🧬Cohomology Theory Unit 4 – Excision and Mayer-Vietoris in Cohomology

Key Concepts and Definitions

• Cohomology groups $H^n(X;G)$ algebraic invariants associated to a topological space $X$ and an abelian group $G$
• Relative cohomology groups $H^n(X,A;G)$ cohomology groups of a pair $(X,A)$ where $A$ is a subspace of $X$
  • Measure the cohomology of $X$ modulo the cohomology of $A$
• Excision a property of cohomology theories stating that the relative cohomology $H^n(X,A;G)$ depends only on a small neighborhood of $A$ in $X$
• Mayer-Vietoris sequence a long exact sequence relating the cohomology of a space $X$ to the cohomology of two subspaces $U$ and $V$ whose union is $X$
  • Allows for the computation of cohomology by breaking a space into simpler pieces
• Connecting homomorphism a map in the Mayer-Vietoris sequence that relates the cohomology of the intersection $U \cap V$ to the cohomology of the union $U \cup V$
• Cup product a bilinear operation on cohomology classes that provides a ring structure on the cohomology of a space
• Cohomological dimension the highest degree in which a space has non-trivial cohomology groups

Historical Context and Development

• Cohomology theories emerged in the 1930s and 1940s as a dual notion to homology theories
  • Homology studies holes and cycles in a space, while cohomology studies the algebraic properties of functions on a space
• Early work on cohomology was done by J.W. Alexander, S. Lefschetz, and J.H.C. Whitehead, among others
• The Mayer-Vietoris sequence was introduced by W. Mayer and L. Vietoris in the 1930s as a tool for computing homology and cohomology
• The Excision Theorem was first proved for singular cohomology by S. Eilenberg in the 1940s
  • Later generalized to other cohomology theories (Čech, sheaf, etc.)
• The development of cohomology was closely tied to the study of algebraic topology and sheaf theory
• Cohomology has found applications in various areas of mathematics, including complex analysis, algebraic geometry, and mathematical physics

Excision Theorem: Statement and Significance

• Excision Theorem: Let $X$ be a topological space and $U \subset A \subset X$ such that the closure of $U$ is contained in the interior of $A$. Then the inclusion $(X \setminus U, A \setminus U) \hookrightarrow (X,A)$ induces an isomorphism $H^n(X,A;G) \cong H^n(X \setminus U, A \setminus U;G)$ for all $n$ and all abelian groups $G$
• Intuitively, excision states that the relative cohomology of a pair $(X,A)$ depends only on a small neighborhood of $A$ in $X$
  • The cohomology is unchanged by "excising" a subset $U$ from both $X$ and $A$, as long as $U$ is "small" relative to $A$
• Excision is a key property of cohomology theories and is used in the proof of the Mayer-Vietoris sequence
• The Excision Theorem allows for the computation of cohomology by breaking a space into simpler pieces and studying the relationships between them
• Excision is a powerful tool for proving invariance properties of cohomology, such as homotopy invariance and the cohomology of a quotient space

Proof Techniques for Excision

• The proof of the Excision Theorem typically involves the following steps:
  1. Construct a short exact sequence of cochain complexes relating the cochains of $(X,A)$, $(X \setminus U, A \setminus U)$, and $(U,U \cap A)$
  2. Show that the cochain complex of $(U,U \cap A)$ is acyclic (has trivial cohomology) using a homotopy argument or a spectral sequence
  3. Apply the long exact sequence in cohomology associated to the short exact sequence of cochain complexes
  4. Conclude that the inclusion $(X \setminus U, A \setminus U) \hookrightarrow (X,A)$ induces an isomorphism on cohomology
• The proof can be carried out in various cohomology theories (singular, Čech, sheaf) with appropriate modifications
• In some cases, excision can be proved using a direct argument involving barycentric subdivision and simplicial approximation
• The Excision Theorem can also be derived as a consequence of the Mayer-Vietoris sequence and the cohomology long exact sequence of a pair

