S-duality
stationary duality, Spanier duality
A duality in homotopy theory which exists (in the absence of restrictions imposed on the dimensions of spaces) for the analogues of ordinary homotopy and cohomotopy groups in the suspension category — for the -homotopy and
-cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or
-category, is a category whose objects are topological spaces
, while its morphisms are classes
of
-homotopic mappings
from a
-fold suspension
into
,
and
being considered as
-homotopic if there exists an
such that the suspensions
and
are homotopic in the ordinary sense. The set
of such classes, which are known as
-mappings, constitutes an Abelian group with respect to the so-called track addition [1], [2], [4], [5]. The group
is the limit of the direct spectrum of the sets
of ordinary homotopy classes with suspension mappings as projections; if
is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism
in which the corresponding elements are represented by one and the same mapping
,
. The
-dual polyhedron of the polyhedron
in a sphere
is an arbitrary polyhedron
in
which is an
-deformation retract of the complement
, i.e. the morphism corresponding to the imbedding
is an
-equivalence. The polyhedron
exists for all
, and
may be considered as
.
For any polyhedra and any polyhedra
and
which are dual to them, there exists a unique mapping
![]() |
satisfying the following conditions:
a) It is an involutory contravariant functorial isomorphism, i.e. is a homomorphism such that if
![]() |
then
![]() |
if
![]() |
then
![]() |
if is an element of
or of
, then
.
b) The following relations are valid:
![]() |
where and
are considered as polyhedra,
-dual to polyhedra
and, correspondingly,
,
this means that it does not depend on
and is stationary with respect to suspension.
c) It satisfies the equation
![]() |
where
![]() |
and
![]() |
are homomorphisms of the above homology and cohomology groups, induced by -mappings
and
, and
![]() |
is an isomorphism obtained from the isomorphism of Alexander duality by replacing the set by its
-deformation retract
.
The construction of is based on the representation of a given mapping as the composition of an imbedding and an
-deformation retract.
The -homotopy group
of a space
is the group
, and the
-cohomotopy group
of
is the group
. As in ordinary homotopy theory, one defines the homomorphisms
![]() |
![]() |
Regarding the spheres and
as
-dual leads to the isomorphisms
![]() |
and to the commutative diagram
![]() |
Thus, the isomorphism connects
-homotopy and
-cohomotopy groups, just as the isomorphism of Alexander duality
connects the homology and cohomology groups. Any duality in the
-category entails a duality of ordinary homotopy classes if the conditions imposed on the space entail the existence of a one-to-one correspondence between the set of the above classes and the set of
-homotopy classes.
Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. converts one of these theorems into the other, which means that
-homotopy groups are replaced by
-cohomotopy groups, homology groups by cohomology groups, the mapping
by the mapping
, the smallest dimension with a non-trivial homology group by the largest dimension with a non-trivial cohomology group, and vice versa. In ordinary homotopy theory the definition of an
-cohomotopy group requires that the dimension of the space does not exceed
(or, more generally, that the space be
-coconnected,
), which impairs the perfectly general nature of duality.
There are several trends of generalization of the theory: e.g. studies are made of spaces with the -homotopy type of polyhedra, the relative case, a theory with supports, etc. [3], [5], , [7]. The theory was one of the starting points in the development of stationary homotopy theory [8].
References
[1] | E.H. Spanier, "Duality and ![]() |
[2] | E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" Mathematika , 2 : 3 (1955) pp. 56–80 |
[3] | E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" Ann. of Math. , 67 : 2 (1958) pp. 203–238 |
[4] | M.G. Barratt, "Track groups 1; 2" Proc. London Math. Soc. , 5 (1955) pp. 71–106; 285–329 |
[5] | E.H. Spanier, J.H.C. Whitehead, "The theory of carriers and ![]() |
[6a] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. Paris , 246 : 17 (1958) pp. 2444–2447 |
[6b] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. Paris , 246 : 18 (1958) pp. 2555–2558 |
[6c] | B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. Paris , 246 : 21 (1958) pp. 2991–2993 |
[6d] | B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. Paris , 247 : 6 (1958) pp. 620–623 |
[6e] | B. Eckmann, P.J. Hilton, "Décomposition homologique d'un polyhèdre simplement connexe" C.R. Acad. Sci. Paris , 248 : 14 (1959) pp. 2054–2056 |
[7] | E.H. Spanier, "Algebraic topology" , McGraw-Hill (1966) |
[8] | G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) |
S-duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=S-duality&oldid=23971