2.4 Induced cohomomorphisms

Induced cohomomorphisms are a key concept in cohomology theory, allowing us to study how cohomology groups change under maps between spaces. They provide a way to relate cohomology groups of different spaces and understand their structure.

These cohomomorphisms are essential for computing cohomology groups and understanding functorial properties. They play a crucial role in constructing long exact sequences, Mayer-Vietoris sequences, and other important tools in algebraic topology.

Induced cohomomorphisms

• Fundamental concept in cohomology theory that allows for the study of how cohomology groups change under maps between spaces
• Induced cohomomorphisms provide a way to relate the cohomology groups of different spaces and understand their structure
• Essential tool for computing cohomology groups and understanding the functorial properties of cohomology

Definition of induced cohomomorphisms

Top images from around the web for Definition of induced cohomomorphisms

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition of induced cohomomorphisms

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Frontiers | Topological Schemas of Memory Spaces View original

  Is this image relevant?

• 

  Cohomology - Wikipedia View original

  Is this image relevant?

1 of 3

• Given a continuous map $f: X \to Y$ between topological spaces, the induced cohomomorphism $f^*: H^n(Y; R) \to H^n(X; R)$ is defined for each cohomology group $H^n$ with coefficients in a ring $R$
• $f^*$ maps cocycles on $Y$ to cocycles on $X$ by pulling back the cocycle along the map $f$
• Induced cohomomorphism preserves the cohomology class, i.e., if $\alpha$ and $\beta$ are cohomologous cocycles on $Y$, then $f^*(\alpha)$ and $f^*(\beta)$ are cohomologous on $X$

Properties of induced cohomomorphisms

• Induced cohomomorphisms are linear maps between the cohomology groups $H^n(Y; R)$ and $H^n(X; R)$
• Composition of maps induces composition of cohomomorphisms: if $f: X \to Y$ and $g: Y \to Z$, then $(g \circ f)^* = f^* \circ g^*$
• Identity map induces the identity cohomomorphism: if $id_X: X \to X$ is the identity map, then $id_X^*: H^n(X; R) \to H^n(X; R)$ is the identity homomorphism

Functoriality of cohomology

• Cohomology is a contravariant functor from the category of topological spaces to the category of graded $R$-modules
• Induced cohomomorphisms are the morphisms in the cohomology functor
• Functoriality allows for the study of how cohomology behaves under maps between spaces and provides a powerful tool for understanding the structure of cohomology groups

Induced cohomomorphisms in long exact sequences

• Induced cohomomorphisms play a crucial role in the construction and analysis of long exact sequences in cohomology
• Long exact sequences relate the cohomology groups of different spaces and provide a means for computing cohomology groups

Connecting homomorphisms

• In a long exact sequence, the connecting homomorphisms are induced by the boundary operator in the cochain complex
• Connecting homomorphisms relate the cohomology groups of different degrees and provide a link between the cohomology of different spaces
• Example: In the long exact sequence of a pair $(X, A)$, the connecting homomorphism $\delta^*: H^n(X, A; R) \to H^{n+1}(A; R)$ is induced by the coboundary operator

Snake lemma

• The snake lemma is a powerful tool for studying the relationship between long exact sequences
• It allows for the construction of a long exact sequence from two short exact sequences of cochain complexes
• The snake lemma is used to derive the long exact sequence of a pair and the Mayer-Vietoris sequence

Induced cohomomorphisms for pairs

• Induced cohomomorphisms can be defined for pairs of spaces $(X, A)$, where $A$ is a subspace of $X$
• These cohomomorphisms relate the cohomology groups of the pair $(X, A)$ to the cohomology groups of $X$ and $A$

Relative cohomology groups

• The relative cohomology groups $H^n(X, A; R)$ are defined as the cohomology groups of the quotient cochain complex $C^*(X, A; R) = C^*(X; R) / C^*(A; R)$
• Relative cohomology groups measure the cohomology of $X$ modulo the cohomology of $A$
• Induced cohomomorphisms for pairs are defined using the relative cohomology groups

Long exact sequence of a pair

• For a pair $(X, A)$, there is a long exact sequence relating the cohomology groups of $X$, $A$, and $(X, A)$: $\cdots \to H^n(X, A; R) \to H^n(X; R) \to H^n(A; R) \to H^{n+1}(X, A; R) \to \cdots$
• The maps in the long exact sequence are induced by the inclusion $A \hookrightarrow X$ and the connecting homomorphisms
• The long exact sequence of a pair is a powerful tool for computing cohomology groups and understanding the relationship between the cohomology of a space and its subspaces

