is a powerful tool in algebraic topology, connecting homology and cohomology groups. It generalizes to a broader class of spaces, including noncompact manifolds and singular spaces. This theorem is crucial for understanding topological properties of algebraic varieties and their complements.
The duality provides an between homology groups of a space and cohomology groups of its complement. For closed subspace A of space X, it states H_i(X, A; R) ≅ H^(n-i)(X \ A; R), where R is a coefficient ring and n is X's . This allows easier computation of homology groups using cohomology.
Lefschetz duality theorem
Fundamental result in algebraic topology establishes a relationship between homology and cohomology groups
Generalizes Poincaré duality to a broader class of spaces, including noncompact manifolds and singular spaces
Plays a crucial role in understanding the topological properties of algebraic varieties and their complements
Relationship between homology and cohomology
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Lefschetz duality provides an isomorphism between the homology groups of a space and the cohomology groups of its complement
Specifically, for a closed subspace A of a X, there is an isomorphism Hi(X,A;R)≅Hn−i(X∖A;R), where R is a ring of coefficients and n is the dimension of X
Allows for the computation of homology groups using cohomology, which is often easier to calculate
For nonsingular algebraic varieties
Lefschetz duality is particularly useful in the context of algebraic geometry, where it applies to nonsingular algebraic varieties
For a nonsingular projective variety X of dimension n over a field k, Lefschetz duality gives an isomorphism Hi(X;k)≅H2n−i(X;k)
Enables the study of the topology of algebraic varieties using cohomological techniques
Over field coefficients
Lefschetz duality is often stated over field coefficients, such as the complex numbers C or finite fields Fq
Working over fields simplifies the algebraic structure of homology and cohomology groups
Allows for the application of powerful tools from linear algebra and representation theory
Poincaré duality as special case
Poincaré duality is a special case of Lefschetz duality that applies to compact oriented manifolds without boundary
States that for a compact oriented manifold M of dimension n, there is an isomorphism Hi(M;R)≅Hn−i(M;R) for any ring of coefficients R
Provides a deep connection between the homology and cohomology of a manifold
For compact oriented manifolds
Poincaré duality holds for compact oriented manifolds, which are topological spaces locally homeomorphic to Euclidean space and have a consistent choice of orientation
Examples of compact oriented manifolds include spheres, tori, and projective spaces
Compact oriented manifolds have a [M]∈Hn(M;R), which plays a crucial role in the formulation of Poincaré duality
Without boundary
Poincaré duality applies to manifolds without boundary, meaning that the manifold has no edges or endpoints
The absence of a boundary simplifies the topological structure and allows for a clean statement of the duality isomorphism
For manifolds with boundary, a more general version called Poincaré-Lefschetz duality is used
Formulation using cup and cap products
Lefschetz duality can be formulated using the cup and cap products, which are fundamental operations in algebraic topology
The cup product is a bilinear map ⌣:Hi(X;R)×Hj(X;R)→Hi+j(X;R) that combines cohomology classes
The cap product is a bilinear map ⌢:Hi(X;R)×Hj(X;R)→Hi−j(X;R) that pairs homology and cohomology classes
Lefschetz duality isomorphism
The Lefschetz duality isomorphism can be expressed as a composition of the cup and cap products
For a closed subspace A of a topological space X, the isomorphism Hi(X,A;R)≅Hn−i(X∖A;R) is given by α↦α⌢[X], where [X] is the fundamental class of X
The inverse isomorphism is given by the cup product with the cohomology class dual to the fundamental class
Via cup product with fundamental class
The Lefschetz duality isomorphism can be described using the cup product with the fundamental class of the ambient space
For a nonsingular projective variety X of dimension n, the isomorphism Hi(X;k)≅H2n−i(X;k) is given by α↦α⌣[X], where [X]∈H2n(X;k) is the fundamental class
The cup product with the fundamental class provides a concrete way to relate homology and cohomology classes
Cap product as dual to cup product
The cap product can be viewed as the dual operation to the cup product, in the sense that it pairs homology and cohomology classes
For a topological space X, the cap product satisfies the relation (α⌣β)⌢γ=α⌢(β⌢γ) for α∈Hi(X;R), β∈Hj(X;R), and γ∈Hk(X;R)
This duality between cup and cap products is essential in the formulation and proof of Lefschetz duality
Proof of Lefschetz duality
The proof of Lefschetz duality relies on several key techniques and results in algebraic topology
Involves the use of long exact sequences, excision, and the properties of cup and cap products
Can be approached from different perspectives, depending on the specific setting and assumptions
Using Poincaré-Lefschetz duality
One approach to proving Lefschetz duality is to use Poincaré-Lefschetz duality, which is a generalization of Poincaré duality to compact manifolds with boundary
Poincaré-Lefschetz duality states that for a compact oriented manifold M with boundary ∂M, there is an isomorphism Hi(M,∂M;R)≅Hn−i(M;R), where n is the dimension of M
By applying Poincaré-Lefschetz duality to a suitable compactification of the space and its complement, one can deduce Lefschetz duality
For compact manifolds with boundary
Poincaré-Lefschetz duality is particularly useful for proving Lefschetz duality in the case of compact manifolds with boundary
