3.3 Excision theorem

The Excision theorem is a key tool in algebraic topology, allowing us to compute homology groups by breaking down spaces into smaller pieces. It relates the homology of a space, a subspace, and their difference, enabling powerful computational techniques.

This theorem is crucial for proving homotopy invariance of homology and developing the Mayer-Vietoris sequence. It has applications in various areas of mathematics, from simplicial complexes to CW complexes, and can be generalized to cohomology theories and K-theory.

Excision theorem overview

• The excision theorem is a fundamental result in algebraic topology that relates the homology groups of a space, a subspace, and their difference
• It allows for the computation of homology groups by breaking down a space into smaller, more manageable pieces
• The excision theorem is a key tool in the development of homology theory and its applications to various areas of mathematics

Importance in algebraic topology

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• The excision theorem plays a crucial role in the computation of homology groups, which are algebraic invariants that capture important topological properties of spaces
• It enables the development of powerful computational techniques, such as the Mayer-Vietoris sequence and the long exact sequence of a pair
• The excision theorem is essential for proving the homotopy invariance of homology, a fundamental property that allows for the study of spaces up to homotopy equivalence

Statement of the theorem

• Let $X$ be a topological space, $A \subset X$ a subspace, and $U \subset A$ an open set in the subspace topology of $A$
• If 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 an isomorphism on homology groups: $H_n(X \setminus U, A \setminus U) \cong H_n(X, A)$ for all $n$

Commutative diagram representation

• The excision theorem can be represented using a commutative diagram of homology groups and induced homomorphisms
• The diagram involves the homology groups of the pairs $(X, A)$, $(X \setminus U, A \setminus U)$, and $(X, X \setminus U)$, along with the appropriate inclusion maps and induced homomorphisms
• The commutativity of the diagram encodes the isomorphism between the homology groups $H_n(X \setminus U, A \setminus U)$ and $H_n(X, A)$

Excision theorem applications

• The excision theorem has numerous applications in algebraic topology and related fields, enabling the computation of homology groups in various contexts
• It serves as a powerful tool for studying the topological properties of spaces and their relationships

Mayer-Vietoris sequence derivation

• The excision theorem is a key ingredient in the derivation of the Mayer-Vietoris sequence, a long exact sequence that relates the homology groups of a space, two subspaces, and their intersection
• By applying the excision theorem to carefully chosen subspaces and their differences, one can construct the Mayer-Vietoris sequence and use it to compute homology groups

Relative homology group calculations

• The excision theorem allows for the calculation of relative homology groups, which measure the homology of a pair of spaces $(X, A)$, where $A$ is a subspace of $X$
• By excising an appropriate open set from both $X$ and $A$, the excision theorem reduces the computation of relative homology groups to the computation of absolute homology groups of simpler spaces

Homotopy invariance of homology

• The excision theorem is essential for proving the homotopy invariance of homology, which states that homotopy equivalent spaces have isomorphic homology groups
• By applying the excision theorem to homotopy equivalent pairs of spaces, one can establish the homotopy invariance of homology, a fundamental property that allows for the study of spaces up to homotopy equivalence

Excision theorem proof

• The proof of the excision theorem involves several key techniques and constructions from algebraic topology
• The main idea is to establish an isomorphism between the homology groups of the pairs $(X \setminus U, A \setminus U)$ and $(X, A)$ by constructing a chain map between their chain complexes and showing that it induces an isomorphism on homology

Barycentric subdivision technique

• The barycentric subdivision technique is used to refine the triangulation of a simplicial complex, allowing for a more fine-grained analysis of the space
• In the proof of the excision theorem, barycentric subdivision is applied to the simplicial complexes associated with the pairs $(X, A)$ and $(X \setminus U, A \setminus U)$ to obtain a common refinement

Connecting homomorphism construction

• The connecting homomorphism is a key component of the long exact sequence of a pair, which relates the homology groups of a space, a subspace, and their quotient
• In the proof of the excision theorem, the connecting homomorphism is constructed explicitly using the barycentric subdivision and the boundary operators of the chain complexes

Exact sequence of chain complexes

• The proof of the excision theorem involves the construction of an exact sequence of chain complexes, which encodes the relationships between the chain complexes of the pairs $(X, A)$, $(X \setminus U, A \setminus U)$, and $(X, X \setminus U)$
• The exactness of this sequence is crucial for establishing the isomorphism between the homology groups of interest

Zigzag lemma application

• The zigzag lemma is a technical result in homological algebra that relates the homology groups of a chain complex to those of a subcomplex and a quotient complex
• In the proof of the excision theorem, the zigzag lemma is applied to the exact sequence of chain complexes to deduce the desired isomorphism between the homology groups

Excision theorem generalizations

• The excision theorem can be generalized to various settings beyond ordinary homology theory, allowing for the study of more sophisticated topological invariants
• These generalizations extend the power and applicability of the excision theorem to a wider range of mathematical contexts

Excision for cohomology theories

• The excision theorem can be formulated for cohomology theories, which are contravariant functors from the category of topological spaces to the category of abelian groups
• In the cohomological setting, the excision theorem relates the cohomology groups of a space, a subspace, and their difference, providing a powerful tool for computing cohomology groups

Čech cohomology excision axiom

• Čech cohomology is a cohomology theory that is defined using open covers of a topological space and their nerve complexes
• The excision axiom for Čech cohomology states that the inclusion map of a pair $(X, A)$ induces an isomorphism on Čech cohomology groups, under suitable conditions on the open covers

Generalized excision for spectra

• The excision theorem can be generalized to the setting of spectra, which are sequences of pointed spaces equipped with structure maps
• In this context, the excision theorem takes the form of a homotopy pushout square, relating the homology theories of a spectrum, a subspectrum, and their homotopy cofiber

Excision in K-theory

• K-theory is a generalized cohomology theory that assigns abelian groups to topological spaces, capturing important geometric and algebraic invariants
• The excision theorem in K-theory relates the K-groups of a space, a subspace, and their difference, providing a powerful computational tool in the study of vector bundles and other K-theoretic objects

Excision theorem examples

• The excision theorem can be applied to various types of topological spaces and their subspaces, demonstrating its versatility and usefulness in computing homology groups

Excision for CW complexes

• CW complexes are built by attaching cells of increasing dimension, providing a convenient framework for studying topological spaces
• The excision theorem can be applied to pairs of CW complexes $(X, A)$, where $A$ is a subcomplex of $X$, to compute the relative homology groups $H_n(X, A)$

Excision for simplicial complexes

• Simplicial complexes are combinatorial objects that can be used to model topological spaces
• The excision theorem holds for pairs of simplicial complexes $(K, L)$, where $L$ is a subcomplex of $K$, allowing for the computation of the relative homology groups $H_n(K, L)$

Excision in singular homology

• Singular homology is a homology theory that assigns abelian groups to topological spaces using singular simplices, which are continuous maps from standard simplices to the space
• The excision theorem in singular homology relates the singular homology groups of a space, a subspace, and their difference, providing a powerful tool for computing homology groups

Excision for pair of spaces

• The excision theorem can be applied to pairs of topological spaces $(X, A)$, where $A$ is a subspace of $X$
• By choosing an appropriate open set $U \subset A$ and applying the excision theorem, one can compute the relative homology groups $H_n(X, A)$ in terms of the homology groups of the simpler pair $(X \setminus U, A \setminus U)$