3.2 Long exact sequence of a pair

The long exact sequence of a pair is a crucial tool in cohomology theory. It connects the cohomology groups of a space, its subspace, and their relative cohomology. This sequence allows us to compute cohomology groups and understand relationships between different topological spaces.

By examining the exactness at each term, we can use this sequence to "chase" elements and apply isomorphism theorems. This powerful technique helps us break down complex spaces into simpler pieces, making cohomology calculations more manageable and revealing important topological properties.

Definition of long exact sequence

• A long exact sequence is a sequence of homomorphisms between abelian groups or modules that is infinite in one direction and exact at each term
• Exactness means the image of each homomorphism is equal to the kernel of the next, forming a chain complex with zero homology
• Long exact sequences arise naturally in algebraic topology, providing a powerful tool for computing cohomology groups and understanding the relationships between them

Connecting homomorphisms

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• Connecting homomorphisms, often denoted by $\delta$, link the cohomology groups in a long exact sequence
• These homomorphisms map from the $i$-th cohomology group to the $(i+1)$-th cohomology group, "connecting" the groups in the sequence
• The existence of connecting homomorphisms is a consequence of the Snake Lemma, which relates the kernels and cokernels of homomorphisms between short exact sequences

Exactness at each term

• Exactness at each term means that for every three consecutive terms $A \stackrel{f}{\rightarrow} B \stackrel{g}{\rightarrow} C$ in the sequence, the image of $f$ is equal to the kernel of $g$
• This property ensures that the composition of any two consecutive homomorphisms in the sequence is zero ($g \circ f = 0$)
• Exactness allows for the computation of cohomology groups by "chasing" elements through the sequence and using the isomorphism theorems

Long exact sequence for a pair

• The long exact sequence of a pair $(X, A)$ relates the cohomology groups of the space $X$, the subspace $A$, and the quotient space $X/A$
• It provides a systematic way to compute the cohomology of a space by breaking it down into simpler pieces and using the relative cohomology groups

Definition of a pair

• A pair $(X, A)$ consists of a topological space $X$ and a subspace $A \subseteq X$
• The inclusion map $i: A \hookrightarrow X$ induces homomorphisms between the cohomology groups of $A$ and $X$
• Pairs are essential in studying the relative cohomology groups and the relationships between the cohomology of a space and its subspaces

Relative cohomology groups

• The relative cohomology groups, denoted by $H^n(X, A)$, measure the cohomology of the quotient space $X/A$ relative to the subspace $A$
• They fit into the long exact sequence of the pair $(X, A)$, connecting the absolute cohomology groups of $X$ and $A$
• Relative cohomology groups can be computed using the relative cochain complex, which consists of cochains on $X$ that vanish on $A$

Induced maps between cohomology groups

• The inclusion map $i: A \hookrightarrow X$ and the quotient map $q: X \rightarrow X/A$ induce homomorphisms $i^*: [H^n(X)](https://www.fiveableKeyTerm:h^n(x)) \rightarrow H^n(A)$ and $q^*: H^n(X/A) \rightarrow H^n(X)$ between the cohomology groups
• These induced maps, along with the connecting homomorphisms, form the long exact sequence of the pair $(X, A)$
• The induced maps are functorial, meaning they commute with the coboundary operators and preserve the exactness of the sequence

Connecting homomorphisms for a pair

• In the long exact sequence of a pair $(X, A)$, the connecting homomorphisms $\delta: H^n(A) \rightarrow H^{n+1}(X, A)$ relate the absolute cohomology of the subspace $A$ to the relative cohomology of the pair
• The connecting homomorphisms are defined using the coboundary operator and the induced maps between the cochain complexes of $X$, $A$, and $(X, A)$

Inclusion and restriction maps

• The inclusion map $i: A \hookrightarrow X$ induces a restriction map $i^*: C^n(X) \rightarrow C^n(A)$ between the cochain groups, which sends a cochain on $X$ to its restriction on $A$
• The restriction map is a cochain map, meaning it commutes with the coboundary operators: $i^* \circ \delta_X = \delta_A \circ i^*$
• The restriction map induces the homomorphism $i^*: H^n(X) \rightarrow H^n(A)$ between the cohomology groups

Coboundary operator

• The coboundary operator $\delta: C^n(X, A) \rightarrow C^{n+1}(X, A)$ is defined on the relative cochain complex, sending a relative cochain to its coboundary
• It satisfies $\delta \circ \delta = 0$, making the relative cochain complex a cochain complex
• The cohomology of the relative cochain complex gives the relative cohomology groups $H^n(X, A)$

