3.1 Relative homology groups

Relative homology groups extend homology to pairs of spaces, revealing the relationship between a subspace and its ambient space. This concept allows us to study the additional homological features present in the larger space but not in the subspace.

The long exact sequence of relative homology groups connects the homology of the ambient space, subspace, and their relative homology. This powerful tool, along with the excision theorem, enables us to compute and understand the relationships between different homology groups.

Relative homology definition

• Relative homology extends the concept of homology groups to pairs of topological spaces, allowing for the study of the homology of a space relative to a subspace
• Provides a way to capture the homological relationship between a subspace and its ambient space, revealing important topological information

Subspace and ambient space

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• Consider a pair $(X, A)$ where $X$ is a topological space and $A$ is a subspace of $X$
• $X$ is referred to as the ambient space, and $A$ is the subspace of interest
• The relative homology groups measure the homology of $X$ modulo the homology of $A$, capturing the additional homological features present in $X$ but not in $A$

Quotient group construction

• The relative homology groups $H_n(X, A)$ are defined as the quotient groups $H_n(X, A) = Z_n(X, A) / B_n(X, A)$
• $Z_n(X, A)$ is the group of relative $n$-cycles, which are chains in $X$ whose boundary lies in $A$
• $B_n(X, A)$ is the group of relative $n$-boundaries, which are chains in $X$ that are boundaries of $(n+1)$-chains in $X$

Relative cycles and boundaries

• A relative $n$-cycle is a chain $c \in C_n(X)$ such that $\partial c \in C_{n-1}(A)$, where $\partial$ is the boundary operator
• Intuitively, a relative cycle is a cycle in $X$ whose boundary lies entirely within the subspace $A$
• A relative $n$-boundary is a chain $c \in C_n(X)$ such that $c = \partial d$ for some $(n+1)$-chain $d \in C_{n+1}(X)$
• Relative boundaries capture the notion of chains in $X$ that are boundaries, even if their boundaries do not necessarily lie in $A$

Long exact sequence

• The relative homology groups fit into a long exact sequence that relates the homology of the ambient space, the subspace, and the relative homology groups
• The long exact sequence provides a powerful tool for computing relative homology groups and understanding the relationships between different homology groups

Connecting homomorphism

• The connecting homomorphism $\partial_*: H_n(X, A) \to H_{n-1}(A)$ is a key component of the long exact sequence
• It maps relative homology classes in $H_n(X, A)$ to absolute homology classes in $H_{n-1}(A)$
• The connecting homomorphism measures how relative cycles in $(X, A)$ relate to cycles in the subspace $A$

Exactness and commutativity

• The long exact sequence is an exact sequence, meaning that the image of each homomorphism is equal to the kernel of the next homomorphism
• Exactness captures the precise relationships between the homology groups in the sequence
• The long exact sequence is also commutative, meaning that following different paths in the sequence yields the same result

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence that relates the homology groups of a space $X$ and its subspaces $A$ and $B$ when $X = A \cup B$
• It allows for the computation of the homology groups of $X$ in terms of the homology groups of $A$, $B$, and their intersection $A \cap B$
• The Mayer-Vietoris sequence is a powerful tool in algebraic topology and can be used in conjunction with relative homology

Excision theorem

• The excision theorem is a fundamental result in algebraic topology that relates the relative homology groups of different pairs of spaces
• It states that, under certain conditions, the relative homology groups are invariant under the excision of a "nice" subspace

Statement and intuition

• Let $X$ be a topological space, and let $A$ and $B$ be subspaces of $X$ such that the closure of $A$ is contained in the interior of $B$
• The excision theorem states that the inclusion map $(X \setminus A, B \setminus A) \hookrightarrow (X, B)$ induces an isomorphism on relative homology groups
• Intuitively, the excision theorem allows us to "cut out" a subspace $A$ from both the ambient space $X$ and the subspace $B$ without changing the relative homology groups

Excisive triad

• An excisive triad $(X; A, B)$ is a triple of spaces satisfying the conditions of the excision theorem
• In an excisive triad, the closure of $A$ is contained in the interior of $B$, and $X = \text{int}(A) \cup \text{int}(B)$
• The excision theorem applies to excisive triads, allowing for the computation of relative homology groups