Induced cohomomorphisms in Mayer-Vietoris sequences

• 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 $X$
• Induced cohomomorphisms play a crucial role in the construction and application of the Mayer-Vietoris sequence

Mayer-Vietoris sequence

• For an open cover $X = U \cup V$, the Mayer-Vietoris sequence is a long exact sequence of the form: $\cdots \to H^n(X; R) \to H^n(U; R) \oplus H^n(V; R) \to H^n(U \cap V; R) \to H^{n+1}(X; R) \to \cdots$
• The maps in the sequence are induced by the inclusions of $U$, $V$, and $U \cap V$ into $X$ and the connecting homomorphisms
• The Mayer-Vietoris sequence is a powerful tool for computing the cohomology groups of a space by breaking it down into simpler pieces

Excision theorem

• The excision theorem is a fundamental result in cohomology theory that relates the relative cohomology groups of a pair $(X, A)$ to the relative cohomology groups of a smaller pair $(U, U \cap A)$, where $U$ is an open subset of $X$
• The excision theorem states that if the closure of $U$ is contained in the interior of $A$, then the inclusion $(U, U \cap A) \hookrightarrow (X, A)$ induces an isomorphism in cohomology: $H^n(X, A; R) \cong H^n(U, U \cap A; R)$
• The excision theorem is a key ingredient in the proof of the Mayer-Vietoris sequence and is used to simplify cohomology computations

Applications of induced cohomomorphisms

• Induced cohomomorphisms have numerous applications in cohomology theory and algebraic topology
• They are used to compute cohomology groups, study the structure of cohomology, and relate cohomology to other algebraic and geometric invariants

Computations using induced cohomomorphisms

• Induced cohomomorphisms can be used to compute the cohomology groups of a space by relating them to the cohomology groups of simpler spaces
• Examples include using the long exact sequence of a pair, the Mayer-Vietoris sequence, or the excision theorem to break down a complex space into simpler pieces
• Induced cohomomorphisms also allow for the comparison of cohomology groups under maps between spaces, which can simplify computations

Induced cohomomorphisms in spectral sequences

• Spectral sequences are powerful algebraic tools that compute cohomology groups by successively approximating them using a sequence of algebraic objects
• Induced cohomomorphisms play a crucial role in the construction and convergence of spectral sequences
• The maps between the pages of a spectral sequence are often induced by cohomomorphisms between different spaces or cochain complexes

Naturality of induced cohomomorphisms

• Induced cohomomorphisms satisfy a naturality property that makes them compatible with maps between spaces and homomorphisms between coefficient rings
• Naturality is a key feature of induced cohomomorphisms that allows for their use in functorial constructions and commutative diagrams

Natural transformations

• A natural transformation between two functors $F, G: \mathcal{C} \to \mathcal{D}$ is a collection of morphisms $\eta_X: F(X) \to G(X)$ for each object $X$ in $\mathcal{C}$ that commutes with the morphisms in $\mathcal{C}$
• Induced cohomomorphisms define a natural transformation between the cohomology functors $H^n(-, R)$ for different coefficient rings $R$
• Naturality allows for the comparison of induced cohomomorphisms under change of coefficients and ensures their compatibility with maps between spaces

Commutative diagrams

• Commutative diagrams are a fundamental tool in category theory and algebraic topology for expressing the compatibility of maps and functors
• Induced cohomomorphisms satisfy naturality, which means they fit into commutative diagrams relating the cohomology groups of different spaces
• Commutative diagrams involving induced cohomomorphisms are used to prove theorems, compute cohomology groups, and understand the functorial properties of cohomology

Induced cohomomorphisms vs induced homomorphisms

• Cohomology theory has a dual theory called homology, which assigns to each space a sequence of homology groups $H_n(X; R)$
• Induced homomorphisms in homology are defined similarly to induced cohomomorphisms but with some key differences

Similarities and differences

• Both induced cohomomorphisms and induced homomorphisms are defined using maps between spaces and preserve the algebraic structure of the respective theories
• Induced cohomomorphisms are contravariant functors, mapping in the opposite direction of the original map, while induced homomorphisms are covariant functors, mapping in the same direction as the original map
• Cohomology groups are computed using cochains (maps from chains to the coefficient ring), while homology groups are computed using chains (formal sums of simplices)

Duality between cohomology and homology

• Cohomology and homology are dual theories, and many concepts and results in one theory have analogues in the other
• The universal coefficient theorem relates the cohomology and homology groups of a space, showing that cohomology groups can be computed from homology groups and vice versa
• Poincaré duality, which holds for manifolds, establishes an isomorphism between the cohomology groups $H^k(M; R)$ and the homology groups $H_{n-k}(M; R)$, where $n$ is the dimension of the manifold $M$