Pontryagin duality

A duality between topological groups and their character groups. The duality theorem states that if $ G $ is a locally compact Abelian group and if $ X ( G) $ is its character group, then the natural homomorphism $ G \rightarrow X ( X ( G) ) $ mapping an $ a \in G $ to the character $ \omega _ {a} : X ( G) \rightarrow T $, given by the formula

$$ \omega _ {a} ( \alpha ) = \alpha ( a) ,\ \ \alpha \in X ( G) , $$

is an isomorphism of topological groups. The following statements result from the above theorem.

I) If $ H $ is a closed subgroup of $ G $ and if

$$ H ^ {*} = \{ {\alpha \in X ( G) } : {\alpha ( H) = 0 } \} $$

is its annihilator in $ X ( G) $, then $ H $ coincides with the annihilator

$$ \{ {a \in G } : {\alpha ( a) = 0 \textrm{ for all } \ \alpha \in H ^ {*} } \} $$

of the subgroup $ H ^ {*} $; moreover, the group $ X ( H) $ is naturally isomorphic to $ X ( G) / H ^ {*} $, and $ X ( G/H) $ is isomorphic to the group $ H ^ {*} $.

II) If $ \phi : G \rightarrow H $ is a continuous homomorphism of locally compact Abelian groups, and $ G $ is identified with $ X ( X ( G) ) $ and $ H $ with $ X ( X ( H) ) $ by the natural isomorphisms, then the homomorphism $ \phi $ can be identified with $ ( \phi ^ {*} ) ^ {*} $.

III) The weight of the group $ X ( G) $( as a topological space, cf. Weight of a topological space) coincides with the weight of the group $ G $.

Pontryagin duality establishes a correspondence between compact groups $ G $ and discrete groups $ X ( G) $, and vice versa. Moreover, a compact group $ G $ is connected if and only if $ X ( G) $ is torsion-free. A compact group $ G $ is of dimension $ n < \infty $ if and only if $ X ( G) $ has finite rank $ n $( see Rank of a group). A compact group $ G $ is locally connected if and only if every finite-rank pure subgroup of $ X ( G) $ is free. For finite groups $ G $, Pontryagin duality coincides with duality between finite Abelian groups considered over the field $ \mathbf C $ 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 [8]. On the characterization of reflexive groups, see [9].

There is an analogue of Pontryagin duality for non-commutative groups (the duality theorem of Tannaka–Krein) (see , [6], [7]). Let $ G $ be a compact topological group, let $ R $ be the algebra of complex-valued functions on $ G $ whose translates span a finite-dimensional vector space and let $ S ( R) $ be the set of all non-zero algebra homomorphisms $ \omega : R \rightarrow \mathbf C $ satisfying the condition $ \omega ( \overline{f}\; ) = \overline{ {\omega ( f ) }}\; $, $ f \in R $. One can define a multiplication on $ S ( R) $ which makes $ S ( R) $ into a topological group with respect to the topology of pointwise convergence. To each $ g \in G $ corresponds the homomorphism $ \alpha _ {g} \in S ( R) $ given by the formula

$$ \alpha _ {g} ( f ) = f ( g) ,\ \ f \in R . $$

Then the correspondence $ g \rightarrow \alpha _ {g} $ is an isomorphism of the topological group $ G $ onto $ S ( R) $. There is also an algebraic description of the category of algebras $ R $, 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 ).

References

[1]	L.S. Pontryagin, "The theory of topological commutative groups" Ann. of Math. , 35 : 2 (1934) pp. 361–388
[2]	L.S. Pontryagin, "Topological groups" , Gordon & Breach (1966) (Translated from Russian)
[3]	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)
[5]	S.A. Morris, "Pontryagin duality and the structure of locally compact Abelian groups" , London Math. Soc. Lecture Notes , 29 , Cambridge Univ. Press (1977)
[6]	M.A. Naimark, "Normed rings" , Reidel (1959) (Translated from Russian)
[7]	E. Hewitt, K.A. Ross, "Abstract harmonic analysis" , 2 , Springer (1970)
[8]	M.F. Smith, "The Pontrjagin duality theorem in linear spaces" Ann. of Math. , 56 : 2 (1952) pp. 248–253
[9]	R. Venkataraman, "A characterization of Pontryagin duality" Math. Z. , 149 : 2 (1976) pp. 109–119

Comments

A compact group $ G $ is arcwise connected if and only if $ \mathop{\rm Ext} ( X( G), \mathbf Z ) = 0 $. A compact group $ G $ is metrizable if and only if $ X( G) $ is countable.

