3.4 Relative cohomology groups

Relative cohomology groups extend cohomology to pairs of spaces (X,A), where A is a subspace of X. They capture cohomological information of X relative to A, providing a more refined view of the space's structure.

These groups fit into a long exact sequence connecting the cohomology of X, A, and (X,A). The excision theorem allows for local computations, while the relative cup product gives a multiplicative structure to relative cohomology.

Definition of relative cohomology groups

• Relative cohomology groups extend the concept of cohomology to pairs of spaces $(X,A)$ where $A$ is a subspace of $X$
• Relative cohomology captures the cohomological information of the space $X$ relative to its subspace $A$
• Relative cohomology groups are denoted by $H^n(X,A)$ for each integer $n$

Cohomology groups of a pair

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• For a pair of spaces $(X,A)$, the relative cohomology groups $H^n(X,A)$ are defined as the cohomology groups of the quotient cochain complex $C^*(X,A) = C^*(X)/C^*(A)$
• The relative cochain complex $C^*(X,A)$ consists of cochains on $X$ that vanish when restricted to $A$
• The relative cohomology groups fit into a long exact sequence relating the cohomology of $X$, $A$, and the pair $(X,A)$

Induced homomorphisms in relative cohomology

• Continuous maps between pairs of spaces $(X,A) \to (Y,B)$ induce homomorphisms between the corresponding relative cohomology groups $H^n(X,A) \to H^n(Y,B)$
• These induced homomorphisms are functorial and compatible with the long exact sequence of pairs
• Induced homomorphisms allow for the study of mappings between spaces at the level of relative cohomology

Long exact sequence of a pair

• For a pair of spaces $(X,A)$, there is a long exact sequence connecting the cohomology groups of $X$, $A$, and the pair $(X,A)$: $\cdots \to H^{n-1}(A) \to H^n(X,A) \to H^n(X) \to H^n(A) \to H^{n+1}(X,A) \to \cdots$
• The connecting homomorphism $H^n(A) \to H^{n+1}(X,A)$ measures the obstruction to extending cohomology classes from $A$ to $X$
• The long exact sequence provides a powerful tool for computing relative cohomology groups and understanding their relationships

Excision theorem

• The excision theorem is a fundamental result in algebraic topology that relates the relative cohomology of a pair $(X,A)$ to the relative cohomology of a smaller pair $(U,U \cap A)$, where $U$ is an open subset of $X$
• Excision allows for the computation of relative cohomology groups by focusing on local information near the subspace $A$
• The excision theorem has important consequences and applications in cohomology theory

Statement of the excision theorem

• Let $X$ be a topological space, $A$ a subspace of $X$, and $U$ an open subset of $X$ such that the closure of $U$ is contained in the interior of $A$. Then the inclusion map $(X \setminus U, A \setminus U) \hookrightarrow (X,A)$ induces isomorphisms in relative cohomology: $H^n(X \setminus U, A \setminus U) \cong H^n(X,A)$ for all integers $n$
• Intuitively, excision states that the relative cohomology of $(X,A)$ depends only on the behavior of the pair near $A$, and the rest of $X$ can be "excised" without changing the relative cohomology

Consequences of the excision theorem

• The excision theorem implies that relative cohomology is a local property, depending only on the behavior of the pair near the subspace
• Excision allows for the computation of relative cohomology groups by decomposing the space into simpler pieces
• The excision theorem is a key ingredient in the proof of other important results, such as the Mayer-Vietoris sequence and the Künneth formula

Applications of the excision theorem

• Excision is used to compute the relative cohomology of pairs of spaces that can be decomposed into simpler pieces (CW complexes)
• The excision theorem is applied in the study of manifolds and their local properties (local cohomology)
• Excision plays a role in the formulation and proof of Poincaré duality for manifolds with boundary

Relative cup product

• The relative cup product is a cohomological operation that extends the cup product to relative cohomology groups
• It provides a way to multiply relative cohomology classes and obtain new relative cohomology classes
• The relative cup product endows the relative cohomology groups with a graded ring structure

Definition of the relative cup product

• Let $(X,A)$ be a pair of spaces and $H^*(X,A)$ the relative cohomology ring. The relative cup product is a bilinear map $\cup : H^p(X,A) \otimes H^q(X,A) \to H^{p+q}(X,A)$ defined by $[\alpha] \cup [\beta] = [\alpha \cup \beta]$, where $\alpha$ and $\beta$ are relative cocycles and $\alpha \cup \beta$ is their cup product as cochains
• The relative cup product is well-defined on cohomology classes and is associative and graded-commutative

