View source for Pontryagin duality

A duality between topological groups and their character groups (cf. [[Character group|Character group]]). The duality theorem states that if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737601.png" /> is a locally compact Abelian group and if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737602.png" /> is its character group, then the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737603.png" /> mapping an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737604.png" /> to the character <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737605.png" />, given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737606.png" /></td> </tr></table>

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

I) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737607.png" /> is a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737608.png" /> and if

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p0737609.png" /></td> </tr></table>

is its annihilator in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376010.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376011.png" /> coincides with the annihilator

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376012.png" /></td> </tr></table>

of the subgroup <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376013.png" />; moreover, the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376014.png" /> is naturally isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376015.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376016.png" /> is isomorphic to the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376017.png" />.

II) If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376018.png" /> is a continuous homomorphism of locally compact Abelian groups, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376019.png" /> is identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376021.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376022.png" /> by the natural isomorphisms, then the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376023.png" /> can be identified with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376024.png" />.

III) The weight of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376025.png" /> (as a topological space, cf. [[Weight of a topological space|Weight of a topological space]]) coincides with the weight of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376026.png" />.

Pontryagin duality establishes a correspondence between compact groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376027.png" /> and discrete groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376028.png" />, and vice versa. Moreover, a compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376029.png" /> is connected if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376030.png" /> is torsion-free. A compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376031.png" /> is of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376032.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376033.png" /> has finite rank <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376034.png" /> (see [[Rank of a group|Rank of a group]]). A compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376035.png" /> is locally connected if and only if every finite-rank pure subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376036.png" /> is free. For finite groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376037.png" />, Pontryagin duality coincides with [[Duality|duality]] between finite Abelian groups considered over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376038.png" /> 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 [[#References|[8]]]. On the characterization of reflexive groups, see [[#References|[9]]].

There is an analogue of Pontryagin duality for non-commutative groups (the duality theorem of Tannaka–Krein) (see , [[#References|[6]]], [[#References|[7]]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376039.png" /> be a compact topological group, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376040.png" /> be the algebra of complex-valued functions on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376041.png" /> whose translates span a finite-dimensional vector space and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376042.png" /> be the set of all non-zero algebra homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376043.png" /> satisfying the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376044.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376045.png" />. One can define a multiplication on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376046.png" /> which makes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376047.png" /> into a topological group with respect to the topology of pointwise convergence. To each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376048.png" /> corresponds the homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376049.png" /> given by the formula

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376050.png" /></td> </tr></table>

Then the correspondence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376051.png" /> is an isomorphism of the topological group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376052.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376053.png" />. There is also an algebraic description of the category of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376054.png" />, 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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.S. Pontryagin,   "The theory of topological commutative groups"  ''Ann. of Math.'' , '''35''' :  2  (1934)  pp. 361–388</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Gordon &amp; Breach  (1966)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E. van Kampen,   "Locally bicompact Abelian groups and their character groups"  ''Ann. of Math.'' , '''36'''  (1935)  pp. 448–463</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  M.G. Krein,   "Hermitian positive kernels on homogeneous spaces, I"  ''Ukrain. Mat. Zh.'' , '''1''' :  4  (1949)  pp. 64–98  (In Russian)</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  M.G. Krein,   "Hermitian positive kernels on homogeneous spaces, II"  ''Ukrain. Mat. Zh.'' , '''2''' :  1  (1950)  pp. 10–59  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S.A. Morris,   "Pontryagin duality and the structure of locally compact Abelian groups" , ''London Math. Soc. Lecture Notes'' , '''29''' , Cambridge Univ. Press  (1977)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  M.A. Naimark,   "Normed rings" , Reidel  (1959)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''2''' , Springer  (1970)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  M.F. Smith,   "The Pontrjagin duality theorem in linear spaces"  ''Ann. of Math.'' , '''56''' :  2  (1952)  pp. 248–253</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  R. Venkataraman,   "A characterization of Pontryagin duality"  ''Math. Z.'' , '''149''' :  2  (1976)  pp. 109–119</TD></TR></table>



====Comments====
A compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376055.png" /> is arcwise connected if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376056.png" />. A compact group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376057.png" /> is metrizable if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376058.png" /> is countable.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  D.L. Armacost,   "The structure of locally compact abelian groups" , M. Dekker  (1981)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G. Hochschild,   "The structure of Lie groups" , Holden-Day  (1965)</TD></TR></table>

Pontryagin duality in topology is an isomorphism between a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376059.png" />-dimensional Aleksandrov–Čech cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376060.png" />, with coefficients in a group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376061.png" />, of a compact set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376062.png" /> lying in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376063.png" />-dimensional compact orientable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376064.png" /> and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376065.png" />-dimensional cohomology group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376066.png" /> of the complement <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376067.png" />, provided that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376068.png" /> (homology and cohomology in dimension zero are reduced; the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376069.png" /> means compact support). In the case when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376070.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376071.png" /> is a finite polyhedron, J.W. Alexander proved the existence of this isomorphism. N. Steenrod established such an isomorphism for an arbitrary open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376072.png" />, and K.A. Sitnikov for an arbitrary subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376073.png" />.

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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376075.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376076.png" /> is the compact character group of the discrete group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376077.png" />. The equivalence of both versions of the duality law follows from the fact that the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376078.png" /> is the character group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376079.png" />. Under the assumption that the manifold is acyclic in dimensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376080.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376081.png" />, since the homology sequence of the pair <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376082.png" /> is exact it follows that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376083.png" />, thus Pontryagin duality is a simple corollary of Poincaré–Lefschetz duality (see [[Poincaré duality|Poincaré duality]]).

The most general form of the considered duality relations is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376084.png" /> be an arbitrary manifold (which may be generalized and need not be compact or orientable), let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376085.png" /> be a locally constant system of coefficients with fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376086.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376087.png" /> be an arbitrary subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376088.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376089.png" /> be the family of closed sets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376090.png" /> contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376091.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376092.png" /> implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376093.png" />. Here the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376094.png" /> are the homology functors with closed supports contained in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376095.png" /> (i.e. direct limits of the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376096.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376097.png" />), and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376098.png" /> is the locally constant system of coefficients generated by the groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p07376099.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p073760100.png" />. In the above equality the cohomology coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p073760101.png" /> can be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073760/p073760102.png" /> if one considers homology with coefficients in some specially defined system.

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

''E.G. Sklyarenko''