References

[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 $ p $- dimensional Aleksandrov–Čech cohomology group $ H ^ {p} ( A ; G ) $, with coefficients in a group $ G $, of a compact set $ A $ lying in an $ n $- dimensional compact orientable manifold $ M ^ {n} $ and the $ ( n- p- 1) $- dimensional cohomology group $ H _ {n-} p- 1 ^ {c} ( B ; G ) $ of the complement $ B = M ^ {n} \setminus A $, provided that $ H ^ {p} ( M ^ {n} ; G ) = H ^ {p+} 1 ( M ^ {n} ; G ) = 0 $( homology and cohomology in dimension zero are reduced; the symbol $ {} ^ {c} $ means compact support). In the case when $ A $ or $ B $ is a finite polyhedron, J.W. Alexander proved the existence of this isomorphism. N. Steenrod established such an isomorphism for an arbitrary open subset $ A \subset M ^ {n} $, and K.A. Sitnikov for an arbitrary subset $ A $.

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 $ H _ {p} ( A ; G ^ {*} ) $ and $ H _ {n-} p- 1 ^ {c} ( B ; G ) $, where $ G ^ {*} $ is the compact character group of the discrete group $ G $. The equivalence of both versions of the duality law follows from the fact that the group $ H _ {p} ( A ; G ^ {*} ) $ is the character group of $ H ^ {p} ( A ; G ) $. Under the assumption that the manifold is acyclic in dimensions $ p $ and $ p+ 1 $, since the homology sequence of the pair $ ( M ^ {n} , A ) $ is exact it follows that $ H ^ {p} ( A ; G ) = H ^ {p+} 1 ( M ^ {n} , A ; G ) $, 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 $ M ^ {n} $ be an arbitrary manifold (which may be generalized and need not be compact or orientable), let $ {\mathcal G} $ be a locally constant system of coefficients with fibres $ G $, let $ A $ be an arbitrary subset of $ M ^ {n} $, and let $ \Phi $ be the family of closed sets of $ M ^ {n} $ contained in $ B = M ^ {n} \setminus A $. Then $ H _ {p} ( M ^ {n} ; {\mathcal G} ) = H _ {p+} 1 ( M ^ {n} ; {\mathcal G} ) = 0 $ implies that $ H ^ {p} ( A ; {\mathcal H} _ {n} ( {\mathcal G} )) = H _ {n-} p- 1 ^ \Phi ( B ; {\mathcal G} ) $. Here the $ H _ {q} ^ \Phi $ are the homology functors with closed supports contained in $ \Phi $( i.e. direct limits of the groups $ H _ {q} ( F ; {\mathcal G} ) $, $ F \in \Phi $), and $ {\mathcal H} _ {n} ( {\mathcal G} ) $ is the locally constant system of coefficients generated by the groups $ H _ {n} ( M ^ {n} , M ^ {n} \setminus x ; {\mathcal G} ) $, $ x \in M ^ {n} $. In the above equality the cohomology coefficients $ {\mathcal H} _ {n} ( {\mathcal G} ) $ can be replaced by $ {\mathcal G} $ if one considers homology with coefficients in some specially defined system.

References

[1]	P.S. Aleksandrov, "Topological duality theorems" Trudy Mat. Inst. Steklov. , 48 (1955) pp. Part 1. Closed sets (In Russian)
[2]	W.S. Massey, "Homology and cohomology theory" , M. Dekker (1978)
[3]	E.G. Sklyarenko, "Homology and cohomology of general spaces" Itogi Nauk. i Tekhn. Sovrem. Probl. Mat. , 50 (1989) pp. 129–266 (In Russian)

E.G. Sklyarenko