Properties of the relative cup product

• The relative cup product is natural with respect to maps of pairs, i.e., if $f: (X,A) \to (Y,B)$ is a map of pairs, then $f^*([\alpha] \cup [\beta]) = f^*([\alpha]) \cup f^*([\beta])$
• The relative cup product satisfies the Leibniz rule with respect to the coboundary operator: $\delta([\alpha] \cup [\beta]) = \delta([\alpha]) \cup [\beta] + (-1)^p [\alpha] \cup \delta([\beta])$
• The relative cup product is compatible with the connecting homomorphism in the long exact sequence of a pair

Relative cohomology ring

• The relative cohomology groups $H^*(X,A)$ form a graded ring with respect to the relative cup product
• The multiplicative structure of the relative cohomology ring encodes additional information about the pair $(X,A)$
• The relative cohomology ring is a powerful tool for studying the algebraic topology of pairs of spaces

Connecting homomorphism

• The connecting homomorphism is a linear map that arises in the long exact sequence of relative cohomology groups
• It measures the obstruction to extending cohomology classes from the subspace to the entire space
• The connecting homomorphism plays a crucial role in understanding the relationship between absolute and relative cohomology

Definition of the connecting homomorphism

• In the long exact sequence of relative cohomology for a pair $(X,A)$: $\cdots \to H^{n-1}(A) \to H^n(X,A) \to H^n(X) \to H^n(A) \xrightarrow{\delta} H^{n+1}(X,A) \to \cdots$ the connecting homomorphism $\delta: H^n(A) \to H^{n+1}(X,A)$ is defined as follows:
  • Let $[\alpha] \in H^n(A)$ be a cohomology class represented by a cocycle $\alpha \in C^n(A)$
  • Extend $\alpha$ to a cochain $\tilde{\alpha} \in C^n(X)$ (not necessarily a cocycle)
  • The coboundary $\delta \tilde{\alpha} \in C^{n+1}(X)$ vanishes when restricted to $A$, hence it defines a relative cocycle in $C^{n+1}(X,A)$
  • The connecting homomorphism maps $[\alpha]$ to the relative cohomology class $[\delta \tilde{\alpha}] \in H^{n+1}(X,A)$

Naturality of the connecting homomorphism

• The connecting homomorphism is natural with respect to maps of pairs, i.e., if $f: (X,A) \to (Y,B)$ is a map of pairs, then the following diagram commutes:

  H^n(A) ---δ---> H^{n+1}(X,A)
    |               |
  f^|             f^|
    v               v
  H^n(B) ---δ---> H^{n+1}(Y,B)
  

• Naturality allows for the study of the connecting homomorphism under mappings between pairs of spaces

Connecting homomorphism in the long exact sequence

• The connecting homomorphism $\delta: H^n(A) \to H^{n+1}(X,A)$ fits into the long exact sequence of relative cohomology
• It connects the absolute cohomology of the subspace $A$ to the relative cohomology of the pair $(X,A)$
• The connecting homomorphism measures the failure of a cohomology class on $A$ to extend to a cohomology class on $X$, and it captures the "difference" between absolute and relative cohomology

Relative cohomology vs absolute cohomology

• Relative cohomology groups $H^*(X,A)$ provide a more refined version of cohomology compared to absolute cohomology groups $H^*(X)$
• Relative cohomology takes into account the presence of a subspace $A$ and captures the cohomological information of $X$ relative to $A$
• Understanding the relationship between relative and absolute cohomology is crucial for many applications in algebraic topology

Comparison of relative and absolute cohomology groups

• There is a natural map from relative cohomology to absolute cohomology, induced by the inclusion $(X,\emptyset) \hookrightarrow (X,A)$: $H^n(X,A) \to H^n(X)$
• This map fits into the long exact sequence of relative cohomology, relating relative cohomology to absolute cohomology of both $X$ and $A$
• In some cases, relative cohomology groups can be computed from absolute cohomology groups using the long exact sequence

Advantages of relative cohomology

• Relative cohomology provides a more detailed picture of the cohomological structure of a space, taking into account the presence of a subspace
• Relative cohomology is better suited for studying spaces with local properties or spaces that are decomposed into simpler pieces
• Relative cohomology is a key tool in the study of manifolds with boundary and in the formulation of Poincaré-Lefschetz duality

Limitations of relative cohomology

• Computing relative cohomology groups can be more challenging than computing absolute cohomology groups, as it involves working with quotient cochain complexes
• Relative cohomology may not capture all the relevant information about a space, and in some cases, absolute cohomology or other invariants may be more appropriate
• The interpretation of relative cohomology classes may be less intuitive compared to absolute cohomology classes

