Mayervietoris SequenceEdit

The Mayer–Vietoris sequence is a central tool in algebraic topology that ties the homology of a space to the homology of two overlapping subspaces. It provides a long exact sequence that lets us infer information about a space from simpler pieces and their intersection. The sequence is named after Walther Mayer and Leopold Vietoris, who introduced ideas of this kind in the early development of the field.

Background

The construction rests on the idea that when a space X is covered by two subspaces A and B (X = A ∪ B), the homology of X can be recovered from the homology of A, B, and their intersection A∩B. This principle of breaking a complex object into manageable parts is a recurring theme in topology and geometry, and the Mayer–Vietoris sequence provides a precise algebraic framework for it. In many expositions, one emphasizes the long exact sequence it induces in homology, and its dual version in cohomology. For the foundational language, see Topological space and Homology.

Statement of the theorem

Informally, assume X is a space that is the union of two subspaces A and B, with A, B, and A∩B satisfying mild niceness conditions so that their homology groups behave well (for example, when A and B are open in X, or when X is a CW complex and A and B are subcomplexes with X = A ∪ B).

There is a natural long exact sequence of homology groups ... → H_n(A∩B) → H_n(A) ⊕ H_n(B) → H_n(X) → H_{n-1}(A∩B) → H_{n-1}(A) ⊕ H_{n-1}(B) → ... where the maps are induced by inclusion and the connecting homomorphism δ carries information from the intersection into lower-dimensional pieces. This is usually written with the middle term as a direct sum of the homology of A and B, and the first map encodes the identification on the overlap.

There is a parallel sequence in cohomology, often written with the appropriate cohomology groups and the corresponding maps, yielding a long exact sequence in cohomology as well. See Cohomology and Singular homology for the standard frameworks.

The chain-level picture

From a computational viewpoint, one can realize the Mayer–Vietoris sequence by assembling chain complexes C_(A), C_(B), and C_(A∩B) into a short exact sequence 0 → C_(A∩B) → C_(A) ⊕ C_(B) → C_(X) → 0, where the middle map sends an element of C_(A∩B) to its images in C_(A) and −C_(B). Taking homology yields the long exact sequence in homology. This construction relies on standard notions from Chain complex theory and the functorial properties of Homology.

In many practical settings (for instance when working with CW complexs or other well-behaved spaces), the hypotheses for the chain-level construction are satisfied, and the resulting homology calculations can be carried out effectively.

Examples and computations

Circle: Let X = S^1 be covered by two contractible subspaces A and B with A∩B also contractible. Then H_0(A) ≅ H_0(B) ≅ Z and H_0(A∩B) ≅ Z, while H_n(A) = H_n(B) = H_n(A∩B) = 0 for n > 0. The Mayer–Vietoris sequence then yields H_1(S^1) ≅ Z, consistent with the well-known fact that the circle has a single 1-dimensional hole. See Circle for related topology.
Torus and sphere decompositions: By choosing A and B to be appropriate subspaces of a space like a torus or a sphere, one can compute their homology by combining information from simpler pieces. This mirrors the general philosophy of gluing in topology and is a standard application pattern discussed in Algebraic topology.

Variants and related tools

Cohomology version: The dual long exact sequence in Cohomology provides a complementary perspective on how information glues between A, B, and X.
Relative homology and excision: The Mayer–Vietoris framework interacts with relative theories and with excision principles used in broader homology calculations.
Other coverings: In more general settings, one can replace the pair (A, B) with more elaborate coverings and obtain spectral sequences (e.g., the Cech-to-derived spectral sequence) that encode gluing data in higher degrees. See discussions of Relative homology and CW complex for context.

Applications and significance

The Mayer–Vietoris sequence is a workhorse in algebraic topology for: - Computing homology of spaces built by gluing, such as unions of simpler subspaces. - Understanding how holes and voids in a space arise from features of parts and their overlap. - Providing a foundation for more advanced techniques in Algebraic topology that rely on decomposing spaces.

It is a standard reference point for both formal proofs and practical computations, and its influence extends into areas like geometric topology and manifold theory. See also Homology for the broader language and Topological space for the ambient setting.