6.2 Gysin homomorphism and push-forward maps
4 min read•july 30, 2024
The and push-forward maps are key tools in cohomology and . They generalize integration along fibers and allow us to move between spaces of different dimensions, preserving important algebraic structures.
These concepts are crucial for understanding the Thom Isomorphism Theorem and its applications. They provide a way to relate of different spaces, enabling calculations and proofs in topology, geometry, and K-theory.
Gysin homomorphism and Thom isomorphism
Definition and relation to cohomology
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- The Gysin homomorphism is a map in cohomology associated to an oriented vector bundle, generalizing the notion of integration along the fiber in de Rham cohomology
- For an oriented vector bundle π: E → B of rank r, the Gysin homomorphism is a map π!: H^(E) → H^{-r}(B), where H^* denotes cohomology with coefficients in a ring R
- The Thom isomorphism is a canonical isomorphism φ: H^(B) → H^{+r}(E, E - B) for an oriented vector bundle π: E → B of rank r
- (E, E - B) denotes the relative cohomology of the pair
- The Gysin homomorphism π! is related to the Thom isomorphism φ by the composition π! = (PD_B)^(-1) ∘ φ ∘ PD_E
- PD_E and PD_B are the Poincaré duality isomorphisms for E and B, respectively
Properties and functoriality
- Push-forward maps in cohomology, also known as or , are associated to proper maps between oriented manifolds
- For a proper map f: X → Y between oriented manifolds of codimension r, the f_!: H^(X) → H^{-r}(Y) satisfies the property
- If g: Y → Z is another proper map, then (g ∘ f)! = g! ∘ f_!
- The push-forward map is compatible with the
- For classes α ∈ H^(Y) and β ∈ H^(X), we have f_!(f^(α) ∪ β) = α ∪ f_!(β), where f^ is the pull-back map in cohomology
- The push-forward map satisfies the
- For a vector bundle π: E → B and classes α ∈ H^(B) and β ∈ H^(E), we have π_!(π^*(α) ∪ β) = α ∪ π_!(β)
Computing the Gysin homomorphism
Specific vector bundles
- For the B × R^r → B, the Gysin homomorphism is the projection onto the cohomology of the base space B
- For the γ^1 over the CP^n, the Gysin homomorphism π!: H^(CP^n) → H^{-2}(CP^{n-1}) is given by π!(x^k) = x^{k-1} for k ≥ 1
- x is the generator of H^*(CP^n)
Embeddings and normal bundles
- For a i: X → Y of codimension r between , the Gysin homomorphism i!: H^(Y) → H^{-r}(X) is the composite of:
- The Thom isomorphism for the of X in Y
- The Poincaré duality isomorphism for X
Push-forward maps in cohomology
Definition and properties
- Push-forward maps in cohomology, also known as Gysin maps or umkehr maps, are associated to proper maps between oriented manifolds
- For a proper map f: X → Y between oriented manifolds of codimension r, the push-forward map f_!: H^(X) → H^{-r}(Y) satisfies the functoriality property
- If g: Y → Z is another proper map, then (g ∘ f)! = g! ∘ f_!
- The push-forward map is compatible with the cup product
- For classes α ∈ H^(Y) and β ∈ H^(X), we have f_!(f^(α) ∪ β) = α ∪ f_!(β), where f^ is the pull-back map in cohomology
Projection formula
- The push-forward map satisfies the projection formula
- For a vector bundle π: E → B and classes α ∈ H^(B) and β ∈ H^(E), we have π_!(π^*(α) ∪ β) = α ∪ π_!(β)
- The projection formula relates the push-forward, pull-back, and cup product operations in cohomology
- It allows for simplifying computations involving these operations
Applications of the Gysin homomorphism and push-forward maps
K-theory
- The Gysin homomorphism and push-forward maps can be defined for vector bundles in K-theory, using the Thom isomorphism in K-theory and the
- For a complex vector bundle π: E → B of rank r, the Gysin homomorphism in K-theory is a map π!: K(E) → K(B) that decreases the dimension by r
- It is related to the Gysin homomorphism in cohomology via the Chern character
- The Gysin homomorphism in K-theory satisfies analogues of the properties and functoriality of push-forward maps in cohomology
- This includes the projection formula and compatibility with the tensor product of vector bundles
Specific applications
- Applications of the Gysin homomorphism in K-theory include:
- Computing the K-theory of (projective spaces, )
- Studying the K-theory of
- Proving the Grothendieck-Riemann-Roch theorem, which relates the Chern character and push-forward maps in K-theory and cohomology
- The Gysin homomorphism and push-forward maps are powerful tools for computing invariants and understanding the structure of vector bundles and manifolds
Key Terms to Review (27)
Alexander Grothendieck: Alexander Grothendieck was a revolutionary French mathematician known for his significant contributions to algebraic geometry, homological algebra, and K-theory. His work fundamentally shaped modern mathematics, particularly through the development of the Grothendieck group and the insights into K-theory that link algebraic structures with topological concepts.
