1.5 Induced homomorphisms

Induced homomorphisms are a key concept in cohomology theory, allowing us to relate cohomology groups of different spaces through continuous maps. They provide a powerful tool for studying how cohomology changes under mappings, crucial for understanding topological properties.

These homomorphisms are defined by pulling back cochains and satisfy important properties like functoriality and naturality. They're compatible with cohomological structures like cup products and long exact sequences, making them essential for computations and applications in algebraic topology.

Definition of induced homomorphisms

• Induced homomorphisms are a fundamental concept in cohomology theory that allow us to relate the cohomology groups of different spaces
• They provide a way to study how the cohomology of a space changes when we have a continuous map between spaces
• Understanding induced homomorphisms is crucial for many applications of cohomology theory, such as studying the topology of manifolds and fiber bundles

Induced maps between cohomology groups

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• Given a continuous map $f: X \to Y$ between topological spaces, we can define induced homomorphisms $f^*: H^n(Y) \to H^n(X)$ between the cohomology groups of $Y$ and $X$
• The induced map $f^*$ is defined by pulling back cochains from $Y$ to $X$ using the map $f$
• Intuitively, the induced map $f^*$ allows us to compare the cohomological information of the spaces $X$ and $Y$ via the map $f$

Functoriality of cohomology

• Cohomology is a contravariant functor from the category of topological spaces to the category of abelian groups
• This means that for any continuous map $f: X \to Y$, we have an induced homomorphism $f^*: H^n(Y) \to H^n(X)$ that is compatible with composition of maps
• Functoriality allows us to study the behavior of cohomology under continuous maps and provides a powerful tool for understanding the relationships between different spaces

Naturality of induced homomorphisms

• Induced homomorphisms satisfy a naturality property, which means they are compatible with the coboundary operators in the cohomology groups
• Specifically, for any continuous map $f: X \to Y$, the induced map $f^*$ commutes with the coboundary operators $\delta$, i.e., $f^* \circ \delta_Y = \delta_X \circ f^*$
• Naturality ensures that induced homomorphisms preserve the cohomological structure and are well-behaved with respect to the long exact sequences associated with cohomology

Construction of induced homomorphisms

• To define induced homomorphisms, we need to understand how to relate the cochains on different spaces via continuous maps
• The construction of induced homomorphisms involves two main operations: pullbacks and pushforwards of cochains
• These operations allow us to transfer cohomological information between spaces in a way that is compatible with the coboundary operators and the cup product

Pullbacks of cochains

• Given a continuous map $f: X \to Y$ and a cochain $\varphi \in C^n(Y)$, we define the pullback of $\varphi$ by $f$ as the cochain $f^*\varphi \in C^n(X)$ given by $(f^*\varphi)(c) = \varphi(f_*c)$ for any $n$-chain $c$ in $X$
• The pullback operation $f^*$ is a cochain map, meaning it commutes with the coboundary operators: $f^* \circ \delta_Y = \delta_X \circ f^*$
• Pullbacks allow us to transfer cochains from the codomain $Y$ to the domain $X$ of the map $f$, providing a way to relate the cohomology groups of these spaces

Pushforwards of cochains

• In some cases, we may also consider the pushforward of cochains, which transfers cochains from the domain $X$ to the codomain $Y$ of a map $f: X \to Y$
• The pushforward operation is more subtle and requires additional conditions on the map $f$, such as being a proper map or a fiber bundle projection
• Pushforwards are particularly useful in the study of cohomology of fiber bundles and the Gysin homomorphism

Cup product and induced homomorphisms

• Induced homomorphisms are compatible with the cup product structure on cohomology
• Given a continuous map $f: X \to Y$ and cochains $\varphi \in C^n(Y)$ and $\psi \in C^m(Y)$, we have the following compatibility relation: $f^*(\varphi \smile \psi) = f^*\varphi \smile f^*\psi$, where $\smile$ denotes the cup product
• This compatibility allows us to study the behavior of cohomology rings under continuous maps and provides a powerful tool for computing cohomology in various situations

Properties of induced homomorphisms

• Induced homomorphisms satisfy several important properties that make them a fundamental tool in cohomology theory
• These properties include compatibility with long exact sequences, induced maps on relative cohomology, and the Mayer-Vietoris sequence
• Understanding these properties is crucial for applying induced homomorphisms to solve problems and compute cohomology in various settings

Compatibility with long exact sequences

• Induced homomorphisms are compatible with the long exact sequences associated with cohomology
• Given a continuous map $f: X \to Y$ and a pair $(Y, A)$ of spaces, we have a commutative diagram relating the long exact sequences of $(Y, A)$ and $(X, f^{-1}(A))$ via the induced maps $f^*$
• This compatibility allows us to study the behavior of cohomology under continuous maps and provides a powerful tool for computing cohomology using long exact sequences

Induced maps on relative cohomology

• Induced homomorphisms can also be defined for relative cohomology groups
• Given a continuous map $f: (X, A) \to (Y, B)$ between pairs of spaces, we have induced homomorphisms $f^*: H^n(Y, B) \to H^n(X, A)$ on the relative cohomology groups
• These induced maps on relative cohomology are compatible with the long exact sequences associated with the pairs $(X, A)$ and $(Y, B)$, providing a way to relate the relative cohomology of different spaces

