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 -theory" Bull. Amer. Math. Soc. , 62 (1956) pp. 194–203 MR0085506
[2]	E.H. Spanier, J.H.C. Whitehead, "Duality in homotopy theory" Mathematika , 2 : 3 (1955) pp. 56–80 MR0074823 Zbl 0064.17202
[3]	E.H. Spanier, J.H.C. Whitehead, "Duality in relative homotopy theory" Ann. of Math. , 67 : 2 (1958) pp. 203–238 MR0105105 Zbl 0092.15701
[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 -theory" , Algebraic geometry and Topology (A Symp. in honor of S. Lefschetz) , Princeton Univ. Press (1957) pp. 330–360 MR0084772
[6a]	B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Groupes absolus" C.R. Acad. Sci. Paris , 246 : 17 (1958) pp. 2444–2447 MR0100261 Zbl 0092.39901
[6b]	B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Suites exactes" C.R. Acad. Sci. Paris , 246 : 18 (1958) pp. 2555–2558 MR0100262 Zbl 0092.40001
[6c]	B. Eckmann, P.J. Hilton, "Groupes d'homotopie et dualité. Coefficients" C.R. Acad. Sci. Paris , 246 : 21 (1958) pp. 2991–2993 MR0100263 Zbl 0092.40101
[6d]	B. Eckmann, P.J. Hilton, "Transgression homotopique et cohomologique" C.R. Acad. Sci. Paris , 247 : 6 (1958) pp. 620–623 MR0100264 Zbl 0092.40102
[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) MR0210112 MR1325242 Zbl 0145.43303
[8]	G.W. Whitehead, "Recent advances in homotopy theory" , Amer. Math. Soc. (1970) MR0309097 Zbl 0217.48601