In
mathematics, Spanier–Whitehead duality is a
duality theory
In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
in
homotopy theory, based on a geometrical idea that a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
''X'' may be considered as dual to its complement in the ''n''-
sphere
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is th ...
, where ''n'' is large enough. Its origins lie in
Alexander duality theory, in
homology theory, concerning complements in
manifolds. 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 theoret ...
of
string theory. It is named for
Edwin Spanier and
J. H. C. Whitehead, who developed it in papers from 1955.
The basic point is that sphere complements determine the homology, but not the
homotopy type
In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a deforma ...
, 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.
Statement
Let ''X'' be a compact
neighborhood retract in
. Then
and
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 objects in arbitrary monoidal categories. It is only a partial generalization, based upon the categorical properties of dual ...
s in the category of pointed
spectra with the smash product as a monoidal structure. Here
is the union of
and a point,
and
are reduced and unreduced
suspension
Suspension or suspended may refer to:
Science and engineering
* Suspension (topology), in mathematics
* Suspension (dynamical systems), in mathematics
* Suspension of a ring, in mathematics
* Suspension (chemistry), small solid particles suspende ...
s respectively.
Taking homology and cohomology with respect to an
Eilenberg–MacLane spectrum In mathematics, specifically algebraic topology, there is a distinguished class of spectra called Eilenberg–Maclane spectra HA for any Abelian group Apg 134. Note, this construction can be generalized to commutative rings R as well from its under ...
recovers
Alexander duality formally.
References
*
*
*
{{DEFAULTSORT:Spanier-Whitehead Duality
Homotopy theory
Duality theories