Computations and examples

• Computing relative cohomology groups is an important task in algebraic topology, as it provides insight into the structure of pairs of spaces
• Examples of relative cohomology computations include pairs of spaces such as discs and their boundaries, spheres and points, and manifolds with boundary
• These computations often rely on the long exact sequence of relative cohomology and the excision theorem

Relative cohomology of a disc and its boundary

• Let $D^n$ be the $n$-dimensional disc and $S^{n-1}$ its boundary sphere. The relative cohomology groups $H^*(D^n, S^{n-1})$ can be computed using the long exact sequence: $\cdots \to H^{k-1}(S^{n-1}) \to H^k(D^n, S^{n-1}) \to H^k(D^n) \to H^k(S^{n-1}) \to \cdots$
• Since $D^n$ is contractible, its absolute cohomology vanishes for $k > 0$, and the long exact sequence yields isomorphisms: $H^k(D^n, S^{n-1}) \cong H^{k-1}(S^{n-1})$
• The relative cohomology groups of $(D^n, S^{n-1})$ are thus determined by the absolute cohomology of the sphere $S^{n-1}$

Relative cohomology of a sphere and a point

• Consider the pair $(S^n, \{pt\})$ consisting of an $n$-dimensional sphere and a point. The relative cohomology groups $H^*(S^n, \{pt\})$ can be computed using the long exact sequence: $\cdots \to H^{k-1}(\{pt\}) \to H^k(S^n, \{pt\}) \to H^k(S^n) \to H^k(\{pt\}) \to \cdots$
• The absolute cohomology of a point vanishes for $k > 0$, and the long exact sequence yields isomorphisms: $H^k(S^n, \{pt\}) \cong H^k(S^n)$ for $k < n$, and a short exact sequence: $0 \to H^n(S^n, \{pt\}) \to H^n(S^n) \to \mathbb{Z} \to 0$
• The relative cohomology groups of $(S^n, \{pt\})$ are closely related to the absolute cohomology of the sphere $S^n$

Relative cohomology of a manifold with boundary

• Let $M$ be a compact oriented $n$-dimensional manifold with boundary $\partial M$. The relative cohomology groups $H^*(M, \partial M)$ are related to the absolute cohomology of $M$ and $\partial M$ via the long exact sequence: $\cdots \to H^{k-1}(\partial M) \to H^k(M, \partial M) \to H^k(M) \to H^k(\partial M) \to \cdots$
• The excision theorem can be used to compute the relative cohomology groups $H^*(M, \partial M)$ by decomposing $M$ into simpler pieces
• Poincaré-Lefschetz duality relates the relative cohomology of $(M, \partial M)$ to the absolute homology of $M$, providing another approach to computing these groups

Applications of relative cohomology

• Relative cohomology has numerous applications in algebraic topology and related fields
• It plays a crucial role in the study of manifolds with boundary, providing a cohomological framework for Poincaré-Lefschetz duality
• Relative cohomology is used in the formulation of the Thom isomorphism theorem and in obstruction theory

Poincaré-Lefschetz duality

• Poincaré-Lefschetz duality is a generalization of Poincaré duality for manifolds with boundary
• It relates the relative cohomology of a compact oriented manifold $(M, \partial M)$ to the absolute homology of $M$: $H^k(M, \partial M) \cong H_{n-k}(M)$ where $n$ is the dimension of $M$
• Poincaré-Lefschetz duality provides a powerful tool for computing relative cohomology groups and understanding the topology of manifolds with boundary

Thom isomorphism theorem

• The Thom isomorphism theorem relates the cohomology of a vector bundle to the compactly supported cohomology of its total space
• It states that for a rank $k$ vector bundle $E$ over a space $X$, there is an isomorphism: $H^{n+k}(E, E_0) \cong H^n(X)$ where $E_0$ is the complement of the zero section of $E$
• The relative cohomology group $H^{n+k}(E, E_0)$ is called the Thom space of the vector bundle, and the isomorphism is given by the Thom class
• The Thom isomorphism theorem has applications in characteristic class theory and in the study of orientability of vector bundles

Obstruction theory and relative cohomology

• Obstruction theory is a framework for studying the existence and uniqueness of continuous mappings between spaces satisfying certain conditions
• Relative cohomology plays a role in obstruction theory by measuring the obstruction to extending maps or homotopies
• The obstruction classes live in relative cohomology groups and provide information about the possibility of constructing mappings with desired properties
• Obstruction theory has applications in the classification of vector bundles, the study of characteristic classes, and in understanding the h