connects 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 extends 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
Top images from around the web for Relationship to Poincaré duality
algebraic topology - Homology Poincare Homology Sphere by Mayer-Vietoris - Mathematics Stack ... View original
Is this image relevant?
1 of 3
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 [Sn](https://www.fiveableKeyTerm:sn), there is an isomorphism H~i(K)≅H~n−i−1(Sn∖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 Sn, the duality relates the topology of K to that of its complement Sn∖K
Examples of compact subsets include closed balls, finite sets of points, and knots embedded in S3
Duality between reduced homology and cohomology
For a compact subset K of a sphere Sn, Alexander duality establishes an isomorphism between the reduced homology groups of K and the reduced cohomology groups of its complement Sn∖K
The isomorphism is given by H~i(K)≅H~n−i−1(Sn∖K), where H~i denotes the i-th reduced homology group and 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 (Sn∖A,Sn∖X)
The isomorphism is given by Hi(X,A)≅Hn−i(Sn∖A,Sn∖X), where Hi denotes the i-th relative homology group and Hn−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∖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∖A) and the cohomology groups of A with coefficients in the orientation sheaf of M
The isomorphism is given by Hi(M,M∖A)≅Hn−i(A;OM), where OM 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
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 K1 and K2 in S3, their linking number lk(K1,K2) can be computed using Alexander duality
The linking number is related to the intersection number of K1 with a surface bounded by K2, 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
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