6.3 Poincaré duality

Poincaré duality is a key result in algebraic topology, linking homology and cohomology groups of oriented manifolds. It reveals deep connections between a manifold's topological structure and its algebraic invariants, enabling easier computation of cohomology groups.

The theorem requires oriented manifolds and uses concepts like cup and cap products. It has various formulations, including for de Rham cohomology, and applications in intersection theory and Morse theory. Generalizations extend its principles to other mathematical settings.

Poincaré duality theorem

• Fundamental result in algebraic topology establishes a relationship between the homology and cohomology groups of an oriented manifold
• Provides a deep connection between the topological structure of a manifold and its algebraic invariants
• Allows for the computation of cohomology groups in terms of homology groups, which are often easier to understand and calculate

Manifolds and orientability

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• Manifolds are topological spaces that locally resemble Euclidean space (e.g., the surface of a sphere or a torus)
• Orientability is a property of manifolds that allows for a consistent choice of orientation (e.g., a Möbius strip is non-orientable, while a cylinder is orientable)
• Poincaré duality requires the manifold to be oriented, which is necessary for the definition of the fundamental class

Homology and cohomology groups

• Homology groups $H_k(M)$ capture information about the $k$-dimensional "holes" in a manifold $M$ (e.g., $H_0$ counts connected components, $H_1$ counts loops, $H_2$ counts voids)
• Cohomology groups $H^k(M)$ are the dual of homology groups and capture information about the $k$-dimensional "cocycles" on a manifold
• Poincaré duality relates the homology and cohomology groups of an oriented manifold

Cup and cap products

• The cup product $\smile: H^k(M) \times H^l(M) \to H^{k+l}(M)$ is a bilinear operation on cohomology groups that combines cocycles
• The cap product $\frown: H_k(M) \times H^l(M) \to H_{k-l}(M)$ is a bilinear operation that pairs homology and cohomology classes
• Cup and cap products play a crucial role in the formulation and proof of Poincaré duality

Fundamental class of manifold

• The fundamental class $[M] \in H_n(M)$ is a generator of the top homology group of an oriented $n$-dimensional manifold $M$
• It represents the orientation of the manifold and is used to define the Poincaré duality isomorphism
• The existence and uniqueness (up to sign) of the fundamental class rely on the orientability of the manifold

Poincaré duality for de Rham cohomology

• De Rham cohomology is a cohomology theory based on differential forms, which are objects that generalize the notion of functions on a manifold
• Poincaré duality for de Rham cohomology relates the de Rham cohomology groups to the homology groups of a manifold
• This formulation of Poincaré duality is particularly useful in differential geometry and physics

De Rham cohomology vs singular cohomology

• Singular cohomology is defined using cochains, which are linear functionals on chains (formal sums of simplices)
• De Rham cohomology is defined using differential forms, which are antisymmetric multilinear functionals on tangent vectors
• On smooth manifolds, the de Rham theorem states that de Rham cohomology is isomorphic to singular cohomology

Integration of differential forms

• Integration of differential forms over chains provides a way to pair de Rham cohomology classes with homology classes
• The integration of an $n$-form over an oriented $n$-dimensional manifold yields a number, which is invariant under cohomologous forms and homologous chains
• This integration pairing is the key ingredient in the proof of Poincaré duality for de Rham cohomology

Hodge star operator and adjoint

• The Hodge star operator $\star: \Omega^k(M) \to \Omega^{n-k}(M)$ is a linear operator that maps $k$-forms to $(n-k)$-forms on an oriented $n$-dimensional manifold $M$
• It is defined using the Riemannian metric and the orientation of the manifold
• The Hodge star operator induces an isomorphism between the de Rham cohomology groups $H^k(M)$ and $H^{n-k}(M)$, which is the Poincaré duality isomorphism in this setting

Poincaré duality in various settings

• Poincaré duality can be formulated and proved in different settings, depending on the type of manifolds and cohomology theories considered
• These variations of Poincaré duality share the same underlying principle of relating homology and cohomology groups but differ in their technical details and applications

Smooth vs topological manifolds

• Smooth manifolds are topological manifolds equipped with a differentiable structure (i.e., a smooth atlas of charts)
• Poincaré duality for smooth manifolds can be formulated using de Rham cohomology, which relies on the differentiable structure
• For topological manifolds, Poincaré duality is formulated using singular homology and cohomology, which do not require a differentiable structure

Compact vs non-compact manifolds

• Poincaré duality holds for compact oriented manifolds without boundary
• For non-compact manifolds, Poincaré duality may fail or require modifications (e.g., using compactly supported cohomology)
• The compactness assumption ensures that the homology and cohomology groups are finitely generated and that the integration pairing is well-defined

Manifolds with vs without boundary

• Poincaré duality, as originally stated, applies to manifolds without boundary
• For manifolds with boundary, a modified version called Poincaré-Lefschetz duality holds, which relates the homology and cohomology groups of the manifold and its boundary
• The presence of a boundary introduces additional technical considerations and requires a careful treatment of the boundary conditions

Applications of Poincaré duality

• Poincaré duality has numerous applications in various areas of mathematics, including algebraic topology, differential geometry, and mathematical physics
• It provides a powerful tool for computing homology and cohomology groups, as well as for understanding the topological structure of manifolds

Intersection theory and cup product

• Poincaré duality allows for the definition of the intersection product on homology classes, which geometrically represents the intersection of submanifolds
• The intersection product is dual to the cup product on cohomology classes via the Poincaré duality isomorphism
• Intersection theory has applications in enumerative geometry, where it is used to count the number of solutions to geometric problems

Lefschetz fixed-point theorem

• The Lefschetz fixed-point theorem relates the fixed points of a continuous map on a manifold to the traces of the induced maps on homology groups
• Poincaré duality is used in the proof of the Lefschetz fixed-point theorem to relate the traces of the induced maps on homology and cohomology groups
• This theorem has applications in dynamical systems and the study of periodic orbits

Morse theory and critical points

• Morse theory studies the relationship between the topology of a manifold and the critical points of a smooth function defined on it
• Poincaré duality is used in Morse theory to relate the critical points of a function to the homology groups of the manifold
• This connection allows for the computation of homology groups using the critical points of a carefully chosen function (Morse function)

Generalizations of Poincaré duality

• Poincaré duality has been generalized and extended to various other settings beyond the realm of oriented manifolds
• These generalizations often involve more sophisticated algebraic and geometric tools, such as sheaves, derived categories, and stacks

Verdier duality in sheaf theory

• Verdier duality is a generalization of Poincaré duality in the context of sheaf theory, which is a framework for studying local-to-global properties of topological spaces
• It relates the derived functors of the direct and inverse image functors for sheaves on a locally compact space
• Verdier duality has applications in the study of perverse sheaves and the Riemann-Hilbert correspondence

Serre duality in complex geometry

• Serre duality is a generalization of Poincaré duality for complex manifolds, relating the cohomology groups of holomorphic vector bundles
• It states that the cohomology groups of a holomorphic vector bundle and its dual are related by a natural pairing
• Serre duality has applications in the study of moduli spaces of complex curves and the theory of abelian varieties

Poincaré-Lefschetz duality for manifolds with boundary

• Poincaré-Lefschetz duality is a generalization of Poincaré duality for manifolds with boundary, relating the homology and cohomology groups of the manifold and its boundary
• It involves the concept of relative homology and cohomology groups, which take into account the presence of the boundary
• Poincaré-Lefschetz duality has applications in the study of cobordisms and the classification of manifolds with boundary