Commutative diagrams

• The induced maps and connecting homomorphisms fit into commutative diagrams, which illustrate the relationships between the various cochain complexes and cohomology groups
• These diagrams help in understanding the functoriality of the long exact sequence and the naturality of the connecting homomorphisms
• Commutative diagrams are essential tools in homological algebra and algebraic topology for studying the properties of long exact sequences and other algebraic structures

Exactness of the long exact sequence

• The long exact sequence of a pair $(X, A)$ is exact at each term, meaning the image of each homomorphism is equal to the kernel of the next
• Exactness allows for the computation of cohomology groups by "chasing" elements through the sequence and using the isomorphism theorems

Kernels and images

• The kernel of a homomorphism $f: A \rightarrow B$ is the subgroup $\ker(f) = \{a \in A | f(a) = 0\}$, consisting of elements that map to zero under $f$
• The image of a homomorphism $f: A \rightarrow B$ is the subgroup $\text{im}(f) = \{f(a) | a \in A\}$, consisting of elements that are the result of applying $f$ to elements of $A$
• In an exact sequence, the image of each homomorphism is equal to the kernel of the next, forming a chain complex with zero homology

Proof of exactness

• The proof of exactness for the long exact sequence of a pair $(X, A)$ involves showing that the composition of any two consecutive homomorphisms is zero and that the image of each homomorphism is equal to the kernel of the next
• This is typically done by using the definitions of the induced maps and connecting homomorphisms, along with the properties of the cochain complexes and the Snake Lemma
• The proof relies on the commutativity of certain diagrams and the exactness of short exact sequences of cochain complexes

Snake lemma

• The Snake Lemma is a powerful tool in homological algebra that relates the kernels, cokernels, and homology groups of homomorphisms between short exact sequences
• It is named after the "snake-like" diagram that arises when applying the lemma, which connects the various groups and homomorphisms involved
• The Snake Lemma is crucial in proving the exactness of the long exact sequence of a pair and in constructing the connecting homomorphisms

Applications of the long exact sequence

• The long exact sequence of a pair has numerous applications in algebraic topology, including the computation of cohomology groups, the study of topological invariants, and the proof of important theorems

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence that relates the cohomology groups of a space $X$ to the cohomology groups of two open subsets $U$ and $V$ that cover $X$
• It is constructed using the long exact sequences of the pairs $(X, U)$, $(X, V)$, and $(X, U \cap V)$, along with the induced maps between them
• The Mayer-Vietoris sequence is a powerful tool for computing the cohomology of spaces that can be decomposed into simpler pieces (CW complexes, manifolds)

Excision theorem

• The Excision Theorem states that if $(X, A)$ is a pair and $U \subseteq A$ is a subset such that the closure of $U$ is contained in the interior of $A$, then the inclusion map $(X - U, A - U) \hookrightarrow (X, A)$ induces an isomorphism $H^n(X - U, A - U) \cong H^n(X, A)$ between the relative cohomology groups
• This theorem allows for the computation of relative cohomology groups by "excising" a suitable subset and working with a simpler pair
• The Excision Theorem is a key ingredient in the proof of the Mayer-Vietoris sequence and other long exact sequences in algebraic topology

Cohomology of spheres and projective spaces

• The long exact sequence of a pair can be used to compute the cohomology groups of spheres and projective spaces, which are fundamental examples in algebraic topology
• For the $n$-sphere $S^n$, the long exact sequence of the pair $(D^{n+1}, S^n)$, where $D^{n+1}$ is the $(n+1)$-dimensional disk, yields the cohomology groups $H^i(S^n) \cong \mathbb{Z}$ for $i = 0, n$ and $H^i(S^n) \cong 0$ otherwise
• For the real projective space $\mathbb{R}P^n$, the long exact sequence of the pair $(\mathbb{R}P^n, \mathbb{R}P^{n-1})$, along with the cohomology of spheres, can be used to compute the cohomology groups $H^i(\mathbb{R}P^n)$ for all $i$

Functoriality of the long exact sequence

• The long exact sequence of a pair is functorial, meaning that it is compatible with induced maps between pairs and commutes with the connecting homomorphisms
• This functoriality allows for the study of the relationships between long exact sequences and the comparison of cohomology groups across different pairs

Induced maps between long exact sequences

• Given a map of pairs $f: (X, A) \rightarrow (Y, B)$, there are induced maps $f^*: H^n(Y) \rightarrow H^n(X)$, $f^*: H^n(B) \rightarrow H^n(A)$, and $f^*: H^n(Y, B) \rightarrow H^n(X, A)$ between the corresponding cohomology groups
• These induced maps fit into a commutative diagram relating the long exact sequences of the pairs $(X, A)$ and $(Y, B)$
• The commutativity of this diagram expresses the functoriality of the long exact sequence and allows for the comparison of cohomology groups across different pairs

