6.4 Lefschetz duality

Lefschetz duality is a powerful tool in algebraic topology, connecting homology and cohomology groups. It generalizes Poincaré duality 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 isomorphism 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 dimension. 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 topological space $X$, there is an isomorphism $H_i(X, A; R) \cong H^{n-i}(X \setminus A; R)$, 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 $H_i(X; k) \cong H^{2n-i}(X; k)$
• 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 $\mathbb{C}$ or finite fields $\mathbb{F}_q$
• 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 $H_i(M; R) \cong H^{n-i}(M; R)$ 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 fundamental class $[M] \in H_n(M; R)$, 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 $\smile: H^i(X; R) \times H^j(X; R) \to H^{i+j}(X; R)$ that combines cohomology classes
• The cap product is a bilinear map $\frown: H_i(X; R) \times H^j(X; R) \to H_{i-j}(X; R)$ 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 $H_i(X, A; R) \cong H^{n-i}(X \setminus A; R)$ is given by $\alpha \mapsto \alpha \frown [X]$, 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 $H_i(X; k) \cong H^{2n-i}(X; k)$ is given by $\alpha \mapsto \alpha \smile [X]$, where $[X] \in H^{2n}(X; k)$ 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 $(\alpha \smile \beta) \frown \gamma = \alpha \frown (\beta \frown \gamma)$ for $\alpha \in H^i(X; R)$, $\beta \in H^j(X; R)$, and $\gamma \in H_k(X; R)$
• 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 $\partial M$, there is an isomorphism $H_i(M, \partial M; R) \cong H^{n-i}(M; R)$, 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 $\partial M$ and its complement $X \setminus M$, one can establish the Lefschetz duality isomorphism $H_i(X, M; R) \cong H^{n-i}(X \setminus M; R)$
• 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 $\mathbb{CP}^n$, Lefschetz duality gives an isomorphism $H_i(\mathbb{CP}^n; \mathbb{C}) \cong H^{2n-i}(\mathbb{CP}^n; \mathbb{C})$
• Combined with the cellular decomposition of projective spaces, this allows for the explicit computation of their cohomology groups

Lefschetz hyperplane theorem

• The Lefschetz hyperplane theorem 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 $H \subset X$, the inclusion map $H \hookrightarrow X$ induces an isomorphism $H_i(H; \mathbb{C}) \cong H_i(X; \mathbb{C})$ 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

• 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

• 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 $\mathcal{F}$ on $X$, Verdier duality provides an isomorphism $\mathbf{D}(\mathcal{F}) \cong \mathbf{R}\mathcal{H}om(\mathcal{F}, \omega_X)$, where $\mathbf{D}$ is the duality functor and $\omega_X$ 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 $(\otimes, \mathcal{H}om, f^*, f_*, f_!, f^!)$ 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