# 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 $S$- homotopy and $S$- cohomotopy groups or stationary homotopy and cohomotopy groups, forming extra-ordinary (generalized) homology and cohomology theories. A suspension category, or $S$- category, is a category whose objects are topological spaces $X$, while its morphisms are classes $\{ f \}$ of $S$- homotopic mappings $f$ from a $p$- fold suspension $S ^ { p } X _ { 1 }$ into $S ^ { p } X _ { 2 }$, $f$ and $g: S ^ { q } X _ { 1 } \rightarrow S ^ { q } X _ { 2 }$ being considered as $S$- homotopic if there exists an $r \geq \max { ( p, q) }$ such that the suspensions $S ^ { {r-p} } f$ and $S ^ { {r-q} } g$ are homotopic in the ordinary sense. The set $\{ X _ { 1 } , X _ { 2 } \}$ of such classes, which are known as $S$- mappings, constitutes an Abelian group with respect to the so-called track addition [1], [2], [4], [5]. The group $\{ X _ { 1 } , X _ { 2 } \}$ is the limit of the direct spectrum of the sets $[ S ^ { k } X _ { 1 } , S ^ { k } X _ { 2 } ]$ of ordinary homotopy classes with suspension mappings as projections; if $k$ is sufficiently large, it is a group spectrum with homomorphisms. There exists an isomorphism $S: \{ X _ { 1 } , X _ { 2 } \} \rightarrow \{ SX _ { 1 } , SX _ { 2 } \}$ in which the corresponding elements are represented by one and the same mapping $S ^ { p } X _ { 1 } \rightarrow S ^ { p } X _ { 2 }$, $p \geq 1$. The $n$- dual polyhedron of the polyhedron $X$ in a sphere $S ^ { n }$ is an arbitrary polyhedron $D _ { n } X$ in $S ^ { n }$ which is an $S$- deformation retract of the complement $S ^ { n } \setminus X$, i.e. the morphism corresponding to the imbedding $D _ { n } X \subset S ^ { n } \setminus X$ is an $S$- equivalence. The polyhedron $D _ { n } X$ exists for all $X$, and $X$ may be considered as $D _ { n } ^ { 2 } X$.

For any polyhedra $X _ { 1 } , X _ { 2 }$ and any polyhedra $D _ { n } X _ { 1 }$ and $D _ { n } X _ { 2 }$ which are dual to them, there exists a unique mapping

$$D _ { n } : \{ X _ { 1 } , X _ { 2 } \} \rightarrow \ \{ D _ { n } X _ { 2 } , D _ { n } X _ { 1 } \}$$

satisfying the following conditions:

a) It is an involutory contravariant functorial isomorphism, i.e. $D _ { n }$ is a homomorphism such that if

$$i : X _ { 1 } \subset X _ { 2 } \ \textrm{ and } \ i ^ \prime : D _ { n } X _ { 2 } \subset D _ { n } X _ { 1 } ,$$

then

$$D _ { n } \{ i \} = \{ i ^ \prime \} ;$$

if

$$\{ f _ { 1 } \} \in \{ X _ { 1 } , X _ { 2 } \} \ \textrm{ and } \ \ \{ f _ { 2 } \} \in \{ X _ { 2 } , X _ { 3 } \} ,$$

then

$$D _ { n } { ( \{ f _ { 2 } \} \cdot \{ f _ { 1 } \} ) } = \ D _ { n } \{ f _ { 1 } \} \cdot D _ { n } \{ f _ { 2 } \} ;$$

if $\theta$ is an element of $\{ X _ { 1 } , X _ { 2 } \}$ or of $\{ D _ { n } X _ { 2 } , D _ { n } X _ { 1 } \}$, then $D _ { n } D _ { n } \theta = \theta$.

b) The following relations are valid:

$$SD _ { n } = D _ { {n+1} } \ \textrm{ and } \ D _ { {n+1} } S = D _ { n } ,$$

where $SD _ { n } X _ { i }$ and $D _ { n } X _ { i }$ are considered as polyhedra, ${ ( {n+1} ) }$- dual to polyhedra $X _ { i }$ and, correspondingly, $SX _ { i }$, $i = 1, 2;$ this means that it does not depend on $n$ and is stationary with respect to suspension.

