4.1 Excision theorem

The Excision theorem is a powerful tool in cohomology theory, allowing us to simplify complex spaces while preserving cohomological information. It relates the cohomology groups of a space to those of certain subspaces, enabling the computation of cohomology groups by removing or "excising" specific areas.

This theorem plays a crucial role in developing long exact sequences and the Mayer-Vietoris sequence, essential for computing cohomology groups of complex spaces. It's key for studying local properties of spaces by focusing on smaller, more manageable subspaces and their cohomology groups.

Excision theorem overview

• The excision theorem is a fundamental result in algebraic topology and cohomology theory that relates the cohomology groups of a space to those of certain subspaces
• It allows for the computation of cohomology groups by "excising" or removing certain subspaces, simplifying the overall space while preserving the cohomological information
• The theorem plays a crucial role in the development of long exact sequences and the Mayer-Vietoris sequence, which are essential tools in computing cohomology groups of complex spaces

Importance in cohomology theory

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• The excision theorem is a key tool in computing cohomology groups of topological spaces, especially those that can be decomposed into simpler subspaces
• It enables the study of local properties of spaces by focusing on smaller, more manageable subspaces and their cohomology groups
• The theorem is essential in proving the homotopy invariance of cohomology and establishing the relationship between cohomology and other invariants, such as homology and homotopy groups

Excision theorem statement

• Let $X$ be a topological space and $A, B \subset X$ be subspaces such that the closure of $A$ is contained in the interior of $B$, i.e., $\overline{A} \subset \text{int}(B)$
• Then the inclusion map $i: (X \setminus A, B \setminus A) \to (X, B)$ induces an isomorphism on cohomology groups: $i^*: H^n(X, B) \to H^n(X \setminus A, B \setminus A)$ for all $n \geq 0$

Topological spaces and subspaces

• The excision theorem deals with a topological space $X$ and two subspaces $A$ and $B$, where $A$ is "excisable" with respect to $B$
• The condition $\overline{A} \subset \text{int}(B)$ ensures that $A$ is completely contained within $B$ and does not intersect the boundary of $B$
• Examples of such spaces include:
  • $X = \mathbb{R}^2$, $A =$ open disk, $B =$ larger open disk containing $A$
  • $X =$ torus, $A =$ small open set, $B =$ larger open set containing $A$

Inclusion maps and induced homomorphisms

• The inclusion map $i: (X \setminus A, B \setminus A) \to (X, B)$ is a continuous map between the pairs of spaces $(X \setminus A, B \setminus A)$ and $(X, B)$
• This map induces a homomorphism $i^*: H^n(X, B) \to H^n(X \setminus A, B \setminus A)$ between the corresponding cohomology groups for each degree $n$
• The excision theorem states that these induced homomorphisms are isomorphisms, meaning that the cohomology groups of $(X, B)$ and $(X \setminus A, B \setminus A)$ are isomorphic

Excision theorem proof

• The proof of the excision theorem relies on the construction of a long exact sequence of cohomology groups, known as the Mayer-Vietoris sequence
• This sequence relates the cohomology groups of the space $X$, the subspaces $A$ and $B$, and their intersection $A \cap B$
• The proof involves showing that the inclusion map induces an isomorphism between certain terms in the Mayer-Vietoris sequence, which in turn implies the isomorphism stated in the excision theorem

Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence of cohomology groups that arises from the decomposition of a space $X$ into two subspaces $A$ and $B$: $\cdots \to H^n(A \cap B) \to H^n(A) \oplus H^n(B) \to H^n(X) \to H^{n+1}(A \cap B) \to \cdots$
• This sequence relates the cohomology groups of $X$, $A$, $B$, and $A \cap B$, and is a powerful tool in computing cohomology groups of spaces that can be decomposed into simpler subspaces

Long exact sequence of cohomology groups

• The proof of the excision theorem involves the construction of a long exact sequence of cohomology groups for the pairs $(X, B)$ and $(X \setminus A, B \setminus A)$: $\cdots \to H^n(X, B) \to H^n(X \setminus A, B \setminus A) \to H^{n+1}(X, X \setminus A) \to \cdots$
• By analyzing this sequence and the maps between the cohomology groups, one can show that the inclusion map induces an isomorphism between $H^n(X, B)$ and $H^n(X \setminus A, B \setminus A)$

Commutative diagram and naturality

• The proof of the excision theorem also involves the construction of a commutative diagram relating the long exact sequences of cohomology groups for different pairs of spaces
• This diagram helps to establish the naturality of the isomorphism induced by the inclusion map, meaning that it is compatible with the maps between cohomology groups arising from continuous maps between spaces

Excision theorem applications

• The excision theorem has numerous applications in algebraic topology and cohomology theory, as it allows for the computation of cohomology groups of spaces by breaking them down into simpler subspaces
• Some notable applications include the computation of relative cohomology groups, the cohomology of spheres and projective spaces, and the cohomology of CW complexes

