mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Spanier–Whitehead duality is a duality theory in

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

, based on a geometrical idea that a topological space ''X'' may be considered as dual to its complement in the ''n''- sphere, where ''n'' is large enough. Its origins lie in

Alexander duality 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 Pavel Alexandrov and Lev Pontryagin. It applies to the homology theory properties of ...

theory, in

homology theory In mathematics, homology is a general way of associating a sequence of algebraic objects, such as abelian groups or modules, with other mathematical objects such as topological spaces. Homology groups were originally defined in algebraic topolog ...

, 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 of

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 interac ...

. It is named for

Edwin Spanier Edwin Henry Spanier (August 8, 1921 – October 11, 1996) was an American mathematician at the University of California at Berkeley, working in algebraic topology. He co-invented Spanier–Whitehead duality and Alexander–Spanier cohomology, ...

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 Princeton, ...

, who developed it in papers from 1955. The basic point is that sphere complements determine the homology, but not the homotopy type, in general. What is determined, however, is the

stable homotopy type A stable is a building in which livestock, especially horses, are kept. It most commonly means a building that is divided into separate stalls for individual animals and livestock. There are many different types of stables in use today; the ...

, 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 F ...

Statement

Let ''X'' be a compact

neighborhood retract In topology, a branch of mathematics, a retraction is a continuous mapping from a topological space into a subspace that preserves the position of all points in that subspace. The subspace is then called a retract of the original space. A deformat ...

\R^n

. Then

X^+

and

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

are dual objects 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

formally.

References

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