Mayer-Vietoris Sequence: Construction and Properties

• Let $X$ be a topological space and $U,V \subset X$ open subsets such that $X = U \cup V$. The Mayer-Vietoris sequence is a long exact sequence of the form: $\cdots \to H^n(U \cap V;G) \xrightarrow{\delta} H^n(U;G) \oplus H^n(V;G) \xrightarrow{\alpha} H^n(X;G) \xrightarrow{\beta} H^{n+1}(U \cap V;G) \to \cdots$
• The maps in the sequence are induced by inclusions and restrictions of cochains
  • $\delta$ is the connecting homomorphism, $\alpha$ is the difference of restrictions, and $\beta$ is the sum of restrictions
• The Mayer-Vietoris sequence is natural with respect to maps of spaces and abelian groups
• The sequence is exact, meaning that the kernel of each map is equal to the image of the previous map
• The Mayer-Vietoris sequence can be derived from a short exact sequence of cochain complexes using the snake lemma or the zig-zag lemma
• The sequence provides a powerful tool for computing cohomology by breaking a space into simpler pieces (usually open sets) and studying the relationships between them

Applications of Excision and Mayer-Vietoris

• Computing the cohomology of a space that can be decomposed into simpler pieces (CW complexes, simplicial complexes, manifolds)
  • Example: the cohomology of a torus can be computed using a decomposition into two annuli
• Proving the homotopy invariance of cohomology
  • Excision allows for the comparison of cohomology groups of homotopic spaces
• Establishing the cohomology of a quotient space
  • Excision can be used to relate the cohomology of a space to that of a quotient space (lens spaces, projective spaces)
• Studying the cohomology of complements and deleted neighborhoods
  • The Mayer-Vietoris sequence can be applied to compute the cohomology of the complement of a subspace or a deleted neighborhood
• Analyzing the cohomology of a space with a group action
  • Excision and Mayer-Vietoris can be used in conjunction with spectral sequences to study equivariant cohomology
• Computing the cohomology of a fiber bundle or a covering space
  • The Mayer-Vietoris sequence can be adapted to the setting of fibrations and covering spaces (Leray-Serre spectral sequence)

Computational Examples and Exercises

• Example: Use the Mayer-Vietoris sequence to compute the cohomology of the real projective plane $\mathbb{RP}^2$
  • Decompose $\mathbb{RP}^2$ into two open sets (a disk and a Möbius band) and analyze the resulting sequence
• Example: Apply excision to show that the cohomology of a space is invariant under the attachment of a cell
  • Relate the cohomology of a space $X$ to that of the space $X \cup_f D^n$ obtained by attaching an $n$-cell via a map $f: S^{n-1} \to X$
• Exercise: Compute the cohomology of the Klein bottle using a suitable decomposition and the Mayer-Vietoris sequence
• Exercise: Use excision to prove the homotopy invariance of cohomology for CW complexes
• Exercise: Apply the Mayer-Vietoris sequence to compute the cohomology of a wedge sum of spaces in terms of the cohomology of the individual spaces
• Exercise: Use excision and the Mayer-Vietoris sequence to analyze the cohomology of a space with a group action (lens spaces, Grassmannians)

Advanced Topics and Further Directions

• Relative cup product and the cohomology ring structure
  • The relative cup product provides a ring structure on the relative cohomology $H^*(X,A;R)$ when $R$ is a commutative ring
• Cap product and Poincaré duality
  • The cap product is a bilinear pairing between cohomology and homology that generalizes Poincaré duality for manifolds
• Cohomology with compact supports and Borel-Moore homology
  • Cohomology with compact supports is a variant of cohomology that is better suited for studying non-compact spaces and is dual to Borel-Moore homology
• Sheaf cohomology and the de Rham theorem
  • Sheaf cohomology is a generalization of singular cohomology that is adapted to the study of sheaves on a space and is related to de Rham cohomology for smooth manifolds
• Spectral sequences and the Leray-Serre spectral sequence
  • Spectral sequences are algebraic tools for computing cohomology by successive approximations and are particularly useful in the study of fiber bundles and covering spaces
• Equivariant cohomology and localization theorems
  • Equivariant cohomology is a generalization of cohomology that takes into account the action of a group on a space and is related to localization theorems in equivariant topology
• Intersection cohomology and perverse sheaves
  • Intersection cohomology is a variant of cohomology that is adapted to the study of singular spaces and is closely related to the theory of perverse sheaves in algebraic geometry