Algebraic Varieties: Algebraic varieties are geometric objects that are the solutions to systems of polynomial equations. They serve as a central concept in algebraic geometry, bridging the gap between algebra and geometry by providing a way to study the solutions of polynomial equations using geometric methods. They come in various forms, such as affine varieties, projective varieties, and more, allowing for a wide range of applications, including connections to cohomology and K-theory.
Chern character: The Chern character is an important topological invariant associated with complex vector bundles, which provides a connection between K-theory and cohomology. It captures information about the curvature of the vector bundle and its underlying geometric structure, serving as a bridge in various applications, from fixed point theorems to differential geometry.
Closed embedding: A closed embedding is a type of morphism in algebraic geometry that represents an inclusion of one algebraic variety into another, such that the image of the included variety is closed in the ambient variety. This concept is crucial for understanding how varieties can be related and how push-forward maps operate in the context of Gysin homomorphisms, where one often needs to consider the properties of varieties under these embeddings.
Cohomology classes: Cohomology classes are equivalence classes of cochains, which provide algebraic invariants that help describe the topological structure of a space. They capture information about the global properties of spaces through the dual nature of cohomology, linking it closely with concepts like homology and the Gysin homomorphism. Understanding these classes is essential for applying push-forward maps, as they allow for the transfer of cohomological information across different spaces.
Complex projective space: Complex projective space, denoted as $$ ext{CP}^n$$, is a fundamental space in algebraic geometry and topology that consists of lines through the origin in $$ ext{C}^{n+1}$$. It serves as a model for projective geometry and is instrumental in various mathematical areas, including Gysin homomorphism, push-forward maps, and K-Theory, where it helps in understanding the relationships between different cohomology theories and vector bundles over complex manifolds.
Cup product: The cup product is a fundamental operation in cohomology that combines two cohomology classes to produce a new cohomology class, playing a critical role in the algebraic topology of manifolds. It serves as a way to multiply cohomology classes, allowing for the exploration of various topological properties and structures. The cup product connects to important concepts such as Gysin homomorphisms and K-theory, as it helps in understanding how different classes interact under push-forward maps and contributes to the richness of both complex and real K-theory.
Exact Sequences: Exact sequences are sequences of algebraic objects and morphisms where the image of one morphism equals the kernel of the next. This concept is crucial in understanding how different spaces or structures interact with one another, highlighting relationships such as cohomology and homology. In various contexts, exact sequences can provide powerful tools for studying properties like K-theory and Gysin homomorphisms, as well as their connections to algebraic structures.
Flag varieties: Flag varieties are geometric structures that parameterize chains of subspaces within a vector space, capturing the essence of how these subspaces can be arranged in relation to one another. They play a crucial role in various areas of algebraic geometry and representation theory, providing a framework for studying vector bundles and their properties. The study of flag varieties connects directly to important concepts like Gysin homomorphisms, K-groups, and Bott periodicity, all of which utilize these structures to derive deeper insights into topological and algebraic properties.
Functoriality: Functoriality refers to the principle that relationships between mathematical structures can be preserved through functors, which are mappings between categories that respect the structures involved. This concept is essential in understanding how various K-Theories relate to each other and how different constructions or operations can yield consistent results across different contexts.
Grassmannians: Grassmannians are mathematical spaces that parameterize all possible k-dimensional subspaces of a vector space, typically denoted as $G(k, n)$ for k-dimensional subspaces of an n-dimensional space. They play a crucial role in various areas of mathematics, including topology and algebraic geometry, and serve as essential tools in understanding vector bundles and their relations to K-theory.
Gysin Homomorphism: The Gysin homomorphism is a fundamental concept in algebraic topology and K-theory that provides a way to relate the cohomology of a space to the cohomology of its submanifolds, particularly when dealing with fiber bundles and push-forward maps. This homomorphism captures how the inclusion of a submanifold affects the overall topological structure of the manifold, allowing mathematicians to translate geometric information into algebraic data.
Gysin maps: Gysin maps are homomorphisms in algebraic topology that arise from the integration along the fiber of a proper map between two manifolds. These maps play a crucial role in relating the cohomology of the base space to the cohomology of the total space, facilitating push-forward operations in K-theory. They help in understanding how certain topological features behave under projection, particularly when examining fiber bundles and their associated classes.
Index Theory: Index theory is a branch of mathematics that studies the relationship between the analytical properties of differential operators and topological invariants of manifolds. It provides a powerful tool for understanding how various geometric and topological aspects influence the behavior of solutions to differential equations, linking analysis, topology, and geometry.