Naturality of connecting homomorphisms

• The connecting homomorphisms in the long exact sequence of a pair are natural, meaning they commute with the induced maps between long exact sequences
• This naturality is expressed by the commutativity of certain squares in the diagram relating the long exact sequences of two pairs
• The naturality of connecting homomorphisms is crucial in proving the functoriality of the long exact sequence and in studying the relationships between cohomology theories

Commutative diagrams of long exact sequences

• Commutative diagrams involving long exact sequences are powerful tools for understanding the relationships between cohomology groups and induced maps
• These diagrams can be used to prove the functoriality of the long exact sequence, the naturality of connecting homomorphisms, and the compatibility of long exact sequences with other constructions in algebraic topology (Mayer-Vietoris sequence, Künneth formula)
• The ability to manipulate and interpret commutative diagrams is essential for working with long exact sequences and other algebraic structures in cohomology theory

Relationship to other cohomology theories

• The long exact sequence of a pair is a fundamental tool in algebraic topology and is closely related to other cohomology theories, such as singular cohomology, cellular cohomology, de Rham cohomology, and Čech cohomology
• Understanding the relationships between these cohomology theories and their respective long exact sequences is crucial for computing cohomology groups and studying the properties of topological spaces

Singular vs. cellular cohomology

• Singular cohomology is defined using cochains on the singular complex of a space, which consists of maps from standard simplices to the space
• Cellular cohomology, on the other hand, is defined using cochains on the cellular complex of a CW complex, which is built by attaching cells of increasing dimension
• For CW complexes, singular and cellular cohomology are isomorphic, and the long exact sequences of pairs in both theories are compatible under this isomorphism

de Rham cohomology

• De Rham cohomology is a cohomology theory for smooth manifolds, defined using differential forms and the exterior derivative
• The long exact sequence of a pair in de Rham cohomology relates the cohomology groups of a manifold, a submanifold, and the relative cohomology groups
• For smooth manifolds, de Rham cohomology is isomorphic to singular cohomology, and the long exact sequences in both theories are compatible under this isomorphism

Čech cohomology

• Čech cohomology is a cohomology theory defined using open covers of a space and the nerve complex associated with each cover
• The long exact sequence of a pair in Čech cohomology relates the cohomology groups of a space, a subspace, and the relative cohomology groups
• For paracompact Hausdorff spaces, Čech cohomology is isomorphic to singular cohomology, and the long exact sequences in both theories are compatible under this isomorphism

Computational examples and exercises

• Computing cohomology groups and long exact sequences for specific pairs is essential for developing a deep understanding of the concepts and techniques in cohomology theory
• Examples and exercises help to illustrate the key ideas, such as the functoriality of the long exact sequence, the naturality of connecting homomorphisms, and the relationships between different cohomology theories

Calculating relative cohomology groups

• To calculate relative cohomology groups $H^n(X, A)$, one typically uses the long exact sequence of the pair $(X, A)$ and the known absolute cohomology groups of $X$ and $A$
• Example: Let $X = S^2$ and $A = \{(0, 0, 1), (0, 0, -1)\}$ be the north and south poles. Using the long exact sequence of the pair $(S^2, A)$ and the known cohomology groups of $S^2$ and $A$, compute the relative cohomology groups $H^n(S^2, A)$ for all $n$

Determining the long exact sequence for specific pairs

• Given a pair $(X, A)$, one can determine the long exact sequence by computing the induced maps between the absolute cohomology groups and the connecting homomorphisms
• Example: Let $X = \mathbb{R}P^2$ and $A = \mathbb{R}P^1$. Using the known cohomology groups of $\mathbb{R}P^2$ and $\mathbb{R}P^1$, determine the long exact sequence of the pair $(\mathbb{R}P^2, \mathbb{R}P^1)$ and compute the relative cohomology groups $H^n(\mathbb{R}P^2, \mathbb{R}P^1)$ for all $n$

Solving problems using the long exact sequence

• The long exact sequence of a pair is a powerful tool for solving problems in algebraic topology, such as computing the cohomology groups of spaces, proving topological invariants, and understanding the relationships between spaces
• Example: Use the long exact sequence of the pair $(S^n, S^{n-1})$ to prove that the reduced cohomology groups of the $n$-sphere are given by $\tilde{H}^i(S^n) \cong \mathbb{Z}$ for $i = n$ and $\tilde{H}^i(S^n) \cong 0$ otherwise