Induced maps and the Mayer-Vietoris sequence

• Induced homomorphisms play a crucial role in the Mayer-Vietoris sequence, which relates the cohomology of a space to the cohomology of its subspaces
• Given a continuous map $f: X \to Y$ and an open cover $\{U, V\}$ of $Y$, we have a commutative diagram relating the Mayer-Vietoris sequences of $\{U, V\}$ and $\{f^{-1}(U), f^{-1}(V)\}$ via the induced maps $f^*$
• This compatibility allows us to use the Mayer-Vietoris sequence to compute the cohomology of spaces by breaking them down into simpler subspaces and studying the induced maps between them

Applications of induced homomorphisms

• Induced homomorphisms have numerous applications in algebraic topology and related fields
• They are used to study the invariance of cohomology under homotopy equivalence, compute the cohomology of fiber bundles, and understand characteristic classes
• These applications demonstrate the power and versatility of induced homomorphisms in solving problems and understanding the topology of spaces

Invariance of cohomology under homotopy equivalence

• One of the most important applications of induced homomorphisms is in proving the invariance of cohomology under homotopy equivalence
• If two spaces $X$ and $Y$ are homotopy equivalent, then there exist continuous maps $f: X \to Y$ and $g: Y \to X$ such that $f \circ g$ and $g \circ f$ are homotopic to the identity maps on $Y$ and $X$, respectively
• Using the induced homomorphisms $f^*$ and $g^*$, we can show that the cohomology groups of $X$ and $Y$ are isomorphic, establishing the invariance of cohomology under homotopy equivalence

Cohomology of fiber bundles

• Induced homomorphisms play a crucial role in computing the cohomology of fiber bundles
• Given a fiber bundle $\pi: E \to B$ with fiber $F$, we can use the induced homomorphisms associated with the projection map $\pi$ to relate the cohomology of the total space $E$ to the cohomology of the base space $B$ and the fiber $F$
• The Leray-Serre spectral sequence, which is a powerful tool for computing the cohomology of fiber bundles, relies heavily on the properties of induced homomorphisms

Characteristic classes and induced homomorphisms

• Characteristic classes, such as Chern classes and Pontryagin classes, are cohomological invariants that provide important information about vector bundles and manifolds
• Induced homomorphisms are used to study the behavior of characteristic classes under continuous maps and to prove important properties, such as the naturality of characteristic classes
• The compatibility of induced homomorphisms with the cup product is crucial for understanding the ring structure of characteristic classes and their applications in geometry and topology

Examples and computations

• To illustrate the power and utility of induced homomorphisms, it is helpful to consider specific examples and computations in various cohomology theories
• These examples demonstrate how induced homomorphisms can be used to solve problems, compute cohomology groups, and understand the relationships between different spaces
• Some common cohomology theories where induced homomorphisms play a significant role include singular cohomology, de Rham cohomology, and Čech cohomology

Induced maps for singular cohomology

• In singular cohomology, induced homomorphisms are defined using the pullback of cochains
• Given a continuous map $f: X \to Y$, the induced map $f^*: H^n(Y) \to H^n(X)$ is defined by pulling back cochains from $Y$ to $X$ using the map $f$
• Singular cohomology provides a general framework for studying the cohomology of topological spaces and is particularly useful for spaces that are not necessarily manifolds or CW complexes

Induced maps for de Rham cohomology

• In de Rham cohomology, which is defined for smooth manifolds, induced homomorphisms are defined using the pullback of differential forms
• Given a smooth map $f: M \to N$ between manifolds, the induced map $f^*: H^n_{dR}(N) \to H^n_{dR}(M)$ is defined by pulling back differential forms from $N$ to $M$ using the map $f$
• De Rham cohomology is particularly useful for studying the topology of smooth manifolds and has important connections to differential geometry and physics

Induced maps in Čech cohomology

• Čech cohomology is another cohomology theory that is particularly useful for studying sheaves and vector bundles
• In Čech cohomology, induced homomorphisms are defined using the pullback of Čech cochains associated with open covers
• Given a continuous map $f: X \to Y$ and an open cover $\mathcal{U}$ of $Y$, the induced map $f^*: \check{H}^n(Y, \mathcal{U}) \to \check{H}^n(X, f^{-1}\mathcal{U})$ is defined by pulling back Čech cochains from $Y$ to $X$ using the map $f$

Induced homomorphisms in advanced topics

• Induced homomorphisms play a significant role in various advanced topics in algebraic topology and related fields
• These topics include equivariant cohomology, spectral sequences, and Steenrod operations
• Understanding the behavior of induced homomorphisms in these contexts is crucial for solving complex problems and uncovering deeper connections between different areas of mathematics

Equivariant cohomology and induced maps

• Equivariant cohomology is a generalization of ordinary cohomology that takes into account the action of a group on a space
• In equivariant cohomology, induced homomorphisms are defined for equivariant maps, which are continuous maps that are compatible with the group action
• The properties of induced homomorphisms in equivariant cohomology are similar to those in ordinary cohomology, but with additional compatibility conditions related to the group action

Induced maps in spectral sequences

• Spectral sequences are powerful algebraic tools that allow us to compute cohomology groups by successive approximations
• Induced homomorphisms play a crucial role in the construction and convergence of spectral sequences
• In particular, the induced maps between the pages of a spectral sequence are essential for understanding the relationships between different cohomology groups and for computing the limit of the spectral sequence

Steenrod operations and induced homomorphisms

• Steenrod operations are a family of cohomology operations that provide additional structure on the cohomology rings of spaces
• Induced homomorphisms are compatible with Steenrod operations, meaning that they commute with the action of these operations on cohomology
• The compatibility of induced homomorphisms with Steenrod operations is crucial for studying the behavior of cohomology under continuous maps and for understanding the deeper algebraic structure of cohomology rings