Invariants in Algebraic Topology: Invariants in algebraic topology are properties of topological spaces that remain unchanged under continuous transformations, serving as tools to classify and distinguish between different spaces. These invariants help mathematicians understand the essential structure of a space, such as its shape, holes, and connectedness, even when the space is deformed. They are crucial for identifying equivalence classes of spaces and have significant implications in various areas of mathematics.
K-Theory: K-Theory is a branch of mathematics that studies vector bundles and their generalizations through the construction of K-groups, which provide a way to classify and understand vector bundles up to isomorphism. It connects various areas of mathematics, including topology, algebra, and geometry, offering insights into fixed point theorems, quantum field theory, and even string theory.
Michael Atiyah: Michael Atiyah was a prominent British mathematician known for his significant contributions to topology, geometry, and K-Theory. His work laid the groundwork for several important theories and concepts that link abstract mathematics to physical applications, especially in areas like quantum field theory and differential geometry.
Normal bundle: A normal bundle is a vector bundle that describes the directions in which one can move away from a submanifold within a larger manifold. It captures how the submanifold sits inside the ambient space, providing crucial information about its geometry and topology. The normal bundle plays an important role in various mathematical contexts, including the study of Gysin homomorphisms and push-forward maps as well as the foundational concepts of vector bundles.
Oriented cohomology: Oriented cohomology is a refined version of cohomology theory that takes into account the orientation of manifolds and other topological spaces. It allows for the definition of cohomology classes that are sensitive to the orientation, meaning it can distinguish between different orientations of the same underlying space. This is particularly important when studying phenomena like Gysin homomorphisms and push-forward maps, as these concepts rely on how one can relate the cohomology of a manifold to that of its submanifolds or images under continuous maps.
Projection Formula: The projection formula is a crucial concept in K-Theory that relates the Gysin homomorphism to push-forward maps when dealing with a proper map between spaces. It allows one to compute the push-forward of a cohomology class via the Gysin map, bridging the gap between cohomology classes on a manifold and its submanifolds. This formula is significant because it helps in understanding how topological properties can be transferred through mappings.
Projective bundles: Projective bundles are geometric constructions that associate a projective space to a vector bundle, allowing one to study the properties of the bundle by examining its points in a projective setting. This concept connects closely with the Gysin homomorphism and push-forward maps as it facilitates the computation of K-theory classes associated with vector bundles, while also demonstrating the functorial properties of K-theory through its interactions with these geometrical structures.
Proper Push-Forward: Proper push-forward is a mathematical operation that generalizes the concept of pushing forward classes from a space to another through a continuous map, specifically in the context of proper maps. This operation is significant in the study of Gysin homomorphisms and plays a vital role in the intersection theory and cohomology of manifolds, linking various spaces while preserving their topological features.
Push-forward map: A push-forward map is a mathematical tool used in algebraic topology that takes a cohomology class on one space and 'pushes it forward' to another space via a continuous function. This concept is vital in the study of Gysin homomorphisms, where it helps relate the cohomology of a manifold to that of its submanifolds through a proper morphism. The push-forward map facilitates understanding how topological properties are preserved or transformed across different spaces.
Smooth manifolds: Smooth manifolds are mathematical spaces that locally resemble Euclidean space and are equipped with a smooth structure, allowing for the differentiation of functions. They provide a setting where concepts from calculus can be applied in a more generalized context, enabling the study of geometric and topological properties. This concept is crucial for understanding how Gysin homomorphisms and push-forward maps operate in algebraic topology, particularly in the context of smooth mappings between manifolds.
Tautological line bundle: The tautological line bundle is a specific line bundle associated with projective spaces, where each fiber over a point in the base space is precisely the one-dimensional vector space spanned by that point. This concept connects to various operations in K-theory, especially regarding Gysin homomorphisms and push-forward maps, which relate cohomology classes across different spaces.
Trivial vector bundle: A trivial vector bundle is a type of vector bundle that is globally isomorphic to a product of a base space and a typical fiber. This means that it can be thought of as simply taking the base space and attaching a vector space to each point in that space without any twists or complicated structures. The concept of trivial vector bundles connects to the foundational ideas of vector bundles, how they behave under push-forward maps, and their classification in terms of continuity and smoothness.
Umkehr Maps: Umkehr maps are a type of map in K-Theory that relate the K-theory groups of a space to its subspaces through a process called push-forward. They can be viewed as dual to the Gysin homomorphisms and play a significant role in calculating K-theory for fiber bundles and related structures. These maps help facilitate the transfer of information about the cohomology of spaces, especially when considering bundles or complex projective spaces.