6.5 Alexander duality

Alexander duality connects homology and cohomology groups of compact subsets in spheres or manifolds to their complements. It's a powerful tool for computing topological invariants and understanding space structures in cohomology theory.

This duality extends Poincaré duality to compact subsets and their complements. It establishes an isomorphism between reduced homology groups of a compact subset and reduced cohomology groups of its complement, allowing for computation of one from the other.

Basics of Alexander duality

• Alexander duality establishes a relationship between the homology and cohomology groups of a compact subset of a sphere or manifold and its complement
• Provides a powerful tool for computing topological invariants and understanding the structure of topological spaces in Cohomology Theory

Relationship to Poincaré duality

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• Poincaré duality relates the homology and cohomology groups of a compact oriented manifold without boundary
• Alexander duality extends this idea to compact subsets of spheres or manifolds and their complements
• Both dualities highlight the deep connections between homology and cohomology in algebraic topology

Duality between homology and cohomology

• Alexander duality establishes an isomorphism between the reduced homology groups of a compact subset and the reduced cohomology groups of its complement
• Specifically, for a compact subset $K$ of a sphere $[S^n](https://www.fiveableKeyTerm:s^n)$, there is an isomorphism $\tilde{H}_i(K) \cong \tilde{H}^{n-i-1}(S^n \setminus K)$
• This duality allows for the computation of one set of invariants (homology or cohomology) from the other

Alexander duality for compact subsets

Compact subsets of spheres

• Alexander duality is most commonly stated for compact subsets of spheres
• Given a compact subset $K$ of a sphere $S^n$, the duality relates the topology of $K$ to that of its complement $S^n \setminus K$
• Examples of compact subsets include closed balls, finite sets of points, and knots embedded in $S^3$

Duality between reduced homology and cohomology

• For a compact subset $K$ of a sphere $S^n$, Alexander duality establishes an isomorphism between the reduced homology groups of $K$ and the reduced cohomology groups of its complement $S^n \setminus K$
• The isomorphism is given by $\tilde{H}_i(K) \cong \tilde{H}^{n-i-1}(S^n \setminus K)$, where $\tilde{H}_i$ denotes the $i$-th reduced homology group and $\tilde{H}^{n-i-1}$ denotes the $(n-i-1)$-th reduced cohomology group
• This duality allows for the computation of the homology or cohomology of a compact subset from the cohomology or homology of its complement, respectively

Relative version for compact pairs

• Alexander duality can be extended to compact pairs $(X, A)$, where $A$ is a closed subset of $X$
• In this case, the duality relates the relative homology groups of $(X, A)$ to the relative cohomology groups of $(S^n \setminus A, S^n \setminus X)$
• The isomorphism is given by $H_i(X, A) \cong H^{n-i}(S^n \setminus A, S^n \setminus X)$, where $H_i$ denotes the $i$-th relative homology group and $H^{n-i}$ denotes the $(n-i)$-th relative cohomology group

Alexander duality in manifolds

Generalizing to manifolds

• Alexander duality can be generalized to compact subsets of manifolds
• Given a compact subset $K$ of an $n$-dimensional manifold $M$, the duality relates the homology of $K$ to the cohomology of $M \setminus K$ with coefficients in the orientation sheaf of $M$
• This generalization allows for the application of Alexander duality to a wider range of topological spaces

Duality for closed subsets

• In the case of a closed subset $A$ of a manifold $M$, Alexander duality takes the form of an isomorphism between the relative homology groups of $(M, M \setminus A)$ and the cohomology groups of $A$ with coefficients in the orientation sheaf of $M$
• The isomorphism is given by $H_i(M, M \setminus A) \cong H^{n-i}(A; \mathcal{O}_M)$, where $\mathcal{O}_M$ denotes the orientation sheaf of $M$
• This version of the duality is particularly useful in the study of submanifolds and their complements

Relationship to Lefschetz duality

• Lefschetz duality is another important duality theorem in algebraic topology that relates the homology and cohomology of a compact oriented manifold with boundary
• Alexander duality can be seen as a special case of Lefschetz duality when the manifold is a sphere and the subset is compact
• Both dualities highlight the deep connections between homology, cohomology, and the topology of manifolds

Computational aspects and applications

Explicit formulas and calculations

• Alexander duality provides explicit formulas for computing the homology or cohomology of a compact subset from the cohomology or homology of its complement
• These formulas involve the connecting homomorphisms in the long exact sequences of pairs and the cap product with the fundamental class of the ambient sphere or manifold
• In practice, these calculations can be carried out using techniques from algebraic topology, such as the Mayer-Vietoris sequence and the cup product

Connection to linking numbers

• Alexander duality has a close connection to the concept of linking numbers in knot theory
• Given two disjoint oriented closed curves $K_1$ and $K_2$ in $S^3$, their linking number $\operatorname{lk}(K_1, K_2)$ can be computed using Alexander duality
• The linking number is related to the intersection number of $K_1$ with a surface bounded by $K_2$, which can be expressed in terms of the cup product and the duality isomorphism

Applications in knot theory and topology

• Alexander duality has numerous applications in knot theory and low-dimensional topology
• It can be used to compute the homology groups of knot complements, which are important invariants in knot theory
• The duality also plays a role in the study of link concordance and the construction of knot invariants such as the Alexander polynomial
• In topology, Alexander duality is used to study the relationships between subspaces of manifolds and their complements, as well as to compute topological invariants such as the Euler characteristic and the intersection form

Variants and generalizations

Alexander-Pontryagin duality

• Alexander-Pontryagin duality is a generalization of Alexander duality that applies to locally compact Hausdorff spaces
• It establishes a duality between the cohomology of a locally compact space with compact support and the homology of its one-point compactification
• This duality extends the ideas of Alexander duality to a broader class of topological spaces and provides a unified framework for studying duality in topology

Duality in extraordinary cohomology theories

• Alexander duality can be generalized to extraordinary cohomology theories, such as K-theory and cobordism theory
• In these settings, the duality relates the extraordinary cohomology groups of a compact subset to the extraordinary homology groups of its complement
• These generalizations provide powerful tools for studying the topology of manifolds and their subspaces using advanced algebraic and geometric techniques

Duality in algebraic topology vs algebraic geometry

• Duality is a central theme in both algebraic topology and algebraic geometry
• In algebraic topology, dualities like Alexander, Poincaré, and Lefschetz duality relate the homology and cohomology of spaces and manifolds
• In algebraic geometry, dualities such as Serre duality and Grothendieck duality relate the cohomology of algebraic varieties and sheaves
• While the underlying principles of duality are similar in both fields, the specific formulations and applications differ due to the distinct nature of the objects studied in each area