Alexander duality
Encyclopedia
In mathematics
, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander
, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin. It applies to the homology theory
properties of the complement of a subspace X in Euclidean space
, a sphere
, or other manifold.
, locally contractible subspace of the sphere S of dimension n. Let Y be the complement of X in S. Then if H stands for reduced homology
or reduced cohomology, with coefficients in a given abelian group
, there is an isomorphism between
and
Note that we can drop "local contractibility" as part of the hypothesis, if we use Čech cohomology
, which is designed to deal with local pathologies.
.
Alexander had little of the modern apparatus, and his result was only for the Betti number
s, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus
construction in the 3-sphere
shows that the complement of a solid torus
is another solid torus; which will be open if the other is closed, but this doesn't affect its homology. Each of the solid tori is from the homotopy
point of view a circle
. If we just write down the Betti numbers
of the circle (up to H3, since we are in the 3-sphere), then reverse as
and then shift one to the left to get
there is a difficulty, since we are not getting what we started with. On the other hand the same procedure applied to the reduced Betti numbers, for which the initial Betti number is decremented by 1, starts with
and gives
whence
This does work out, predicting the complement's reduced Betti numbers.
The prototype here is the Jordan curve theorem
, which topologically
concerns the complement of a circle
in the Riemann sphere
. It also tells the same story. We have the honest Betti numbers
of the circle, and therefore
by flipping over and
by shifting to the left. This gives back something different from what the Jordan theorem states, which is that there are two components, each contractible (Schoenflies theorem, to be accurate about what is used here). That is, the correct answer in honest Betti numbers is
Once more, it is the reduced Betti numbers that work out. With those, we begin with
to finish with
From these two examples, therefore, Alexander's formulation can be inferred: reduced Betti numbers b*i are related in complements by
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, Alexander duality refers to a duality theory presaged by a result of 1915 by J. W. Alexander
James Waddell Alexander II
James Waddell Alexander II was a mathematician and topologist of the pre-World War II era and part of an influential Princeton topology elite, which included Oswald Veblen, Solomon Lefschetz, and others...
, and subsequently further developed, particularly by P. S. Alexandrov and Lev Pontryagin. It applies to the homology theory
Homology theory
In mathematics, homology theory is the axiomatic study of the intuitive geometric idea of homology of cycles on topological spaces. It can be broadly defined as the study of homology theories on topological spaces.-The general idea:...
properties of the complement of a subspace X in Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, a sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
, or other manifold.
Modern statement
Let X be a compactCompact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, locally contractible subspace of the sphere S of dimension n. Let Y be the complement of X in S. Then if H stands for reduced homology
Reduced homology
In mathematics, reduced homology is a minor modification made to homology theory in algebraic topology, designed to make a point have all its homology groups zero...
or reduced cohomology, with coefficients in a given abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, there is an isomorphism between
- Hq(Y)
and
- Hn − q − 1(X).
Note that we can drop "local contractibility" as part of the hypothesis, if we use Čech cohomology
Cech cohomology
In mathematics, specifically algebraic topology, Čech cohomology is a cohomology theory based on the intersection properties of open covers of a topological space. It is named for the mathematician Eduard Čech.-Motivation:...
, which is designed to deal with local pathologies.
Alexander's 1915 result
To go back to Alexander's original work, it is assumed that X is a simplicial complexSimplicial complex
In mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their n-dimensional counterparts...
.
Alexander had little of the modern apparatus, and his result was only for the Betti number
Betti number
In algebraic topology, a mathematical discipline, the Betti numbers can be used to distinguish topological spaces. Intuitively, the first Betti number of a space counts the maximum number of cuts that can be made without dividing the space into two pieces....
s, with coefficients taken modulo 2. What to expect comes from examples. For example the Clifford torus
Clifford torus
In geometric topology, the Clifford torus is a special kind of torus sitting inside R4. Alternatively, it can be seen as a torus sitting inside C2 since C2 is topologically the same space as R4...
construction in the 3-sphere
3-sphere
In mathematics, a 3-sphere is a higher-dimensional analogue of a sphere. It consists of the set of points equidistant from a fixed central point in 4-dimensional Euclidean space...
shows that the complement of a solid torus
Solid torus
In mathematics, a solid torus is a topological space homeomorphic to S^1 \times D^2, i.e. the cartesian product of the circle with a two dimensional disc endowed with the product topology. The solid torus is a connected, compact, orientable 3-dimensional manifold with boundary...
is another solid torus; which will be open if the other is closed, but this doesn't affect its homology. Each of the solid tori is from the homotopy
Homotopy
In topology, two continuous functions from one topological space to another are called homotopic if one can be "continuously deformed" into the other, such a deformation being called a homotopy between the two functions...
point of view a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
. If we just write down the Betti numbers
- 1, 1, 0, 0
of the circle (up to H3, since we are in the 3-sphere), then reverse as
- 0, 0, 1, 1
and then shift one to the left to get
- 0, 1, 1, 0
there is a difficulty, since we are not getting what we started with. On the other hand the same procedure applied to the reduced Betti numbers, for which the initial Betti number is decremented by 1, starts with
- 0, 1, 0, 0
and gives
- 0, 0, 1, 0
whence
- 0, 1, 0, 0.
This does work out, predicting the complement's reduced Betti numbers.
The prototype here is the Jordan curve theorem
Jordan curve theorem
In topology, a Jordan curve is a non-self-intersecting continuous loop in the plane, and another name for a Jordan curve is a "simple closed curve"...
, which topologically
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
concerns the complement of a circle
Circle
A circle is a simple shape of Euclidean geometry consisting of those points in a plane that are a given distance from a given point, the centre. The distance between any of the points and the centre is called the radius....
in the Riemann sphere
Riemann sphere
In mathematics, the Riemann sphere , named after the 19th century mathematician Bernhard Riemann, is the sphere obtained from the complex plane by adding a point at infinity...
. It also tells the same story. We have the honest Betti numbers
- 1, 1, 0
of the circle, and therefore
- 0, 1, 1
by flipping over and
- 1, 1, 0
by shifting to the left. This gives back something different from what the Jordan theorem states, which is that there are two components, each contractible (Schoenflies theorem, to be accurate about what is used here). That is, the correct answer in honest Betti numbers is
- 2, 0, 0.
Once more, it is the reduced Betti numbers that work out. With those, we begin with
- 0, 1, 0
to finish with
- 1, 0, 0.
From these two examples, therefore, Alexander's formulation can be inferred: reduced Betti numbers b*i are related in complements by
- b*i → b*n − i − 1.
Further reading
- Ezra Miller, Bernd Sturmfels, Combinatorial Commutative Algebra (2005), Ch. 5 Alexander Duality