By considering a compact manifold M with boundary ∂M and its complement X∖M, one can establish the Lefschetz duality isomorphism Hi(X,M;R)≅Hn−i(X∖M;R)
The proof involves relating the homology of the manifold with boundary to the cohomology of its interior using excision and long exact sequences
Excision and long exact sequences
Excision is a fundamental property in algebraic topology that allows for the computation of homology groups by cutting out a subspace and considering its complement
Long exact sequences are powerful tools that relate the homology or cohomology groups of a space, a subspace, and their relative versions
In the proof of Lefschetz duality, excision and long exact sequences are used to relate the homology of a compact manifold with boundary to the cohomology of its complement
Applications and examples
Lefschetz duality has numerous applications in various areas of mathematics, particularly in algebraic geometry and topology
Provides a powerful tool for computing homology and cohomology groups of spaces and their complements
Leads to important results and insights in the study of algebraic varieties and their topological properties
In algebraic geometry
Lefschetz duality is extensively used in algebraic geometry to study the topology of algebraic varieties
Allows for the computation of the homology and cohomology groups of projective varieties and their complements
Plays a crucial role in the study of the geometry of algebraic curves, surfaces, and higher-dimensional varieties
Computing cohomology of projective spaces
Lefschetz duality can be used to compute the cohomology groups of projective spaces, which are fundamental examples of algebraic varieties
For the complex projective space CPn, Lefschetz duality gives an isomorphism Hi(CPn;C)≅H2n−i(CPn;C)
Combined with the cellular decomposition of projective spaces, this allows for the explicit computation of their cohomology groups
Lefschetz hyperplane theorem
The is a classic result in algebraic geometry that relies on Lefschetz duality
States that for a smooth projective variety X of dimension n and a hyperplane H⊂X, the inclusion map H↪X induces an isomorphism Hi(H;C)≅Hi(X;C) for i<n−1 and a surjection for i=n−1
Provides a powerful tool for understanding the topology of a variety by studying its hyperplane sections
Generalization to singular varieties
Lefschetz duality can be generalized to singular varieties, which are algebraic varieties that may have singularities or non-smooth points
The generalization requires the use of more advanced tools and concepts from homology and sheaf theory
Allows for the study of the topology of singular spaces and their relationship with their smooth counterparts
Borel-Moore homology
is a homology theory designed to work well with non-compact and singular spaces
Defined using locally finite chains, which are infinite chains with certain finiteness conditions
Allows for the formulation of a version of Lefschetz duality for singular varieties
Verdier duality
is a far-reaching generalization of Poincaré duality and Lefschetz duality to the setting of sheaves on topological spaces
Relates the cohomology of a sheaf to the homology of its dual sheaf, providing a powerful framework for studying the topology of singular spaces
Plays a central role in the modern approach to intersection cohomology and the study of perverse sheaves
Relationship with intersection cohomology
Intersection cohomology is a cohomology theory designed to capture the topological information of singular spaces while satisfying certain desirable properties, such as Poincaré duality
Lefschetz duality and Verdier duality provide the foundation for the development of intersection cohomology
The generalized Lefschetz duality for intersection cohomology allows for the study of the topology of singular varieties and their relationship with their smooth counterparts
Lefschetz duality in sheaf theory
Sheaf theory provides a powerful language for formulating and studying Lefschetz duality in a more general and abstract setting
Allows for the treatment of Lefschetz duality for sheaves and complexes of sheaves, which encompass a wide range of topological and algebraic objects
Provides a unifying framework for understanding the relationship between homology and cohomology on various spaces
Verdier duality for constructible sheaves
Verdier duality is particularly important for constructible sheaves, which are sheaves that exhibit certain finiteness and regularity properties
For a locally compact space X and a constructible sheaf F on X, Verdier duality provides an isomorphism D(F)≅RHom(F,ωX), where D is the duality functor and ωX is the dualizing complex
This isomorphism generalizes Lefschetz duality to the setting of sheaves and allows for the study of the topology of singular spaces
On locally compact spaces
Lefschetz duality in sheaf theory is often formulated for locally compact spaces, which include a wide range of topological spaces encountered in practice
Locally compact spaces, such as algebraic varieties and manifolds, provide a natural setting for studying sheaves and their cohomology
The local compactness condition ensures that the sheaf-theoretic machinery, such as the duality functor and the dualizing complex, is well-behaved
Compatibility with six operations
Lefschetz duality in sheaf theory is compatible with the six operations of Grothendieck, which are fundamental functors in the theory of sheaves and derived categories
The six operations (⊗,Hom,f∗,f∗,f!,f!) provide a powerful toolkit for manipulating sheaves and studying their relationships
The compatibility of Lefschetz duality with these operations allows for the derivation of various duality statements and the study of the functorial properties of homology and cohomology