Homotopy invariance

• The relative homology groups are homotopy invariant, meaning that they are preserved under homotopy equivalences of pairs
• If $(X, A)$ and $(Y, B)$ are homotopy equivalent pairs, then their relative homology groups are isomorphic
• Homotopy invariance is a crucial property that allows for the simplification and comparison of relative homology groups

Relative homology computations

• Computing relative homology groups often involves using the long exact sequence, excision theorem, and other algebraic tools
• By breaking down a pair of spaces into simpler components or utilizing known results, one can determine the relative homology groups

Relative homology of a pair

• To compute the relative homology groups of a pair $(X, A)$, one can use the long exact sequence relating the homology groups of $X$, $A$, and $(X, A)$
• By knowing the homology groups of $X$ and $A$, and using the exactness of the sequence, the relative homology groups can be determined
• The connecting homomorphism plays a crucial role in relating the different homology groups in the sequence

Relative homology of a triple

• A triple of spaces $(X, A, B)$ consists of a space $X$ and two subspaces $A$ and $B$ such that $B \subseteq A \subseteq X$
• The relative homology groups of a triple fit into a long exact sequence involving the relative homology groups of the pairs $(X, A)$, $(A, B)$, and $(X, B)$
• By exploiting the exactness of the sequence and known information about the pairs, the relative homology groups of the triple can be computed

Künneth formula for relative homology

• The Künneth formula is a theorem that relates the homology groups of a product space to the homology groups of its factors
• In the context of relative homology, there is a relative version of the Künneth formula
• The relative Künneth formula expresses the relative homology groups of a product pair in terms of the tensor products and torsion products of the relative homology groups of the factor pairs

Applications of relative homology

• Relative homology has numerous applications in algebraic topology and related fields
• It provides a framework for studying the homological properties of spaces and their subspaces, leading to important results and insights

Homology of CW complexes

• CW complexes are a class of topological spaces built by attaching cells of increasing dimension
• The homology groups of a CW complex can be computed using a cellular chain complex, which is based on the cellular structure of the complex
• Relative homology can be used to study the homology of subcomplexes and quotient complexes, providing a more refined analysis of the homological structure

Lefschetz duality

• Lefschetz duality is a theorem that relates the homology groups of a compact, oriented manifold to the cohomology groups of its complement
• It establishes an isomorphism between the relative homology groups of the manifold and the absolute cohomology groups of its complement
• Lefschetz duality has important applications in studying the topology of manifolds and their embeddings

Poincaré duality for manifolds with boundary

• Poincaré duality is a fundamental result that relates the homology and cohomology groups of a compact, oriented manifold without boundary
• For manifolds with boundary, a relative version of Poincaré duality holds
• Relative Poincaré duality relates the relative homology groups of a manifold with boundary to the absolute cohomology groups of the manifold and the cohomology groups of its boundary

Relative vs absolute homology

• Relative homology and absolute homology are two related but distinct concepts in algebraic topology
• While absolute homology studies the homological properties of a single space, relative homology considers pairs of spaces and their relationships

Comparison of properties

• Absolute homology groups $H_n(X)$ measure the $n$-dimensional holes in a space $X$
• Relative homology groups $H_n(X, A)$ measure the $n$-dimensional holes in $X$ that are not present in the subspace $A$
• Relative homology provides a more refined analysis of the homological structure, taking into account the presence of a subspace

Long exact sequence connecting relative and absolute

• There is a long exact sequence that connects the relative and absolute homology groups of a pair $(X, A)$
• The sequence relates the absolute homology groups of $X$ and $A$ with the relative homology groups of $(X, A)$
• The connecting homomorphism in the sequence maps relative homology classes to absolute homology classes, capturing the relationship between the two concepts

Excision in absolute homology

• The excision theorem also holds for absolute homology groups under certain conditions
• If $(X, A)$ is an excisive pair, then the inclusion map $X \setminus A \hookrightarrow X$ induces an isomorphism on absolute homology groups
• Excision in absolute homology allows for the computation of homology groups by decomposing a space into simpler pieces and studying their homological relationships