Relative cohomology groups

• The excision theorem is used to define relative cohomology groups, which measure the cohomological difference between a space $X$ and a subspace $A$
• The relative cohomology group $H^n(X, A)$ is defined as the $n$-th cohomology group of the pair $(X, A)$, and the excision theorem allows for its computation by excising certain subspaces
• Relative cohomology groups are essential in the study of cohomology theories and the formulation of important results, such as the long exact sequence of a pair and the cohomology of quotient spaces

Cohomology of spheres and projective spaces

• The excision theorem is used to compute the cohomology groups of spheres and projective spaces, which are fundamental examples in algebraic topology
• For example, the cohomology groups of the $n$-sphere $S^n$ can be computed by excising a small open disk and applying the excision theorem: $H^k(S^n) \cong \begin{cases} \mathbb{Z}, & k = 0, n \\ 0, & \text{otherwise} \end{cases}$
• Similarly, the cohomology groups of real projective spaces $\mathbb{R}P^n$ can be computed using the excision theorem and the long exact sequence of a pair

Cohomology of CW complexes

• CW complexes are a class of topological spaces built by attaching cells of increasing dimension, and their cohomology groups can be computed using the excision theorem
• The theorem allows for the computation of the cohomology groups of a CW complex by inductively excising the cells of lower dimension and applying the long exact sequence of a pair
• This process leads to the cellular cohomology of a CW complex, which is a powerful tool in the study of the cohomological properties of spaces and their relationship with homotopy theory

Excision theorem generalizations

• The excision theorem admits various generalizations and analogues in different cohomology theories and settings, extending its applicability and usefulness
• Some notable generalizations include the excision theorem for homology groups, excision in sheaf cohomology, and the relationship between Čech cohomology and excision

Excision for homology groups

• The excision theorem also holds for homology groups, which are the dual notion to cohomology groups and measure the "holes" in a topological space
• In the homology setting, the excision theorem states that the inclusion map $i: (X \setminus A, B \setminus A) \to (X, B)$ induces an isomorphism on homology groups: $i_*: H_n(X \setminus A, B \setminus A) \to H_n(X, B)$ for all $n \geq 0$
• The proof of the excision theorem for homology groups is similar to that for cohomology groups and relies on the construction of the Mayer-Vietoris sequence and the long exact sequence of a pair

Excision in sheaf cohomology

• Sheaf cohomology is a generalization of singular cohomology that associates cohomology groups to sheaves on a topological space
• The excision theorem in sheaf cohomology states that the cohomology groups of a sheaf on a space $X$ can be computed by excising certain subspaces, provided that the sheaf satisfies certain conditions (e.g., being flasque or soft)
• This generalization allows for the computation of sheaf cohomology groups in various settings, such as in the study of coherent sheaves on algebraic varieties and the de Rham cohomology of smooth manifolds

Čech cohomology and excision

• Čech cohomology is another cohomology theory that associates cohomology groups to a topological space by considering its open covers and their nerve complexes
• The excision theorem in Čech cohomology states that the Čech cohomology groups of a space $X$ can be computed by excising certain subspaces, provided that the open covers satisfy certain conditions (e.g., being fine enough or having a suitable refinement)
• The relationship between Čech cohomology and singular cohomology is established through the de Rham theorem, which relates the de Rham cohomology of a smooth manifold to its singular cohomology via integration of differential forms

Excision theorem vs other theorems

• The excision theorem is one of several fundamental results in algebraic topology and cohomology theory, and it is closely related to other important theorems and concepts
• Some notable comparisons and relationships include the Mayer-Vietoris sequence, homotopy invariance of cohomology, and the snake lemma

Comparison with Mayer-Vietoris sequence

• The Mayer-Vietoris sequence is a long exact sequence of cohomology groups that arises from the decomposition of a space into two subspaces, and it is closely related to the excision theorem
• While the excision theorem deals with the isomorphism induced by the inclusion map between certain pairs of spaces, the Mayer-Vietoris sequence provides a more general relationship between the cohomology groups of a space and its subspaces
• The excision theorem can be seen as a special case of the Mayer-Vietoris sequence, where one of the subspaces is "excisable" with respect to the other

Relation to homotopy invariance

• Homotopy invariance is a fundamental property of cohomology theories, stating that the cohomology groups of a space are invariant under homotopy equivalence
• The excision theorem is a key ingredient in proving the homotopy invariance of singular cohomology, as it allows for the comparison of cohomology groups of spaces that are homotopy equivalent by excising certain subspaces
• In particular, the excision theorem is used to show that the cohomology groups of a space are isomorphic to those of its homotopy equivalent spaces, by considering the mapping cylinder and mapping cone of a homotopy equivalence

Excision and the snake lemma

• The snake lemma is a powerful result in homological algebra that relates the kernels, cokernels, and images of certain maps in a commutative diagram of exact sequences
• In the proof of the excision theorem, the snake lemma is used to analyze the long exact sequences of cohomology groups and establish the isomorphism induced by the inclusion map
• The snake lemma helps to keep track of the maps between the cohomology groups and their properties, such as injectivity, surjectivity, and exactness, which are crucial in proving the excision theorem and other related results in cohomology theory