c) It satisfies the equation

$$D _ { a } ^ { n } \theta _ { * } = { ( D _ { n } \theta ) } ^ { * } D _ { a } ^ { n } ,$$

where

$$\theta _ { * } : H _ { p } { ( X _ { 1 } ) } \rightarrow H _ { p } { ( X _ { 2 } ) }$$

and

$${ ( D _ { n } \theta ) } ^ { * } : H ^ { { {n-p} -1} } { ( D _ { n } X _ { 1 } ) } \rightarrow H ^ { { {n-p} -1} } { ( D _ { n } X _ { 2 } ) }$$

are homomorphisms of the above homology and cohomology groups, induced by $S$- mappings $\theta \in \{ X _ { 1 } , X _ { 2 } \}$ and $D _ { n } \theta$, and

$$D _ { a } : H _ { p } { ( X _ { i } ) } \rightarrow H ^ { { {n-p} -1} } { ( D _ { n } X _ { i } ) } ,\ {i=1} , 2 ,$$

is an isomorphism obtained from the isomorphism of Alexander duality by replacing the set $S ^ { n } \setminus X _ { i }$ by its $S$- deformation retract $D _ { n } X _ { i }$.

The construction of $D _ { n }$ is based on the representation of a given mapping as the composition of an imbedding and an $S$- deformation retract.

The $S$- homotopy group $\Sigma _ { p } { ( X) }$ of a space $X$ is the group $\{ S ^ { p } , X \}$, and the $S$- cohomotopy group $\Sigma ^ { p } { ( X) }$ of $X$ is the group $\{ X, S ^ { p } \}$. As in ordinary homotopy theory, one defines the homomorphisms

$$\phi _ { p } : \Sigma _ { p } { ( X) } \rightarrow H _ { p } { ( X) } ,$$

$$\phi ^ { p } : \Sigma ^ { p } { ( X) } \rightarrow H ^ { p } { ( X) } .$$

Regarding the spheres $S ^ { p }$ and $S ^ { { {n-p} -1} }$ as $n$- dual leads to the isomorphisms

$$D _ { n } : \Sigma _ { p } { ( X) } \rightarrow \Sigma ^ { { {n-p} -1} } { ( D _ { n } X) }$$

and to the commutative diagram

$$\begin{array}{ccc} {\Sigma _ { p } { ( X) } } & \stackrel{ \phi _ p }{\rightarrow} &{H _ { p } { ( X) } } \\ { { {D _ { n } } } \downarrow } &{} &{\downarrow { {D _ { a } ^ { n } } } } \\ {\Sigma ^ { { {n-p} -1} } { ( D _ { n } ^ { X } ) } } & \stackrel{\phi ^{n-p-1}}{\rightarrow} &{H ^ { { {n-p} -1} } { ( D _ { n } X) } } \\ \end{array}$$

Thus, the isomorphism $D _ { n }$ connects $S$- homotopy and $S$- cohomotopy groups, just as the isomorphism of Alexander duality $D _ { a } ^ { n }$ connects the homology and cohomology groups. Any duality in the $S$- 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 $S$- homotopy classes.

Examples of dual assumptions in this theory include Hurewicz's isomorphism theorem and Hopf's classification theorem. $D _ { n }$ converts one of these theorems into the other, which means that $S$- homotopy groups are replaced by $S$- cohomotopy groups, homology groups by cohomology groups, the mapping $\phi _ { p }$ by the mapping $\phi ^ { { {n-p} -1} }$, 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 $n$- cohomotopy group requires that the dimension of the space does not exceed ${2n-2}$( or, more generally, that the space be ${ { ( 2n-1) }}$- coconnected, $n > 1$), 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 $S$- 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
How to Cite This Entry:
S-duality. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=S-duality&oldid=49677
This article was adapted from an original article by G.S. Chogoshvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article