2.4 Mayer-Vietoris sequence

The Mayer-Vietoris sequence is a powerful tool in algebraic topology. It connects the homology groups of a space to those of its subspaces, helping us understand complex structures by breaking them down into simpler parts.

This sequence is particularly useful when dealing with spaces that can be split into two simpler subspaces. By analyzing the relationships between these subspaces, we can uncover important information about the original space's homology groups.

Mayer-Vietoris Sequence for Homology

Definition and Structure

• The Mayer-Vietoris sequence relates the homology groups of a space $X$ to the homology groups of two subspaces $A$ and $B$ whose union is $X$

• Given a space $X$ and two subspaces $A$ and $B$ such that $X = A \cup B$, the Mayer-Vietoris sequence is: $\dots \to H_n(A \cap B) \to H_n(A) \oplus H_n(B) \to H_n(X) \to H_{n-1}(A \cap B) \to \dots$

• The maps in the sequence are induced by the inclusions of $A \cap B$ into $A$ and $B$, and the inclusions of $A$ and $B$ into $X$

• The sequence is infinite in both directions, with the homology groups repeating with a shift in dimension (e.g., $H_n(X)$ is followed by $H_{n-1}(A \cap B)$)

Properties and Naturality

• The Mayer-Vietoris sequence is an exact sequence, meaning that the kernel of each map is equal to the image of the preceding map
• The sequence is natural, meaning it is compatible with maps between spaces that preserve the subspace structure
  • If $f: X \to Y$ is a continuous map and $A, B \subset X$, $C, D \subset Y$ are subspaces such that $f(A) \subset C$ and $f(B) \subset D$, then $f$ induces a map between the corresponding Mayer-Vietoris sequences
• The Mayer-Vietoris sequence is functorial, meaning it respects the composition of maps between spaces

Applying the Mayer-Vietoris Sequence

Computing Homology Groups

• The Mayer-Vietoris sequence can be used to compute the homology groups of a space $X$ by splitting it into two subspaces $A$ and $B$ and using the known homology groups of $A$, $B$, and $A \cap B$
• The sequence relates the homology groups of $X$ to the homology groups of $A$, $B$, and $A \cap B$ through a series of maps, which can be analyzed to determine the homology groups of $X$
• To apply the Mayer-Vietoris sequence:
  1. Start with the known homology groups of $A$, $B$, and $A \cap B$
  2. Use the exactness of the sequence to determine the unknown homology groups of $X$
  3. Analyze the maps in the sequence to derive relationships between the homology groups
• The Mayer-Vietoris sequence is particularly useful when the space $X$ has a complicated structure but can be decomposed into simpler subspaces $A$ and $B$ (e.g., a torus can be decomposed into two cylinders)

Simplifying the Computation

• In some cases, the Mayer-Vietoris sequence may split into short exact sequences, which can simplify the computation of the homology groups
• A short exact sequence is a sequence of the form $0 \to A \to B \to C \to 0$, where the map $A \to B$ is injective, the map $B \to C$ is surjective, and the image of $A \to B$ is equal to the kernel of $B \to C$
• When the Mayer-Vietoris sequence splits into short exact sequences, the homology groups of $X$ can be determined by studying the short exact sequences and using algebraic techniques (e.g., the splitting lemma)
• Splitting of the Mayer-Vietoris sequence often occurs when the space $X$ and its subspaces $A$ and $B$ have a particularly nice structure (e.g., when they are all contractible spaces)

Exactness of the Mayer-Vietoris Sequence

Proving Exactness

• The proof of the exactness of the Mayer-Vietoris sequence involves showing that the kernel of each map in the sequence is equal to the image of the preceding map
• The proof typically uses the snake lemma, a result in homological algebra that relates the kernels and cokernels of maps between long exact sequences
• To prove the exactness of the Mayer-Vietoris sequence:
  1. Construct a short exact sequence of chain complexes from the inclusions of $A$, $B$, and $A \cap B$ into $X$
  2. Apply the snake lemma to the resulting long exact sequence in homology
  3. Identify the connecting homomorphism in the Mayer-Vietoris sequence with the boundary map in the long exact sequence obtained from the snake lemma
• The proof requires careful tracking of the maps involved and their compatibility with the boundary maps in the chain complexes

Key Steps and Techniques

• The short exact sequence of chain complexes is constructed using the inclusions $i_A: A \to X$, $i_B: B \to X$, and $i_{A \cap B}: A \cap B \to A, B$
• The short exact sequence is of the form $0 \to C_*(A \cap B) \to C_*(A) \oplus C_*(B) \to C_*(X) \to 0$, where $C_*$ denotes the chain complex functor
• Applying the snake lemma to the short exact sequence of chain complexes yields a long exact sequence in homology involving the homology groups of $A$, $B$, $A \cap B$, and $X$
• The connecting homomorphism in the long exact sequence obtained from the snake lemma is identified with the boundary map in the Mayer-Vietoris sequence
• The exactness of the Mayer-Vietoris sequence follows from the exactness of the long exact sequence obtained from the snake lemma

Generalizing the Mayer-Vietoris Sequence

Multiple Subspaces

• The Mayer-Vietoris sequence can be generalized to a space $X$ that is the union of more than two subspaces

• For a space $X$ that is the union of three subspaces $A$, $B$, and $C$, the generalized Mayer-Vietoris sequence involves the homology groups of the pairwise intersections $A \cap B$, $A \cap C$, and $B \cap C$, as well as the triple intersection $A \cap B \cap C$

• The generalized Mayer-Vietoris sequence for three subspaces is: $\dots \to H_n(A \cap B \cap C) \to H_n(A \cap B) \oplus H_n(A \cap C) \oplus H_n(B \cap C) \to H_n(A) \oplus H_n(B) \oplus H_n(C) \to H_n(X) \to \dots$

• The maps in the generalized sequence are induced by the appropriate inclusions of the intersections into the subspaces and the subspaces into $X$

Arbitrary Finite Number of Subspaces

• The generalized Mayer-Vietoris sequence can be extended to any finite number of subspaces
• For a space $X$ that is the union of subspaces $A_1, A_2, \dots, A_n$, the generalized Mayer-Vietoris sequence involves the homology groups of all possible intersections of the subspaces
• The sequence includes terms for the homology groups of the subspaces $H_*(A_i)$, the pairwise intersections $H_*(A_i \cap A_j)$, the triple intersections $H_*(A_i \cap A_j \cap A_k)$, and so on, up to the intersection of all subspaces $H_*(A_1 \cap A_2 \cap \dots \cap A_n)$
• The maps in the sequence are induced by the appropriate inclusions of the intersections into the subspaces and the subspaces into $X$
• The generalized Mayer-Vietoris sequence for $n$ subspaces can be constructed inductively by applying the Mayer-Vietoris sequence for two subspaces to the union of the first $n-1$ subspaces and the $n$-th subspace