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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. James Waddell Alexander II ( September 19, 1888 – September 23, 1971) was an important Topologist of the pre-WWII era and part of Pavel Sergeyevich Alexandrov (Па́вел Серге́евич Алекса́ндров sometimes romanized Aleksandroff or Aleksandrov ( November 16 Lev Semenovich Pontryagin ( Russian Лев Семёнович Понтрягин ( 3 September 1908 &ndash 3 May 1988) was a It applies to the homology theory properties of the complement of a subspace X in Euclidean space, a sphere, or other manifold. In Mathematics, homology theory is the Axiomatic study of the intuitive geometric idea of homology of cycles on Topological spaces It can be broadly "Globose" redirects here See also Globose nucleus. A sphere (from Greek σφαίρα - sphaira, "globe
Modern statement
Let X be a compact, locally contractible subspace of Euclidean space E of dimension n. In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i Let Y be the complement of X in E. Then if H stands for reduced homology or reduced cohomology, with coefficients in a given abelian group, there is an isomorphism between
- Hq(X)
and
- Hn − q − 1(Y). In Mathematics, reduced homology is a minor modification made to Homology theory in Algebraic topology, designed to make a point have all its 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 An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the
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. Čech cohomology is a particular type of Cohomology in Mathematics.
Alexander's 1915 result
The statement above is from Spanier, Algebraic Topology (p. 296). To go back to Alexander's original work, it is assumed first that X is a simplicial complex, and secondly that complements are taken in the n-sphere, i. In Mathematics, a simplicial complex is a Topological space of a particular kind constructed by "gluing together" points Line segments e. the one-point compactification of E. (Taking out one point from the complement of the compact set makes no difference to the homotopy type, as long as we remove it far enough away from X. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical )
Alexander had little of the modern apparatus, and his result was only for the Betti numbers, with coefficients taken modulo 2. In Algebraic topology, the Betti number of a Topological space is in intuitive terms a way of counting the maximum number of cuts that can be made without dividing 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. In Geometric topology, the Clifford torus is a special kind of Torus sitting inside R 4 In Mathematics, a 3-sphere is a higher-dimensional analogue of a Sphere. In Mathematics, a solid torus is a Topological space Homeomorphic to S^1 \times D^2 i Each of the solid tori is from the homotopy point of view a circle. In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the 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, which topologically concerns the complement of a circle in the Riemann sphere. In Topology, the Jordan curve theorem states that every non-self-intersecting loop in the plane divides the plane into an "inside" and an "outside" Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of Circles are simple Shapes of Euclidean geometry consisting of those points in a plane which are at a constant Distance, called the In Mathematics, the Riemann sphere is a way of extending the plane of Complex numbers with one additional Point at infinity, in a way that 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). In Mathematics, a Topological space X is contractible if the Identity map on X is Null-homotopic, i In Mathematics, the Jordan–Schönflies theorem, or simply the Schönflies theorem, of Geometric topology is a sharpening of the Jordan curve theorem 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.
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