mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, Spanier–Whitehead duality is a duality theory in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which Map (mathematics), maps can come with homotopy, homotopies between them. It originated as a topic in algebraic topology, but nowadays is learned as an independent discipli ...

, based on a geometrical idea that a

topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...

''X'' may be considered as dual to its complement in the ''n''-

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

, where ''n'' is large enough. Its origins lie in Alexander duality theory, in

homology theory In mathematics, the term homology, originally introduced in algebraic topology, has three primary, closely-related usages. The most direct usage of the term is to take the ''homology of a chain complex'', resulting in a sequence of abelian grou ...

, concerning complements in

manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a N ...

s. The theory is also referred to as ''S-duality'', but this can now cause possible confusion with the

S-duality In theoretical physics, S-duality (short for strong–weak duality, or Sen duality) is an equivalence of two physical theories, which may be either quantum field theories or string theories. S-duality is useful for doing calculations in theore ...

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

. It is named for Edwin Spanier and

J. H. C. Whitehead John Henry Constantine Whitehead FRS (11 November 1904 – 8 May 1960), known as "Henry", was a British mathematician and was one of the founders of homotopy theory. He was born in Chennai (then known as Madras), in India, and died in Princet ...

, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the

homotopy type In topology, two continuous functions from one topological space to another are called homotopic (from and ) if one can be "continuously deformed" into the other, such a deformation being called a homotopy ( ; ) between the two functions. A ...

, in general. What is determined, however, is the stable homotopy type, which was conceived as a first approximation to homotopy type. Thus Spanier–Whitehead duality fits into

stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the ...

Statement

Let ''X'' be a compact neighborhood retract in

\R^n

. Then

X^+

and

\Sigma^\Sigma'(\R^n \setminus X)

are

dual object In category theory, a branch of mathematics, a dual object is an analogue of a dual vector space from linear algebra for Object (category theory), objects in arbitrary Monoidal category, monoidal categories. It is only a partial generalization, base ...

s in the category of pointed spectra with the smash product as a monoidal structure. Here

X^+

is the union of

X

and a point,

\Sigma

and

\Sigma'

are reduced and unreduced suspensions respectively. Taking homology and cohomology with respect to an Eilenberg–MacLane spectrum recovers Alexander duality formally.

References

* * * {{DEFAULTSORT:Spanier-Whitehead Duality Homotopy theory Duality theories