A duality between topological groups and their character groups (cf. Character group). The duality theorem states that if is a locally compact Abelian group and if is its character group, then the natural homomorphism mapping an to the character , given by the formula
is an isomorphism of topological groups. The following statements result from the above theorem.
I) If is a closed subgroup of and if
is its annihilator in , then coincides with the annihilator
of the subgroup ; moreover, the group is naturally isomorphic to , and is isomorphic to the group .
II) If is a continuous homomorphism of locally compact Abelian groups, and is identified with and with by the natural isomorphisms, then the homomorphism can be identified with .
III) The weight of the group (as a topological space, cf. Weight of a topological space) coincides with the weight of the group .
Pontryagin duality establishes a correspondence between compact groups and discrete groups , and vice versa. Moreover, a compact group is connected if and only if is torsion-free. A compact group is of dimension if and only if has finite rank (see Rank of a group). A compact group is locally connected if and only if every finite-rank pure subgroup of is free. For finite groups , Pontryagin duality coincides with duality between finite Abelian groups considered over the field of complex numbers.
Topological groups for which the duality theorem is valid are called reflexive. Locally compact groups are not the only reflexive groups, since any reflexive Banach space, regarded as a topological group, is reflexive . On the characterization of reflexive groups, see .
There is an analogue of Pontryagin duality for non-commutative groups (the duality theorem of Tannaka–Krein) (see , , ). Let be a compact topological group, let be the algebra of complex-valued functions on whose translates span a finite-dimensional vector space and let be the set of all non-zero algebra homomorphisms satisfying the condition , . One can define a multiplication on which makes into a topological group with respect to the topology of pointwise convergence. To each corresponds the homomorphism given by the formula
Then the correspondence is an isomorphism of the topological group onto . There is also an algebraic description of the category of algebras , which thus turns out to be dual to the category of compact topological groups. This theory admits a generalization to the case of homogeneous spaces of compact topological groups (see ).
|||L.S. Pontryagin, "The theory of topological commutative groups" Ann. of Math. , 35 : 2 (1934) pp. 361–388|
|||L.S. Pontryagin, "Topological groups" , Gordon & Breach (1966) (Translated from Russian)|
|||E. van Kampen, "Locally bicompact Abelian groups and their character groups" Ann. of Math. , 36 (1935) pp. 448–463|
|[4a]||M.G. Krein, "Hermitian positive kernels on homogeneous spaces, I" Ukrain. Mat. Zh. , 1 : 4 (1949) pp. 64–98 (In Russian)|
|[4b]||M.G. Krein, "Hermitian positive kernels on homogeneous spaces, II" Ukrain. Mat. Zh. , 2 : 1 (1950) pp. 10–59 (In Russian)|
|||S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977)|
|||M.A. Naimark, "Normed rings" , Reidel (1959) (Translated from Russian)|
|||E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1970)|
|||M.F. Smith, "The Pontrjagin duality theorem in linear spaces" Ann. of Math. , 56 : 2 (1952) pp. 248–253|
|||R. Venkataraman, "A characterization of Pontryagin duality" Math. Z. , 149 : 2 (1976) pp. 109–119|
A compact group is arcwise connected if and only if . A compact group is metrizable if and only if is countable.
|[a1]||D.L. Armacost, "The structure of locally compact abelian groups" , M. Dekker (1981)|
|[a2]||G. Hochschild, "The structure of Lie groups" , Holden-Day (1965)|
Pontryagin duality in topology is an isomorphism between a -dimensional Aleksandrov–Čech cohomology group , with coefficients in a group , of a compact set lying in an -dimensional compact orientable manifold and the -dimensional cohomology group of the complement , provided that (homology and cohomology in dimension zero are reduced; the symbol means compact support). In the case when or is a finite polyhedron, J.W. Alexander proved the existence of this isomorphism. N. Steenrod established such an isomorphism for an arbitrary open subset , and K.A. Sitnikov for an arbitrary subset .
In the form cited above the Pontryagin duality law was formulated by P.S. Aleksandrov. In the original version the duality was established in the sense of the theory of characters between the groups and , where is the compact character group of the discrete group . The equivalence of both versions of the duality law follows from the fact that the group is the character group of . Under the assumption that the manifold is acyclic in dimensions and , since the homology sequence of the pair is exact it follows that , thus Pontryagin duality is a simple corollary of Poincaré–Lefschetz duality (see Poincaré duality).
The most general form of the considered duality relations is as follows. Let be an arbitrary manifold (which may be generalized and need not be compact or orientable), let be a locally constant system of coefficients with fibres , let be an arbitrary subset of , and let be the family of closed sets of contained in . Then implies that . Here the are the homology functors with closed supports contained in (i.e. direct limits of the groups , ), and is the locally constant system of coefficients generated by the groups , . In the above equality the cohomology coefficients can be replaced by if one considers homology with coefficients in some specially defined system.
|||P.S. Aleksandrov, "Topological duality theorems" Trudy Mat. Inst. Steklov. , 48 (1955) pp. Part 1. Closed sets (In Russian)|
|||W.S. Massey, "Homology and cohomology theory" , M. Dekker (1978)|
|||E.G. Sklyarenko, "Homology and cohomology of general spaces" Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 50 (1989) pp. 129–266 (In Russian)|
Pontryagin duality. A.L. Onishchik (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Pontryagin_duality&oldid=19147