when n < k, but not much is known when n > k.

[edit] Notes

  1. ^ R. BOTT, L. W. TU – Differential Forms in Algebraic Topology, Graduate Texts in Mathematics 82, Springer.

[edit] See also